Numerical investigation and application of

fractional dynamical systems

Libo Feng

Master of Science, Xiamen University

Bachelor of Science, Inner Mongolia University

A thesis submitted tothe science and engineering faculty

of Queensland University of Technologyin fulfilment of the requirement for the degree of

Doctor of Philosophy

Principal Supervisor: Prof. Fawang LiuAssociate Supervisors: Prof. Ian Turner

Dr. Qianqian Yang

School of Mathematical SciencesScience and Engineering Faculty

Queensland University of Technology

2019

QUT Verified Signature

To my family

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Abstract

The use of fractional differential equations (FDEs) to model the non-locality, memory,spatial heterogeneity and anomalous diffusion inherent in many real-world problems hasattracted considerable attention in many fields of science including physics, electricalsystems, bioengineering, hydrology, chemistry, biochemistry, and finance. FDEs helpin the computational modelling of these complex systems by interpolating between theinteger orders of differential equations to capture nonlocal relations in time and spaceusing power-law memory kernels. This growing interest has led to an intense worldwideresearch focus to uncover new theoretical and numerical methods for solving fractionaldynamical systems.

An important example of a fractional dynamical system arises in the study of non-Newtonian fluids where the heterogeneity of the internal temperature and the distri-bution of fluid density cause the transport in a viscoelastic non-Newtonian fluid to be-come ‘anisotropic’ when an applied external force by shear flow is applied. The classicalresearch methods of universal application and mathematical characterization used in s-tudying Newtonian fluids are no longer applicable because non-Newtonian fluids involvecomplex multi-term time fractional dynamical systems with both diffusion and time frac-tional diffusion terms, as well as a nonlinear reaction term. In addition, many complexfractional dynamical models involve a Riesz fractional operator and typically must besolved over irregular domains, such as the time-space fractional Bloch-Torrey equation,which can be applied to fit the MRI diffusion images obtained from microenvironmentssuch as human brain tissue and human articular cartilage.

The aim of this thesis is to develop new computational fractional dynamical models forkey application areas of science and engineering and solve them using novel numericalmethods. This overall aim will include the following three main objectives:

• to develop new numerical methods and analytical techniques for simulating complexfractional dynamical models to reduce computational cost, which is achieved byutilising high-order numerical methods and fast algorithms;• to develop numerical methods for complex fractional dynamical models with the

Riesz fractional operator on irregular domains, which is achieved by using the fi-nite element method and finite volume method combining the unstructured meshcapability to approximate the space fractional derivative;• to develop numerical methods for viscoelastic non-Newtonian fluid models, such as

the generalised Maxwell fluid model, the generalised Oldroyd-B fluid model and thegeneralised Burgers’ fluid model, which is achieved by applying mixed differenceschemes to discretise the different time fractional derivatives.

One original contribution of this thesis is the treatment of the Riesz space fractionalderivative on irregular convex domains. Based on the Galerkin finite element method(FEM) with an unstructured mesh, a novel numerical technique to treat the Riesz spacefractional derivative on irregular convex and non-convex domains is developed and thetheoretical analysis is presented, which is more flexible compared to the finite differencemethod. In addition, a novel unstructured mesh control volume method to calculate the

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space fractional derivative on irregular convex domains is developed, which can reduceCPU time significantly while retaining the same accuracy and approximation property asthe unstructured mesh finite element method.

Another important contribution of this thesis is that a unified numerical scheme to solvea class of novel multi-term time fractional diffusion-wave and sub-diffusion equations is p-resented and the rigorous stability and convergence analysis is established, which not onlycan be extended to solve the generalised Maxwell fluid model, the generalised Oldroyd-Bfluid model and the generalised Burgers’ fluid model but also to solve the general multi-term time fractional diffusion-wave or sub-diffusion equation, the time-fractional telegraphequation, the fractional cable equation and the fractional Cattaneo diffusion equation.

A series of six published papers and one paper under review is presented on the high-order numerical methods for the Riesz space fractional advection-dispersion equation,a fast second-order accurate method for a two-sided space-fractional diffusion equationwith variable coefficients, unstructured mesh finite element method for the 2D time-spaceRiesz fractional diffusion equation on irregular convex domains, unstructured mesh controlvolume method for two-dimensional space fractional diffusion equations with variablecoefficients on convex domains, finite difference method for the generalised Oldroyd-Bfluid and MHD Couette flow of a generalised Oldroyd-B fluid and finite element methodfor a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equationon convex domains, respectively. There are also two published papers presented on thenovel finite volume method for the Riesz space distributed-order diffusion and advection-diffusion equation in the appendix, respectively.

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Keywords

Caputo fractional derivative Control volume methodCouette flow Crank-Nicolson schemeDistributed-order equation Fast Bi-CGSTAB algorithmFast iterative solver Finite difference methodFinite element method Finite volume methodFractional advection-dispersion equation Fractional diffusion equationFractional diffusion-wave equation Fractional non-Newtonian fluidsFractional sub-diffusion equation Generalized Oldroyd-B fluidHigh-order numerical methods Irregular convex domainsMulti-term time fractional derivative Richardson extrapolation methodRiemann-Liouville fractional derivative Riesz fractional derivativeSpace fractional derivative Stability and convergence analysisTime-space fractional diffusion equation Two-dimensionalUnstructured mesh Variable coefficients

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Acknowledgements

Firstly I would like to give my sincere thanks to my supervisors Prof. Fawang Liu andProf. Ian Turner for their advice, guidance, patience, encouragement, and support alongthis Ph.D. project in the past three years, without which this dissertation could not havebeen completed. I have been extremely lucky to choose them as my supervisors for theynot only possess vast knowledge and scientific passion, but also have a genuine attitude tothe scientific research, from whom I learned a lot. I would also like to thank my associatedsupervisor Dr. Qianqian Yang, who is young and talented and excellent in academia andshared her experience a lot with me.

I am indebted to the Queensland University of Technology for providing me the QUTPostgraduate Research Award (QUTPRA) and the QUT Higher Degree Research TuitionFee Sponsorship to support my study. I am grateful to the School of MathematicalSciences (SMS) and Science and Engineering Faculty (SEF) for providing the HDR fundsto support me to participate in the conference (ICFDA’18) held in Amman, Jordan.

I also specially thank Prof. Pinghui Zhuang from Xiamen University and Prof. YangLiu from Inner Mongolia University for their helpful suggestion and discussion during myPh.D. study. I am also grateful to Dr. Shanlin Qin, Associate Prof. Qingxia Liu andother fellows in our fractional group for their help and encouragement.

My thanks also go to the heart-warmed students and staff at the School of MathematicalSciences and the resources and services provided by Queensland University of Technology(QUT).

Finally, and most importantly, I want to give my special thanks to my family: my parentsand my sister for supporting me spiritually in the pursuit of this project. In particular, Iwill give my deepest gratitude to my parents for their unconditional and constant supportand love.

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List of Publications

1. L. Feng, P. Zhuang, F. Liu, I. Turner, J. Li, High-order numerical methods for theRiesz space fractional advection-dispersion equations, Computers and Mathematicswith Applications, (2016), In press. http://dx.doi.org/10.1016/j.camwa.2016.01.015.

2. L. Feng, P. Zhuang, F. Liu, I. Turner, V. Anh, J. Li, A fast second-order accuratemethod for a two-sided space-fractional diffusion equation with variable coefficients,Computers and Mathematics with Applications 73 (2017) 1155-1171.

3. J. Li, F. Liu, L. Feng, I. Turner, A novel finite volume method for the Riesz spacedistributed-order diffusion equation, Computers and Mathematics with Applications74 (2017) 772-783.

4. J. Li, F. Liu, L. Feng, I. Turner, A novel finite volume method for the Riesz spacedistributed-order advection-diffusion equation, Applied Mathematical Modelling 46(2017) 536-553.

5. L. Feng, F. Liu, I.Turner, P. Zhuang, Numerical methods and analysis for simu-lating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigidplates, International Journal of Heat and Mass Transfer 115 (2017) 1309-1320.

6. L. Feng, F. Liu, I. Turner, Q. Yang, P. Zhuang, Unstructured mesh finite differ-ence/finite element method for the 2D time-space Riesz fractional diffusion equationon irregular convex domains, Applied Mathematical Modelling 59 (2018) 441-463.

7. L. Feng, F. Liu, I. Turner, L. Zheng, Novel numerical analysis of multi-term timefractional viscoelastic non-Newtonian fluid models for simulating unsteady MHDCouette flow of a generalized Oldroyd-B fluid, Fractional Calculus and AppliedAnalysis 21(4) (2018) 1073-1103.

8. L. Feng, F. Liu, I. Turner, Finite difference/finite element method for a novel2D multi-term time fractional mixed sub-diffusion and diffusion-wave equation onconvex domains, Communications in Nonlinear Science and Numerical Simulation70 (2019) 354-371.

9. L. Feng, F. Liu, I. Turner, An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convexdomains, submitted to Journal of Computational and Applied Mathematics, 2018.

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List of Abbreviations

2D-SFDE-VC two-dimensional space fractional diffusion equation with variable coefficients2D-TSRFDE two-dimensional time-space Riesz fractional diffusion equationADE advection-dispersion equationBi-CGSTAB bi-conjugate gradient stabilized methodCGS conjugate gradient squaredCVM control volume methodFADE fractional advection-dispersion equationFBi-CGSTAB fast bi-conjugate gradient stabilizedFDE fractional diffusion/differential equationFDM finite difference methodFEM finite element methodFFT fast Fourier transformFPDE fractional partial differential equationFVM finite volume methodGOBF generalised Oldroyd-B fluidMHD magnetohydrodynamicMHD-GOBF MHD generalised Oldroyd-B fluidREM Richardson extrapolation methodRSDOADE Riesz space distributed-order advection-diffusion equationRSDODE Riesz space distributed-order diffusion equationRSFADE Riesz space fractional advection-dispersion equationRSFDE Riesz space fractional diffusion/differential equationTFMSDWE time-fractional mixed sub-diffusion and diffusion-wave equationTS-FBTE time-space fractional Bloch-Torrey equationsTSSFDE-VC two-sided space fractional diffusion equation with variable coefficientsWSGD weighted and shifted Grunwald differenceWSGL weighted shifted Grunwald-Letnikov

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Contents

Abstract iii

Keywords v

Acknowledgements vi

List of Publications vii

List of Abbreviations viii

Chapter 1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Numerical methods for fractional dynamical models . . . . . . . . 61.2.2 Fractional dynamical models involving the Riesz fractional operator

implemented on irregular domains . . . . . . . . . . . . . . . . . . 71.2.3 Complex viscoelastic non-Newtonian fluid models . . . . . . . . . . 8

1.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.1 Chapter 2: High-order numerical methods for the RSFADE . . . . 131.4.2 Chapter 3: A fast second-order accurate method for a TSSFDE-VC 141.4.3 Chapter 4: Unstructured mesh FD/FEM for the 2D-TSRFDE on

irregular convex domains . . . . . . . . . . . . . . . . . . . . . . . 151.4.4 Chapter 5: An unstructured mesh control volume method for 2D-

SFDE-VC on convex domains . . . . . . . . . . . . . . . . . . . . . 161.4.5 Chapter 6: Numerical methods and analysis for simulating the flow

of a GOBF between two infinite parallel rigid plates . . . . . . . . 171.4.6 Chapter 7: Novel numerical analysis of multi-term time fractional

viscoelastic non-Newtonian fluid models for simulating unsteadyMHD Couette flow of a GOBF . . . . . . . . . . . . . . . . . . . . 18

1.4.7 Chapter 8: FD/FEM for a novel 2D multi-term TFMSDWE . . . 191.4.8 Chapter 9: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 201.4.9 Appendix A: FVM for the Riesz space distributed-order diffusion

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.10 Appendix B: FVM for the Riesz space distributed-order advection-

diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 2 High-order numerical methods for the Riesz space fractional advection-dispersion equations 23

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 The approximation for the Riemann-Liouville fractional derivative . . . . 252.3 The finite difference method for the RSFADE . . . . . . . . . . . . . . . . 292.4 Theoretical analysis of the finite difference method . . . . . . . . . . . . . 30

2.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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2.4.3 Improving the convergence order . . . . . . . . . . . . . . . . . . . 342.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter 3 A fast second-order accurate method for a two-sided space-fractionaldiffusion equation with variable coefficients 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Preliminary knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 The finite difference method for the FDE . . . . . . . . . . . . . . . . . . 463.4 Theoretical analysis of the finite difference method . . . . . . . . . . . . . 51

3.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5 A fast iterative algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.5.1 Efficient storage of matrix Q . . . . . . . . . . . . . . . . . . . . . 573.5.2 A fast bi-conjugate gradient stabilized method . . . . . . . . . . . 58

3.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 4 Unstructured mesh finite difference/finite element method for the 2Dtime-space Riesz fractional diffusion equation on irregular convex do-mains 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Preliminary knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 The fully discrete finite element scheme . . . . . . . . . . . . . . . 724.3.2 The implementation of FEM with an unstructured mesh . . . . . . 734.3.3 Second order temporal numerical scheme . . . . . . . . . . . . . . 77

4.4 Stability and convergence of the fully discrete scheme . . . . . . . . . . . 794.5 FEM for the 2D-TSRFDE on non-convex domains . . . . . . . . . . . . . 834.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Chapter 5 An unstructured mesh control volume method for two-dimensional s-pace fractional diffusion equations with variable coefficients on convexdomains 95

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Control volume finite element method . . . . . . . . . . . . . . . . . . . . 1005.3 Discussion of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 1085.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Chapter 6 Numerical methods and analysis for simulating the flow of a generalisedOldroyd-B fluid between two infinite parallel rigid plates 118

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.2 Formulation of the flow problem . . . . . . . . . . . . . . . . . . . . . . . 1216.3 Preliminary knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4 Derivation and solvability of the numerical scheme . . . . . . . . . . . . . 1276.5 Stability and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.5.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.5.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.6 Improve the time order of the scheme . . . . . . . . . . . . . . . . . . . . 1316.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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Chapter 7 Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of ageneralised Oldroyd-B fluid 139

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.2 Preliminary knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.3 Derivation of the numerical schemes . . . . . . . . . . . . . . . . . . . . . 148

7.3.1 Scheme I: first order implicit scheme . . . . . . . . . . . . . . . . . 1487.3.2 Scheme II: mixed L scheme . . . . . . . . . . . . . . . . . . . . . . 149

7.4 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.4.1 Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.4.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.4.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Chapter 8 Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex do-mains 161

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1618.2 Preliminary knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1648.3 The finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.3.1 Finite element scheme I . . . . . . . . . . . . . . . . . . . . . . . . 1688.3.2 Implementation of the finite element method . . . . . . . . . . . . 1698.3.3 Finite element scheme II . . . . . . . . . . . . . . . . . . . . . . . . 171

8.4 Stability and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Chapter 9 Conclusions 1849.1 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.2 Recommendations for Future Research . . . . . . . . . . . . . . . . . . . . 185

Appendix A A novel finite volume method for the Riesz space distributed-order dif-fusion equation 188

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188A.2 Finite volume method for the distributed-order diffusion equation . . . . . 191A.3 Stability and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 193A.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198A.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Appendix B A novel finite volume method for the Riesz space distributed-order advection-diffusion equation 202

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202B.2 The Crank-Nicolson scheme with the finite volume method . . . . . . . . 205B.3 Stability and convergence of the Crank-Nicolson scheme . . . . . . . . . . 209B.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215B.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

References 223

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List of Tables

2.1 The error and convergence order of the Crank-Nicolson scheme of theRSFDE for different β at t = 1 with τ = h . . . . . . . . . . . . . . . . . . 36

2.2 The error and convergence order of the Crank-Nicolson scheme of the RS-FADE for different α and β at t = 1 with τ = h . . . . . . . . . . . . . . . 37

2.3 The error and convergence order of the RSFADE by applying the REM . 37

3.1 The error and convergence order of the Crank-Nicolson scheme of theTSSFDE for different α at t = 1 with τ = h . . . . . . . . . . . . . . . . . 60

3.2 The error and convergence order of the Crank-Nicolson scheme of theTSSFDE for different K−(x), K+(x) and α at t = 1 with τ = h . . . . . . 61

3.3 Comparison of the convergence property of FBi-CGSTAB versus Gausselimination and Bi-CGSTAB for K−(x) = 2 − x, K+(x) = 2 + x at t = 1with τ = h and α = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 Comparison of the consumed CPU time of FBi-CGSTAB versus Gausselimination and Bi-CGSTAB for K−(x) = 2 − x, K+(x) = 2 + x at t = 1with τ = h and α = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 The error and convergence order of the Crank-Nicolson scheme of TSSFDEfor different β and α at t = 1 with τ = h . . . . . . . . . . . . . . . . . . . 63

3.6 The error and convergence order of the Crank-Nicolson scheme of theRSFDE for different α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 The L(α,β) error, L2 error and convergence order of h for different α, β att = 1 with γ = 0.7, τ = 1/1000 . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2 The L2 error and convergence order of τ for γ = 0.7 at t = 1 with α = β =0.8 and h2 ≈ τ2−γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3 The L2 error and convergence order of τ = h for the second order numericalscheme with γ = 0.7, α = β = 0.8 at t = 1 . . . . . . . . . . . . . . . . . . 89

4.4 The L2 error and convergence order of h for α = β = 0.8, γ = 0.9, τ =1/1000 at t = 1.0 on the general irregular convex domain Ω . . . . . . . . 90

4.5 The L2 error and convergence order of Mxy for different h with β = 0.8,γ = 0.95, τ = 1/20 at t = 50 . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.6 The L2 error and convergence order of h for α = β = 0.8, γ = 0.9, τ =1/1000 at t = 1.0 on the multiply-connected domain . . . . . . . . . . . . 93

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5.1 The size and density of matrix M for different h on a square domain [0, 1]×[0, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2 The L2 error, L∞ error, convergence order and CPU time of h with τ =10−3 for the linear coefficient case at t = 1 . . . . . . . . . . . . . . . . . 110

5.3 The L2 error, L∞ error, convergence order and CPU time of h with τ =10−3 for the quadratic coefficient case at t = 1 . . . . . . . . . . . . . . . 110

5.4 The L2 error, L∞ error, convergence order and CPU time of h with τ =10−3 for the exponential coefficient case at t = 1 . . . . . . . . . . . . . . 111

5.5 Comparison of the consumed CPU time of Gaussian elimination versusBi-CGSTAB method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.6 The comparison of the density of stiffness matrix generated by FEM andCVM for different h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.7 The L2 error, L∞ error and convergence order of h for FEM with τ = 10−3

at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.8 The L2 error, L∞ error and convergence order of h for CVM with τ = 10−3

at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.9 The comparison of running time between FEM and CVM for different hwith α = β = 0.80, τ = 10−3 at t = 1 . . . . . . . . . . . . . . . . . . . . . 114

6.1 The temporal error and convergence order of Scheme I for different β withγ = 1.1 and h = 1/1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2 The temporal error and convergence order of Scheme I for different γ withβ = 0.3 and h = 1/1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3 The temporal error and convergence order of Scheme II for different β withγ = 1.1 and h = 1/1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.4 The temporal error and convergence order of Scheme II for different γ withβ = 0.3 and h = 1/1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.1 The temporal error and convergence of Scheme I for different α, β and γwith h = 1/1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.2 The temporal error and convergence of Scheme II for different α, β and γwith h = 1/1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.3 The running time of Scheme I and II for different τ , with α = 0.7, β =0.6, γ = 1.5, h = 1/1000 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . 158

8.1 The L2 error and convergence order of h for schemes I and II with γ = 1.6,α = 0.7, β = 0.8, τ = 1

1000 at t = 1 . . . . . . . . . . . . . . . . . . . . . . 178

8.2 The L2 error and convergence order of τ for scheme II with τ = h, γ = 1.6,α = 0.7, β = 0.8 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8.3 The L2 error and convergence order of τ for scheme II at t = 1 . . . . . . 178

xiv

8.4 The L2 error and convergence order of h for the problem on a circular domain180

8.5 The L2 error and convergence order of τ for the problem on a circular domain180

8.6 The L2 error and convergence order of h for α = 0.8, β = 0.7 with τ = 11000

at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

A.1 The error and the convergence order of τ and h for σ = 1/500 at t = 1 . . 199

A.2 The error and the convergence order of σ for τ = h = 1/400 at t = 1 . . . 199

A.3 The error and the convergence order of τ = h for σ = 1/100, l = 2,α1 = 1.255, α1 = 1.755 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . 200

B.1 The errors and convergence orders with respect to τ and h . . . . . . . . . 217

B.2 The errors and convergence orders with respect to σ and % . . . . . . . . . 217

B.3 The errors and convergence orders comparison of FVM and FDM for σ =% = 1/1000 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

B.4 The errors and the convergence orders with σ = % = 1/100, α1 = 0.955,α2 = 1.255 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

xv

List of Figures

1.1 The structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 The comparison of the numerical solution and analytical solution at T = 0.4with fixed α = 0.4, β = 1.8, h = τ = 1/100 . . . . . . . . . . . . . . . . . . 38

2.2 The numerical approximation of u(x, t) for α = 0.1, 0.3, 0.5, 0.7, 0.9 atT = 10.0 with fixed β = 1.7, h = τ = 1/100 . . . . . . . . . . . . . . . . . 39

2.3 The numerical approximation of u(x, t) for β = 1.2, 1.4, 1.6, 1.8, 2.0 atT = 10.0 with fixed α = 0.3, h = τ = 1/100 . . . . . . . . . . . . . . . . . 39

2.4 The numerical approximation of u(x, t) for α = 0.4, β = 1.6 at t = 1.0, 2.0,4.0, 8.0 with h = τ = 1/100 . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 The boundaries of convex domain Ω . . . . . . . . . . . . . . . . . . . . . 67

4.2 The sectional view of human brain and unstructured mesh partition . . . 69

4.3 The illustration of support domains ΩLei , ΩR

ei , ΩDei , ΩU

ei . . . . . . . . . . . 75

4.4 The points of intersection by y = yq with the triangle element of Ωej and ∂Ω 76

4.5 The illustration of a multiply-connected domain . . . . . . . . . . . . . . . 83

4.6 The unstructured triangular meshes used in the calculation of the ellipticaldomain for h = 0.12558, 0.08391 and 0.04531 . . . . . . . . . . . . . . . . 88

4.7 The unstructured triangular meshes used in the calculation of the generalirregular convex domain Ω for h = 0.2825, 0.1454 and 0.0557 . . . . . . . . 89

4.8 The diffusion profiles of u(x, y, t) on the general irregular convex domain Ωat different t with γ = 0.9, α = β = 0.80, h = 0.1454, τ = 1/1000 at t = 1.0 90

4.9 The unstructured triangular meshes used in the calculation of the humanbrain-like domain for h = 0.11624, 0.06943 and 0.03278 . . . . . . . . . . . 91

4.10 Plots of Mx(x, y, t), My(x, y, t) at point (x∗, y∗) = (0.5702, 0.8548) for dif-ferent γ with β = 1.0, h = 0.06943 on the human brain-like domain . . . . 92

4.11 Plots of Mx(x, y, t), My(x, y, t) at point (x∗, y∗) = (0.5702, 0.8548) for dif-ferent β with γ = 0.99, h = 0.06943 on the human brain-like domain . . . 92

4.12 Normalized decay of the transverse magnetization versus t at point (x∗, y∗) =(0.5702, 0.8548) for different γ (with fixed β = 1.0) and β (with fixed γ = 0.99) 92

4.13 The illustration of a multiply-connected domain with triangular partition 93

xvii

4.14 The diffusion profiles of u(x, y, t) on a multiply-connected domain at dif-ferent t with γ = 0.9, α = β = 0.80, τ = 1/1000 at t = 1.0 . . . . . . . . . 94

5.1 An illustration of a solution domain with curved boundary . . . . . . . . . 97

5.2 An illustration of a control volume . . . . . . . . . . . . . . . . . . . . . . 100

5.3 A control volume face and the outward normal unit vector . . . . . . . . . 101

5.4 The illustration of control faces with mid-points . . . . . . . . . . . . . . . 102

5.5 The illustration of line y = yr intersecting nq points with the supportdomain Ωek of lk(x, y), where (xr, yr) locates out of Ωek . . . . . . . . . . 104

5.6 The illustration of line y = yr intersecting nq points with the supportdomain Ωek of lk(x, y), where (xr, yr) locates in Ωek . . . . . . . . . . . . . 106

5.7 Sparsity pattern of matrix M for h = 1.6759 × 10−1. The size of M is64×64. Blue points indicate the nonzero entries . . . . . . . . . . . . . . . 108

5.8 The rectangular domain partitioned by unstructured meshes with controlvolumes for h ≈ 3.1123×10−1, 1.6759×10−1, 8.6682×10−2, 4.3719×10−2,respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.9 The unstructured meshes with control volumes for h ≈ 2.8917×10−1, 1.6444×10−1, 8.6550× 10−2, 4.5873× 10−2, respectively . . . . . . . . . . . . . . . 112

5.10 The comparison of the exact solution u(x, y, t) and numerical solutionuh(x, y, t) for h = 4.5873× 10−2, α = β = 0.8 with τ = 10−3 at t = 1 . . . 114

5.11 The error plot of u(x, y, t)− uh(x, y, t) for h = 4.5873× 10−2, α = β = 0.8with τ = 10−3 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.12 The diffusion profiles of u(x, y, t) at t = 0.5 on different convex domainswith α = β = 0.8 and τ = 10−3 . . . . . . . . . . . . . . . . . . . . . . . . 116

6.1 The illustration of the flow problem . . . . . . . . . . . . . . . . . . . . . 121

6.2 Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different λ with α = 0.3, β = 0.8, θ = 2 at t = 2 . . . . . . . . . . 136

6.3 Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different θ with α = 0.3, β = 0.8, λ = 10 at t = 2 . . . . . . . . . 136

6.4 Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different α with β = 0.8, λ = 5, θ = 3 at t = 2 . . . . . . . . . . . 137

6.5 Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different β with α = 0.3, λ = 5, θ = 3 at t = 2 . . . . . . . . . . . 137

6.6 Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different t with α = 0.3, β = 0.8, λ = 5, θ = 3 . . . . . . . . . . . 137

7.1 The ratio of det(HN )det(HN+1) for different β and N . . . . . . . . . . . . . . . . . 148

xviii

7.2 Numerical solution profiles of velocity u(x, t) for different p (K = 2) andK (p = 1) with λ = 3, θ = 4, α = 0.5, β = 0.6 at t = 2 . . . . . . . . . . . 159

7.3 Numerical solution profiles of velocity u(x, t) for different λ (θ = 4) and θ(λ = 3) with p = 1, α = 0.5, β = 0.6, K = 2 at t = 2 . . . . . . . . . . . . 159

7.4 Numerical solution profiles of velocity u(x, t) for different α (β = 0.6) andβ (α = 0.5) with p = 1, λ = 3, θ = 4, K = 2 at t = 2 . . . . . . . . . . . . 159

7.5 Numerical solution profiles of velocity u(x, t) at different t with p = 1,λ = 3, θ = 4, α = 0.5, β = 0.6, K = 2 . . . . . . . . . . . . . . . . . . . . 160

8.1 The illustration of the MHD Olyroyd-B fluid . . . . . . . . . . . . . . . . 163

8.2 The comparison of exact solution with numerical solution for γ = 1.6,α = 0.7, β = 0.8 with τ = h = 1

50 at t = 1 . . . . . . . . . . . . . . . . . . 179

8.3 The triangulation for the circular domain (h ≈ 1.6444 × 10−1 and h ≈8.6550× 10−2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.4 The comparison of exact solution with numerical solution for the problemon a circular domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.5 Numerical solution profiles of velocity u(y, z, t) at different t for α = 0.8,β = 0.7 with h = 1

40 , τ = 1100 . . . . . . . . . . . . . . . . . . . . . . . . . 182

A.1 Comparison of the exact solution and numerical solution with σ = τ = h =1/80 at t = 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

A.2 Numerical solution profile of u(x, t) at different t with σ = 1/100, τ = h =1/200, l = 2, α1 = 1.255, α1 = 1.755 . . . . . . . . . . . . . . . . . . . . . 200

A.3 Numerical solution profile of u(x, t) for different α1 and α2 with σ = 1/100,τ = h = 1/200, l = 2, at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . 201

A.4 Numerical solution profile of u(x, t) for different l with σ = 1/100, τ = h =1/200, α1 = 1.255, α1 = 1.755 at t = 1 . . . . . . . . . . . . . . . . . . . . 201

B.1 Exact solution and numerical solution with σ = % = τ = h = 1/80 at t = 1.0216

B.2 Numerical solution profile of u(x, t) at different t with σ = % = 1/100,τ = h = 1/200, α1 = 0.955, α2 = 1.255 . . . . . . . . . . . . . . . . . . . . 218

B.3 Numerical solution profile of u(x, t) for different α1 with σ = % = 1/100,τ = h = 1/200, α2 = 1.255 at t = 1 . . . . . . . . . . . . . . . . . . . . . . 219

B.4 Numerical solution profile of u(x, t) for different α2 with σ = % = 1/100,τ = h = 1/200, α1 = 0.955 at t = 1 . . . . . . . . . . . . . . . . . . . . . . 219

B.5 Numerical solution at different times . . . . . . . . . . . . . . . . . . . . . 220

B.6 Numerical solution for different α1 . . . . . . . . . . . . . . . . . . . . . . 220

B.7 Numerical solution for different γ1 . . . . . . . . . . . . . . . . . . . . . . 221

xix

B.8 Numerical solution for different P (α) . . . . . . . . . . . . . . . . . . . . . 221

B.9 Numerical solution for different Q(γ) . . . . . . . . . . . . . . . . . . . . . 221

xx

Chapter 1

Introduction

1.1 Background

The topic of fractional calculus dates back to some speculations of G.W. Leibniz in 1695

[136] in which the meaning of the derivative of 0.5 order was considered. Since then,

the theory of derivatives and integrals of arbitrary order was developed by some pioneers

in the field, such as Liouville, Grunwald, Letnikov, Riemman, and Caputo. In the last

few decades, the application of fractional calculus to real-world problems grew rapidly, in

which the use of dynamical systems described by fractional differential equations (FDEs)

had been one of the ways to understand complex materials and processes. Due to the

power to model the non-locality, memory, spatial heterogeneity and anomalous diffusion

inherent in many real-world problems, the application of FDEs has been attracting much

attention in many fields of science including physics [185, 222, 230, 275], biology [23,

174], chemistry [273, 274], hydrology [14, 15, 148], control [132, 179], signal and image

processing [40, 143, 281], and finance [220, 256]. For more applications of fractional

calculus, the readers can refer to some monographs [8, 60, 103, 128, 139, 156, 186, 203, 218]

or related review articles [173, 237]. FDEs help in the computational modelling of these

complex systems by interpolating between the integer orders of differential equations

to capture nonlocal relations in time and space using power-law memory kernels. This

growing interest has led to an intense worldwide research focus to uncover new theoretical

and numerical methods to solve fractional dynamical systems. In this thesis, we focus

on two main problems: complex fractional dynamical models with the Riesz fractional

operator on irregular domains and fractional dynamical systems arising in the study of

non-Newtonian fluids.

Firstly, we introduce some background of the fractional dynamical models with the Riesz

fractional operator. In clinical practices, diffusion weighted MRI is generally used to

characterize and detect some ischemic, neurodegenerative and malignant diseases, which

improves our understanding of biological materials using a detailed analysis of water d-

iffusion heterogeneity and relaxation time distribution. The conventional method is to

use the the Bloch equation to probe the molecular diffusion, which exhibits a typically

exponential behaviour. However, when it comes to complex multiplicity of tissue compart-

ments, the method fails due to the diffusion processes often being anisotropic. Therefore,

some new models are proposed to treat the anomalous diffusion. In [12, 13], Bennett

et al. proposed a so called ‘stretched exponential’ function to describe the diffusion in

MRI, which provides a new way to observe the nanoscale models of porous materials and

tissues. In [131], Kopf et al. developed a fractional order model to simulate the anoma-

lous diffusion of water molecules in both normal and cancerous tissues in NMR. In [197],

1

Chapter 1 2

Ozarslan et al. used the fractal diffusion theory to analyse the NMR echo intensity in hu-

man erythrocyte ghosts, glioblastoma tissues, and normal human brain gray matter and

connected the ‘heterogeneity index’ to the microscopic tissue structure. In [175], Magin

et al. showed that the stretched exponential model was an extension of the Bloch-Torrey

equation using fractional operators, which is a time-space Riesz fractional diffusion equa-

tion. Furthermore, Magin et al. applied the time-space fractional Bloch-Torrey equation

(TS-FBTE) to fit the signal attenuation of the tissues in the human brain, human artic-

ular cartilage and Sephadex gels and found that the fractional model worked very well.

The time-space fractional Bloch-Torrey equations were also investigated by Yu et al. [272]

and Bueno-Orovio et al. [25].

In biological tissues, impulse propagation is known to be modulated by structural het-

erogeneity. In cardiac muscle, to understand how this heterogeneity influences electrical

spread is a key to improve the interpretation of dispersion of repolarization. In [153],

Liu et al. proposed a fractional FitzHugh-Nagumo monodomain model to describe the

propagation of the electrical potential in heterogeneous cardiac tissue, which is a coupled

fractional Riesz space nonlinear reaction-diffusion model. They found that a fractional

power had a strong effect on the width of the excitation. In [23], Bueno-Orivio et al.

proposed a fractional diffusion model as a new mathematical description of structurally

heterogeneous excitable media. They presented a comparison between fractional models

and experimental data of canine cardiac tissue, which showed that data fitting using the

fractional model with a fractional order α = 1.75 was better than that using a standard d-

iffusion model α = 2. They also considered and analyzed the fractional FitzHugh-Nagumo

monodomain model using either the fractional Laplacian or Riesz derivative [24]. Recent-

ly, Farquhar [75] considered the coupled fractional FitzHugh-Nagumo cell model and the

Beeler-Reuter cell model to study different spatial effects on the transport of electrical

impulses in the heart, which may improve the understanding of how the heart works.

As these problems are raised in complex biological tissues or inhomogeneous media, such

as the human brain [175] and cardiac muscle [23], considering the fractional models on

simple regular domains to model the problems is not enough and may bias its intrinsic

mechanism. In fact, many other problems from science and engineering involve fractional

mathematical models that must be computed on irregular domains and therefore seeking

effective numerical methods to solve FDE on such domains is important. Although there

exist some numerical methods to solve fractional dynamical models with the Riesz frac-

tional operator, the study involving the Riesz fractional operator on irregular domains

is sparse and most appear to lack stability and convergence analyses. Due to the non-

locality of the spatial fractional operators, the calculation on irregular domains is more

complex than that on regular domains. Generally, the calculation of spatial fractional

operators on irregular domains or uniform grids generates a dense and regular coefficient

matrix due to the boundary being a fixed constant. However, on irregular domains, the

boundary of the solution domain is no longer a fixed constant and the definition of the

spatial fractional derivative has to change all the time according to the boundary in the

3 Chapter 1

calculation, which is bound to increase the CPU time cost. Furthermore, the generated

coefficient matrix is dense and not regular due to the non-fixed constant boundary in the

calculation, which also will increase the storage and computational costs. In addition,

to establish the stability and convergence analysis of the method on irregular domains is

more complex than that on regular domains, which is also challenging. Therefore, devel-

oping new numerical methods for fractional dynamical models with the Riesz fractional

operator on irregular domains and giving the stability and convergence analysis will be

one main contribution of the work.

In the following, we present some background of the fractional dynamical systems arising

in the study of non-Newtonian fluids. During the past few decades, fluids have been

widely applied in engineering and industry. Viscoelasticity describes the viscous and e-

lastic properties of a material when it undergoes deformation [51]. Some common and

well-known viscoelastic materials are blood, paint, shampoo, honey and cornstarch. The

common fluids, such as water, oil, air, ethanol, and benzene, exhibit a linear relationship

between the stress tensor and the rate of deformation tensor, which are termed as New-

tonian fluids [51]. Generally, the constitutive equation is used to specify the rheological

properties of the material, which is a relation between the stress and the local properties

of the fluid. A Newtonian constitutive equation is the simplest form where the stress

is linearly proportional to the strain history, which is called a linear viscoelastic model.

For small deformations, low stress, low rate and linear materials, linear viscoelasticity is

usually applicable. However, in industry some fluids do not obey the Newtonian assump-

tion, such as molten plastics, slurries, emulsions, pulps, which are termed non-Newtonian

fluids [51]. This means that the rapport between the stress tensor and the rate of defor-

mation tensor is non-linear. Actually, about 90% of fluids are nonlinear under the large

deformations, therefore nonlinear viscoelastic mathematical models are needed.

Research related to non-Newtonian fluid mechanics is of great realistic significance to

industry. In electronics, solid-state lighting, transportation, and manufacturing, cool-

ing is one of the most challenging technical issues to handle heat transfer. To achieve

high-performance cooling, some conventional methods use extended surfaces for heat

dissipation such as microchannels, which increase the size of the thermal management

systems, or use mechanical rotation and vibration, and impose electrostatic or magnetic

fields to improve heat dissipation, which increase external energy cost. To overcome these

problems, nanofluids has been proved a promising technology to achieve high-performance

cooling [52]. Numerous investigations have suggested different mechanisms for heat trans-

fer enhancement of nanofluids [121, 133, 257], however, a unified scientific basis for these

mechanisms still is lacking. One main characteristic property of nanofluids is the memo-

ry, which means the stress on a fluid element depends on the history of the deformation

imposed on that element. Another dominant property is clustering [94], which means the

fluid shear stress exerting on the nanoparticles leads to complex hydrodynamic interac-

tions between fluid and solid nanoparticles and gives rise to the clustering of nanoparticles.

Moreover, for the diffusion of nanoparticles in the porous medium, some research [190]

Chapter 1 4

shows that the process is anomalous, which cannot be described using the classical Fick’s

second law [199]. In addition, for some complex viscoelastic materials, the relaxation

processes deviate from the classical exponential behaviour and exhibit a stretched ex-

ponential decay (Φ(t) ∝ exp(−(t/τ)α, 0 < α < 1) or a scaling decay (Φ(t) ∝ (t/τ)−β,

0 < β < 1) [221] and their actual mechanical response is modulated by the past, i.e.,

infinite memory property [102]. There are also other special characteristics of viscoelastic

materials such as creep, shear thinning, hysteresis, and energy dissipation. However, the

typical viscoelastic fluid models including the Maxwell model, Kelvin-Voigt model, Jef-

frey’s model, Oldroyd-B model and Burgers’ model involve integer-order derivatives and

local property and cannot deal with the memory effect or nonlocality.

The constitutive equations involving fractional derivatives have proved to be a valuable

tool to treat viscoelastic properties [7, 90] or anomalous diffusion in porous media [198,

199]. In [7], Bagley and Torvik found that a three-parameter fractional derivative model

could describe the frequency-dependent stress-strain relationships of viscoelastic materials

very well for three different polymers. Torvik and Bagley [243] also applied a three-

parameter fractional derivative model to the transient response of an elastomer. They

observed that the retarding force was proportional to a fractional derivative of order 3/2 of

the displacement and the stress was proportional to strain and to the ‘0.56’ time derivative

of strain. They also found that the stresses at any point in the interior of the fluid in

motion were proportional to a fractional derivative of the local velocity and the data

fitting by the fractional model was much better by a Voigt model. In [178], Makris and

Constantinou conducted an experiment of a piston moving in a highly viscous gel and they

found that the fractional derivative Maxwell model could fit the viscoelastic properties

of the viscous damper very well, which could be used in vibration-isolation systems for

pipeworks and industrial machines and in seismic-isolation systems for structures. In

[102], Hernandez-Jimenez et al. carried out an experiment to study the stress relaxation of

two different kinds of polymers Methylmethacrylate (PMMA) and Polytetrafluorethylene

PTFE. They observed that the data fitting of the relaxation process using a fractional

Maxwell model was much better than that using the classic Maxwell model. In [44],

Chen et al. explored the viscoelastic properties of the skin, muscle, and fins of Crucian

carp. They applied a fractional Zener model to fit the relaxation force, which described

the relaxation process very well. In [199], Pan et al. studied the thermal transport of

nanofluids in disordered porous media. They found that the diffusion of nanoparticles

was anomalous and the local Nusselt numbers of four different kinds of nanofluids were

all inversely proportional to the fractional derivative exponent. They concluded that a

space fractional thermal transport equation may serve as a candidate model for simulating

the nanofluids. Compared with an integer order model, a fractional model can accurately

describe the viscoelastic behavior of materials using fewer parameters and can simulate the

damping behavior of viscoelastic materials in larger frequency ranges [7]. The fractional

model can also deal with the memory and non-local property. This motivates us to study

a fractional non-Newtonian fluids model.

5 Chapter 1

Quite different to the feature of Newtonian fluids, the constitutive equations of non-

Newtonian fluids leads to highly nonlinear differential equations, which involve complex

multi-term time fractional dynamical systems with both diffusion and time-fractional d-

iffusion terms, as well as a nonlinear reaction term, which increases their difficulty and

complexity. Although some exact solutions of the constitutive equations can be derived by

Fourier or Laplace transforms, they are in series form with generalised G or R functions,

which are hard to express explicitly. Therefore, we need to develop some effective numer-

ical methods to solve these complex multi-term fractional dynamical systems, which will

be another contribution of the work.

This project aims to develop new computational fractional dynamical models including

new numerical methods, mathematical models, and applications, to help address three

key aspects of these challenges, namely,

• to develop new numerical methods and analytical techniques for simulating complex

fractional dynamical models to reduce computational cost, which will be achieved

by utilising high-order numerical methods and fast algorithms;

• to develop numerical methods for complex fractional dynamical models with the

Riesz fractional operator on irregular domains, which will be achieved by using the

finite element method and finite volume method combining the unstructured mesh

capability to approximate the space fractional derivative;

• to develop numerical methods for viscoelastic non-Newtonian fluid models, such as

the generalised Maxwell fluid model, the generalised Oldroyd-B fluid model and

the generalised Burgers’ fluid model, which will be achieved by applying mixed

difference schemes to discretise the different time fractional derivatives.

The structure of this thesis is shown in Figure 1.1.

Figure 1.1: The structure of this thesis

Chapter 1 6

1.2 Literature Review

Here we briefly address the background literature to the three main themes of this project,

namely (i) numerical methods for fractional dynamical models; (ii) fractional dynamical

models involving the Riesz fractional operator implemented on irregular domains; and

(iii) complex viscoelastic non-Newtonian fluid models.

1.2.1 Numerical methods for fractional dynamical models

Over the past few decades, the theory of fractional calculus and fractional derivatives has

been expanding at a fast rate [8, 60, 103, 128, 139, 156, 186, 203, 218]. Firstly, some

fundamental solution or analytical solution of the fractional dynamical models are con-

sidered [42, 107, 111, 147, 176]. A considerable number of numerical methods have been

proposed for solving the fractional partial differential equation (FPDE), including finite

difference methods [45, 47, 62, 76, 148, 149, 181, 238, 299, 300], finite element methods

[20, 58, 68, 79, 116, 138, 160, 164, 280, 303], finite volume methods [77, 100, 155, 261],

spectral methods [16, 145, 277, 297, 298], and meshless methods [161, 301]. Although ex-

isting numerical methods for FDEs are numerous, most of them are time-consuming due

to the memory of time fractional derivative or storage consuming due to the non-locality

of the fractional derivative and the density of the coefficient matrix generated by the frac-

tional derivative. Therefore seeking effective numerical methods to reduce computational

cost or storage is important. To date, Wang et al. [248] developed a fast finite difference

method for fractional diffusion equations. According to the structure of the coefficient

matrix, they decomposed it into a sum of matrices involving Toeplitz matrices, which

reduced the total memory requirement from O(M2) to O(M), where M is the number of

spatial grid points. Then, the Toeplitz matrix can be embedded into a circulant matrix

and evaluated using fast Fourier transform and inverse fast Fourier transform. Wang

et al. [249, 250, 252] also extend the method to two- and three-dimensional problems.

In [194], Moroney and Yang proposed a fast Poisson preconditioner to solve a class of

two-sided nonlinear space-fractional diffusion equations. They employed backward differ-

entiation formulas and Jacobian-free Newton-Krylov methods to solve the semi-discrete

nonlinear systems. Moroney and Yang [195] also proposed a banded preconditioner for

the nonlinear space-fractional diffusion equation. In [228], Simmons et al. developed a

novel preconditioner that eliminated any requirement to form the dense matrix represen-

tation of the fractional Laplacian operator and also captured the full contribution from

the nonlinear reaction term. In [202], Pestana et al. established a circulant preconditioned

short recurrence Krylov subspace iterative method with minimum residual type for non-

symmetric Toeplitz systems. In [74], Farquhar et al. presented an effective numerical

method for computing matrix function vector products on graphics processing units (G-

PUs) generated in solving fractional reaction-diffusion equations. Due to the nonlocality

and singularity of the fractional operator, the calculation for the time-fractional equations

is more difficult and time-consuming, which has been a worldwide research focus as well

recently [6, 258, 279].

7 Chapter 1

1.2.2 Fractional dynamical models involving the Riesz fractional operator implemented

on irregular domains

Space and time-space fractional dynamical models involving the Riesz fractional opera-

tor implemented on irregular domains find application in medical and biological fields.

Such models include the time-space Bloch-Torrey equation [175], the fractional FitzHugh-

Nagumo monodomain model [157], the fractional version of the Gray-Scott model [24],

and a fractional diffusion model of cardiac electrical propagation [23].

The simplest reaction-diffusion model can be written in the form:

∂u(r, t)

∂t= K∆u(r, t) + f(r, t).

Magin et al. [175] proposed a new diffusion model by solving the Bloch-Torrey equation

using fractional calculus. The time-space fractional Bloch-Torrey equations (ST-FBTE)

with a Riesz fractional operator [272] can be written in the following form:

KγC0 D

γtM(r, t) = Kα(

∂α

∂|x|α+

∂α

∂|y|α+

∂α

∂|z|α)M(r, t) + f(r, t),

where 0 < γ < 1, 1 < α < 2, r = (x, y, z), M can be either the transverse components

of the magnetization Mx or My, and f(r, t) = λGMy(r, t) if M = Mx, and f(r, t) =

−λGMx(r, t) if M = My.

The two-dimensional Riesz space fractional diffusion equation can be written in the fol-

lowing form [259]:

∂u

∂t= Kx

∂αu

∂|x|α+Ky

∂αu

∂|y|α+ f(x, y, t), (x, y) ∈ Ω. (1.1)

For Eq.(1.1), Liu et al. [152] considered the finite difference method using a shifted

Grunwald-Letnikov scheme for the Riesz operators. Liu et al. also [153] proposed an im-

plicit alternating direction method for a 2D Riesz space fractional diffusion equation with

a nonlinear reaction term. Yu et al. [272] discussed the stability and convergence anlysis

of an implicit numerical scheme for the time-space fractional Bloch-Torrey equation. Yu

et al. also [271] considered the alternating direction method for the equation in a 3D

case. Bu et al. [20, 21] proposed the finite element method for a two-dimensional space

and time-space Riesz fractional diffusion equation. Zeng et al. [277] established the A-

DI spectral method for a 2D Riesz space fractional nonlinear reaction-diffusion equation.

In [157], Liu et al. proposed a second-order semi-implicit alternating direction scheme

for a 2D fractional FitzHugh-Nagumo monodomain model on an approximate irregular

domain.

Although there exist some numerical methods to solve fractional dynamical models with

the Riesz fractional operator, they are limited to regular domains and using structured

Chapter 1 8

grids. The study involving the Riesz fractional operator on irregular domains is sparse.

In fact, many other problems from science and engineering involve mathematical models

that must be computed on irregular domains such as a human brain domain and cardiac

tissue and therefore seeking effective numerical methods to solve FDE on such domains

is important, which motivates us to develop and explore new numerical methods and

analytical techniques for these fractional dynamical models.

1.2.3 Complex viscoelastic non-Newtonian fluid models

Complex multi-term time fractional dynamical systems with both diffusion and time-

fractional diffusion terms have been used for modelling many relevant physical processes

in fluid dynamics, particularly for a non-Newtonian fluid. The fluid is called a non-

Newtonian fluid when the linear relationship between the shear stress and shear deforma-

tion rate does not hold. These fluids exhibit many typical anomalous diffusion characteris-

tics, such as clustering, shear thinning, stress relaxation, creep etc., and the transmission

process has memory. The constitutive equations involving fractional derivatives have

been proved to be a valuable tool to handle these properties. Generally, viscoelastic non-

Newtonian fluids are classified as fluids of differential, integral and rate types [65, 212], in

which the differential and rate type models are applied to describe the response of fluids

that have a slight memory and the integral models are utilised to describe the response

of fluids that have a considerable memory. Typical rate-type viscoelastic fluid models

contain the generalised Maxwell fluid model, the generalised Oldroyd-B fluid model, and

the generalised Burgers’ fluid model.

The generalised Maxwell fluid model

The Maxwell model was first proposed by Maxwell to describe the dynamical theory

of gases [180]. It is constructed using a series of spring and dashpots and can reflect

some characteristics of the viscoelastic fluid qualitatively, such as stress relaxation. The

Maxwell model is only proper under the condition of very low strain and stress and leads

to an exponential stress relaxation modulus. However, for many real materials, the stress

relaxation exhibits an algebraic decay [103]. As the fractional derivative models with the

algebraic stress decay can be easily constructed, the generalised Maxwell fluid model is

introduced. The constitutive relation for an incompressible Maxwell fluid with fractional

derivative is given by [293]

T = −pI + S, S + λα(Dαt S + V · ∇S− LS− SLT ) = µA,

where T refers to the Cauchy stress tensor, S is the extra stress tensor, A = L + LT

denotes the first Rivlin-Ericksen tensor, L is a velocity gradient, λα is the relaxation

time. −pI denotes the indeterminate spherical stress, µ is the dynamic viscosity of the

fluid, V refers to the velocity, ∇ is a gradient operator, and Dαt is a fractional differential

9 Chapter 1

operator based on either a Riemann-Liouville or Caputo fractional derivative. Assume

that the velocity and shear stress take the form

V = V(y, t) = u(y, t)i, S = S(y, t),

where u is the velocity and i is the unit vector in the x-direction. Taking account of the

initial condition V(y, 0) = 0, S(y, 0) = 0, we have the following equations [293]

ρ∂u(y, t)

∂t= −∂p

∂x+∂τ(y, t)

∂y,

∂p

∂y=∂p

∂z= 0,

(1 + λαDαt ) τ(y, t) = µ

∂u(y, t)

∂y,

where τ(y, t) = Sxy(y, t) is the non-zero shear stress. By combining the above two equa-

tions, we obtain [293]

(1 + λαDαt )∂u(y, t)

∂t= −1

ρ(1 + λαDα

t )∂p

∂x+µ

ρ

∂2u(y, t)

∂y2.

In the absence of pressure gradient in the flow direction and body forces, we arrive at a

simpler model of the form:

(1 + λαDα

t

)∂u∂t

= ν∂2u

∂y2, (1.2)

where ν = µρ . This model is generally used to describe the flow of an incompressible

Maxwell fluid over an infinitely extended flat plate. The fluid is at rest initially and

starts to move with some velocity at t = 0+. In [293], Zheng et al. considered the case

with oscillatory behaviour and constant acceleration. They found that the velocity is

an increasing function of ω (the frequency of oscillation of the plate) near the plate for

the cosine oscillations, however, decreases for the sine oscillations. In [86], Fetecau et

al. considered the case with initial velocity At, where A is a constant. They obtained

the exact analytical solutions using the Fourier sine and Laplace transforms, which is

expressed as double integrals of double series. In [247], Vieru et al. investigated a two-

dimensional fractional Maxwell model between two side walls perpendicular to a plate.

They established the exact solutions for the velocity field using the Fourier and Laplace

transforms. In [205], Qi et al. studied the generalised Maxwell fluid model in a channel.

They derived the exact solutions for an arbitrary pressure gradient utilising the finite

Fourier cosine transform and the Laplace transform.

The generalised Oldroyd-B fluid model

The Oldroyd-B model is commonly used to describe the rheological property of polymer

liquids composed at moderate shear rates and low concentration for high-viscosity New-

tonian molecular weight polymers. As a special viscoelastic non-Newtonian fluid, the

generalised Oldroyd-B fluid has been applied to fluid flow problems of small relaxation

Chapter 1 10

and retardation times. The constitutive relation for an incompressible Oldroyd-B fluid

with fractional derivative is defined as [204]

T = −pI + S, S + λαDαS

Dtα= µ

[1 + λβ

Dβ

Dtβ

]A,

where the expression of Dα

Dtα and Dβ

Dtβare defined as:

DαS

Dtα= Dα

t S + V · ∇S− LS− SLT,

DβA

Dtβ= Dβ

t A + V · ∇A− LA−ALT,

λα is the relaxation time and λβ is the retardation time. This model can degenerate into

an ordinary Oldroyd-B fluid model for α → 1, β → 1 and degenerate to the generalised

Maxwell model for λβ → 0. Using a similar derivation in the absence of a pressure

gradient in the flow direction and body forces, a simple form can be derived:

(1 + λαDα

t

)∂u∂t

= ν(1 + λβDβ

t

)∂2u

∂y2, (1.3)

where λα is the relaxation time, λβ is the retardation time, µ is the dynamic viscosity,

ρ is the constant density of the fluid and ν = µρ . In [204], Qi et al. considered the

first problem of Stokes for a viscoelastic fluid with the generalised Oldroyd-B model

lying over an infinitely extended flat plate. They found that the effect of the relaxation

and retardation times and the fractional orders in the generalised Oldroyd-B model is

significant. Khan et al. [124] studied the constant acceleration and variable acceleration

flow problem of a generalised Oldroyd-B fluid. They observed that the velocity profiles

for a generalised Oldroyd-B fluid were greater than those for the classical Oldroyd-B fluid

and the shear stress for a generalised Oldroyd-B fluid is greater in magnitude than that for

those of the classical Oldroyd-B fluid. Some researchers also studied the two-dimensional

case [85, 125, 223].

When a magnetic field is imposed on the above flow and a low magnetic Reynolds number

is supposed, the following velocity equation can be derived [163]

(1 + λαDα

t

)∂u∂t

= ν(1 + λβDβ

t

)∂2u

∂y2−K

(1 + λαDα

t

)u, (1.4)

where K =σB2

0ρ and B0 is the magnetic intensity and σ is the electrical conductivity. For

the Eq.(1.4), the analytical solution has been studied by Liu and Zheng et al. under the

general boundary conditions and special boundary condition with slip effects [163, 295,

296].

11 Chapter 1

The generalised Burgers’ fluid model

In general, Burgers’ model is used in the description of the motion of the earth’s mantle

[27], the response of asphalt and asphalt concrete [135], or the propagation of seismic

waves in the interior of the earth [201]. The generalised Burgers’ model is derived from the

classic Burgers’ model by replacing the integer order time derivatives to fractional order

time derivatives. The constitutive equation for the incompressible generalised Burgers’

fluid is defined as [126]

T = −pI + S,(

1 + λα1 Dαt + λα2 D2α

t

)S = µ

[1 + λβ3 Dβ

t

]A,

where the operators Dαt , D2α

t , Dβt are defined by

Dαt S = D

αt S + V · ∇S− LS− SLT,

D2αt S = Dα

t

(Dαt S),

Dβt A = Dβ

t A + V · ∇A− LA−ALT.

Then, the following fractional Burgers’ fluid model can be obtained [126]

(1 + λα1 Dα

t + λα2 D2αt

) ∂u∂t

= ν(

1 + λβ3 Dβt

) ∂2u

∂y2, (1.5)

where λ1, λ2, λ3 are the material constants, µ is the dynamic viscosity, ρ is the constant

density of the fluid and ν = µρ . In [126], Khan et al. considered the model for a fluid

lying over an infinitely extended plate with constant acceleration. They found that the

fractional order α was related to the shear-thinning behaviour whereas β was related to

shear-thickening behaviour and the velocity profiles for Oldroyd-B fluid were much greater

than those of Burgers’ fluid. They also observed that the velocity profiles obtained using

the fractional Burgers’ fluid model are much greater in magnitude than those of the

classical Burgers’ fluid model for both constantly and variable accelerated flows. Khan et

al. [127] also considered the model above a flat plate with oscillating motions. They found

that the amplitude of the fluid oscillation decayed away from the plate and approached

zero. They also showed that the decay of the amplitude of oscillation for fractional

Burgers’ fluid was faster than that for classical Burgers’ fluid.

In recent years, there are also other complex non-Newtonian fluid models being proposed.

In [285], Zhao et al. considered a fractional MHD viscoelastic fluid in a porous medium

with Soret and Dufour effects, in which they found that the fractional derivative parame-

ter enhanced the elastic effect of the viscoelastic fluid and decelerated the convection flow.

Zhao et al. [287] also investigated the unsteady Marangoni convection heat transfer of a

viscoelastic Maxwell fluid over a flat surface and found that the fractional derivative pa-

rameters and power law exponent had a significant influence on the characteristic velocity

and temperature fields. Sin et al. [229] used a fractional K-BKZ constitutive equation to

Chapter 1 12

simulate the unsteady flow of a viscoelastic fluid between two parallel plates. They con-

cluded that the fluid with the fractional Maxwell model gradually lost the viscoelasticity,

while the fractional K-BKZ model can preserve the viscoelasticity continually. Pan et al.

[198] developed a stochastic thermal transport model for simulating the nanofluid flow

in porous media. Zhao et al. [288] considered the unsteady convection heat and mass

transfer of a fractional Oldroyd-B fluid and showed that fractional derivative parameters

had a remarkable effect on the viscoelastic properties of the fluid. Chen et al. [50] stud-

ied the boundary layer flow and heat transfer of fractional viscoelastic MHD fluid over a

stretching sheet and found that magnetic field parameter, fractional derivative and wall

stretching exponent had a strong impact on the skin friction and thermal conductivity.

As Eqs.(1.3)-(1.5) contain similar terms, they can be expressed in a generalised form for a

class of novel multi-term time-fractional mixed sub-diffusion and diffusion-wave equations

s∑j=1

bj Dγjt u+ a1

∂u

∂t+

q∑l=1

clDαlt u+ a2u = a3

∂2u

∂y2+ a4D

βt

∂2u

∂y2+ f, (1.6)

where ai > 0, i = 1, 2, 3, 4, bj ≥ 0, j = 1, 2, . . . , s, cl ≥ 0, l = 1, 2, . . . , q, 0 < β < 1,

1 < γ1 < γ2 < . . . < γs < 2 and 0 < α1 < α2 < . . . < αq < 1.

Eq.(1.6) not only incorporates the generalised Maxwell fluid model (1.3), the generalised

Oldroyd-B fluid model (1.4) and the generalised Burgers’ fluid model (1.5) but also con-

tains the general multi-term time fractional subdiffusion [291] or wave equations [298],

the time-fractional telegraph equation [191], the fractional cable equation [101], and a

heated generalised second grade fluid model [224]. Although the research on the ana-

lytical solution of the non-Newtonian fluid models is numerous, the solution is typically

given in a series form involving special functions, such as the Fox H-function or the mul-

tivariate Mittag-Leffler function, and both of these functions are challenging to calculate

explicitly. Therefore, seeking a numerical solution is of importance to provide insight

into the behaviour of these models. However, research of numerical methods for the non-

Newtonian fluid models is limited [159, 225]. Furthermore, studies of numerical methods

for the multi-term fractional partial differential equations are still under development

[154, 187, 289] and addressing this gap in the literature is an important contribution of

this work. This motivates us to simulate the multi-term time fractional dynamical system-

s (1.6) by developing new effective numerical methods and study the physics associated

with the problem.

1.3 Research Objectives

The primary objectives of this thesis are stated as follows:

• Develop new numerical methods and analytical techniques for simulating complex

fractional models to reduce the computational cost, which will be achieved by util-

ising high-order numerical methods and fast algorithms;

13 Chapter 1

• Develop numerical methods for complex fractional dynamical models with the Riesz

fractional operator on irregular domains, which will be achieved by using the finite

element method and finite volume method combining the unstructured mesh capa-

bility to approximate the space fractional derivative;

• Develop numerical methods for viscoelastic non-Newtonian fluid models, such as

the generalised Maxwell fluid model, the generalised Oldroyd-B fluid model and

the generalised Burgers’ fluid model, which will be achieved by applying mixed

difference schemes to discretise the different time fractional derivatives.

1.4 Thesis Outline

This thesis is presented by publications, in which the original contribution to the literature

is listed in six published papers and one paper under review. There are also two papers

listed in the Appendix (Appendices A and B). The outlines of these papers are organised

as follows.

1.4.1 Chapter 2: High-order numerical methods for the RSFADE

In this chapter, we present the work on the high-order numerical methods for the Riesz

space fractional advection-dispersion equations (RSFADE). This work has been published

in the following paper:

F L. Feng, P. Zhuang, F. Liu, I. Turner, J. Li, High-order numerical methods for the

Riesz space fractional advection-dispersion equations, Computers & Mathematics

with Applications, (2016). http://dx.doi.org/10.1016/j.camwa.2016.01.015.

Statement of Joint Authorship

L. Feng (Candidate) Proposed the high-order numerical methods for the RSFADE, de-

rived the Crank-Nicolson scheme and the Richardson extrapolation method, proved the

stability and convergence of the scheme, developed the numerical codes in MATLAB,

interpreted the results, and wrote the manuscript.

P. Zhuang Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

J. Li Assisted with the interpretation of results and the preparation of the paper, proof-

read the manuscript.

Chapter 1 14

Paper Abstract

In this paper, we propose high-order numerical methods for the Riesz space fraction-

al advection-dispersion equations (RSFADE) on a finite domain. The RSFADE is ob-

tained from the standard advection-dispersion equation by replacing the first-order and

second-order space derivative with the Riesz fractional derivatives of order α ∈ (0, 1) and

β ∈ (1, 2], respectively. Firstly, we utilize the weighted and shifted Grunwald difference

operators to approximate the Riesz fractional derivative and present the finite difference

method for the RSFADE. Specifically, we discuss the Crank-Nicolson scheme and solve

it in matrix form. Secondly, we prove that the scheme is unconditionally stable and con-

vergent with the accuracy of O(τ2 + h2). Thirdly, we use the Richardson extrapolation

method (REM) to improve the convergence order which can be O(τ4 +h4). Finally, some

numerical examples are given to show the effectiveness of the numerical method, and the

results are in excellent with the theoretical analysis.

1.4.2 Chapter 3: A fast second-order accurate method for a TSSFDE-VC

In this chapter, we present the work on a fast second-order accurate method for a two-

sided space-fractional diffusion equation with variable coefficients (TSSFDE-VC). This

work has been published in the following paper:

F L. Feng, P. Zhuang, F. Liu, I. Turner, V. Anh and J. Li, A fast second-order accurate

method for a two-sided space-fractional diffusion equation with variable coefficients,

Computers & Mathematics with Applications 73 (2017) 1155-1171.

Statement of Joint Authorship

L. Feng (Candidate) Proposed the second-order scheme to approximate the Riemann-

Liouville fractional derivative, derived the Crank-Nicolson scheme and the fast accurate

iterative method, proved the stability and convergence of the scheme, developed the

numerical codes in MATLAB, interpreted the results, and wrote the manuscript.

P. Zhuang Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

V. Anh Assisted with the interpretation of results and the preparation of the paper,

proofread the manuscript.

15 Chapter 1

J. Li Assisted with the interpretation of results and the preparation of the paper, proof-

read the manuscript.

Paper Abstract

In this paper, we consider a type of fractional diffusion equation (FDE) with variable

coefficients on a finite domain. Firstly, we utilize a second-order scheme to approxi-

mate the Riemann-Liouville fractional derivative and present the finite difference scheme.

Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Second-

ly, we prove the stability and convergence of the scheme and conclude that the scheme

is unconditionally stable and convergent with the second-order accuracy of O(τ2 + h2).

Furthermore, we develop a fast accurate iterative method for the Crank-Nicolson scheme,

which only requires storage of O(m) and computational cost of O(m logm) while retain-

ing the same accuracy and approximation property as Gauss elimination, where m = 1/h

is the partition number in space direction. Finally, several numerical examples are given

to show the effectiveness of the numerical method, and the results are in excellent with

the theoretical analysis.

1.4.3 Chapter 4: Unstructured mesh FD/FEM for the 2D-TSRFDE on irregular

convex domains

In this chapter, we present the work on the unstructured mesh finite difference/finite

element method for the 2D time-space Riesz fractional diffusion equation on irregular

convex domains (2D-TSRFDE). This work has been published in the following paper:

F L. Feng, F. Liu, I. Turner, Q. Yang, P. Zhuang, Unstructured mesh finite differ-

ence/finite element method for the 2D time-space Riesz fractional diffusion equation

on irregular convex domains, Applied Mathematical Modelling 59 (2018) 441-463.

Statement of Joint Authorship

L. Feng (Candidate) Proposed the Galerkin finite element method (FEM) with an un-

structured mesh to deal with the space fractional derivative, derived the second order

finite difference scheme for the temporal fractional derivative, proved the stability and

convergence of the scheme, extended the theory and develop a computational model for

the case of a multiply-connected domain, developed the numerical codes in MATLAB,

interpreted the results, and wrote the manuscript.

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

Chapter 1 16

Q. Yang Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

P. Zhuang Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

Paper Abstract

Fractional differential equations are powerful tools to model the non-locality and spatial

heterogeneity evident in many real-world problems. Although numerous numerical meth-

ods have been proposed, most of them are limited to regular domains and uniform meshes.

For irregular convex domains, the treatment of space fractional derivative becomes more

challenging and the general methods are no longer feasible. In this work, we propose a

novel numerical technique based on the Galerkin finite element method (FEM) with an

unstructured mesh to deal with space fractional derivative on arbitrarily shaped convex

and non-convex domains, which is the most original and significant contribution of this

paper. Moreover, we present a second order finite difference scheme for the temporal frac-

tional derivative. In addition, the stability and convergence of the method are discussed

and numerical examples on different irregular convex domains and non-convex domains

illustrate the reliability of the method. We also extend the theory and develop a compu-

tational model for the case of a multiply-connected domain. Finally, to demonstrate the

versatility and applicability of our method, we solve the coupled two-dimensional frac-

tional Bloch-Torrey equation on a human brain-like domain and exhibit the effects of the

time and space fractional indices on the behaviour of the transverse magnetization.

1.4.4 Chapter 5: An unstructured mesh control volume method for 2D-SFDE-VC on

convex domains

In this chapter, we present the work on an unstructured mesh control volume method for

two-dimensional space fractional diffusion equations with variable coefficients (2D-SFDE-

VC) on convex domains. This work has been submitted to the following journal:

F L. Feng, F. Liu, I. Turner, An unstructured mesh control volume method for two-

dimensional space fractional diffusion equations with variable coefficients on convex

domains, submitted to Journal of Computational and Applied Mathematics, 2018.

Statement of Joint Authorship

L. Feng (Candidate) Proposed the unstructured mesh control volume method to deal

with the space fractional derivative on arbitrarily shaped convex domains, derived the

full implementation of the control volume method on triangular elements, explored the

property of the stiffness matrix and derived the fast iterative method, developed the

numerical codes in MATLAB, interpreted the results, and wrote the manuscript.

17 Chapter 1

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

Paper Abstract

In this paper, we propose a novel unstructured mesh control volume method to deal

with space fractional derivative on arbitrarily shaped convex domains, which to the best

of our knowledge is a new contribution to the literature. Firstly, we present the finite

volume scheme for the two-dimensional space fractional diffusion equation with variable

coefficients and provide the full implementation details for the case where the background

interpolation mesh is based on triangular elements. Secondly, we explore the property

of the stiffness matrix generated by the integral of space fractional derivative. We find

that the stiffness matrix is sparse and not regular. Therefore, we choose a suitable sparse

storage format for the stiffness matrix and develop a fast iterative method to solve the

linear system, which is more efficient than using the Gaussian elimination method. Finally,

we present several examples to verify our method, in which we make a comparison of

our method with the finite element method for solving a Riesz space fractional diffusion

equation on a circular domain. The numerical results demonstrate that our method

can reduce CPU time significantly while retaining the same accuracy and approximation

property as the finite element method. The numerical results also illustrate that our

method is effective and reliable and can be applied to problems on arbitrarily shaped

convex domains.

1.4.5 Chapter 6: Numerical methods and analysis for simulating the flow of a GOBF

between two infinite parallel rigid plates

In this chapter, we present the work on the numerical methods and analysis for simulating

the flow of a generalised Oldroyd-B fluid (GOBF) between two infinite parallel rigid plates.

This work has been published in the following paper:

F L. Feng, F. Liu, I. Turner, P. Zhuang, Numerical methods and analysis for simulating

the flow of a generalised Oldroyd-B fluid between two infinite parallel rigid plates,

International Journal of Heat and Mass Transfer 115 (2017) 1309-1320.

Statement of Joint Authorship

L. Feng (Candidate) Proposed the finite difference method, derived two implicit numer-

ical schemes for the equation, proved the stability and convergence of the scheme, devel-

oped the numerical codes in MATLAB, interpreted the results, and wrote the manuscript.

Chapter 1 18

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

P. Zhuang Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

Paper Abstract

In recent years, non-Newtonian fluids have been widely applied in a number of engi-

neering applications. One particular subclass of non-Newtonian fluids is the generalised

Oldroyd-B fluids with fractional derivative constitutive equations, which can be used to

approximate the response of many dilute polymeric liquids. Different to the general time

fractional diffusion equation, the constitutive equation not only has a multi-term time

derivative but also possesses a special time fractional operator on the spatial derivative,

which is challenging to approximate. The literature reported on the numerical solution

of this model is extremely sparse. In this paper, we will consider the finite difference

method for its discretisation and propose a new scheme to approximate the time frac-

tional derivative. Then we derive an implicit finite difference scheme and establish some

new theoretical analysis of the stability and convergence. Furthermore, we present a nu-

merical scheme to improve the time order. Finally, we present two numerical examples

to show the effectiveness of our method and apply it to solve the generalised Oldroyd-B

fluid model.

1.4.6 Chapter 7: Novel numerical analysis of multi-term time fractional viscoelastic

non-Newtonian fluid models for simulating unsteady MHD Couette flow of a

GOBF

In this chapter, we present the work on the novel numerical analysis of multi-term time

fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette

flow of a GOBF. This work has been published in the following paper:

F L. Feng, F. Liu, I. Turner, L. Zheng, Novel numerical analysis of multi-term time

fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD

Couette flow of a generalised Oldroyd-B fluid, Fractional Calculus and Applied

Analysis 21(4) (2018) 1073-1103.

Statement of Joint Authorship

L. Feng (Candidate) Proposed the finite difference method, derived two implicit nu-

merical schemes for the novel multi-term time fractional viscoelastic non-Newtonian fluid

19 Chapter 1

model, proved the stability and convergence of the scheme, developed the numerical codes

in MATLAB, interpreted the results, and wrote the manuscript.

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

L. Zheng Assisted with the interpretation of results and the preparation of the paper,

proofread the manuscript.

Paper Abstract

In this paper, we consider the application of the finite difference method for a class of

novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important

contribution of the work is that the new model not only has a multi-term time derivative,

of which the fractional order indices range from 0 to 2 but also possesses a special time

fractional operator on the spatial derivative that is challenging to approximate. There

appears to be no literature reported on the numerical solution of this type of equation.

We derive two new different finite difference schemes to approximate the model. Then

we establish the stability and convergence analysis of these schemes based on the discrete

H1 norm and prove that their accuracy is of O(τ + h2) and O(τmin3−γs,2−αq ,2−β + h2),

respectively. Finally, we verify our methods using two numerical examples and apply

the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a

generalised Oldroyd-B fluid model. Our methods are effective and can be extended to

solve other non-Newtonian fluid models such as the generalised Maxwell fluid model, the

generalised second grade fluid model and the generalised Burgers fluid model.

1.4.7 Chapter 8: FD/FEM for a novel 2D multi-term TFMSDWE

In this chapter, we present the work on the finite difference/finite element method for

a novel 2D multi-term time fractional mixed sub-diffusion and diffusion-wave equation

(TFMSDWE) on convex domains. This work has been submitted to the following paper:

F L. Feng, F. Liu, I. Turner, Finite difference/finite element method for a novel 2D

multi-term time fractional mixed sub-diffusion and diffusion-wave equation on con-

vex domains, Communications in Nonlinear Science and Numerical Simulation 70

(2019) 354-371.

Statement of Joint Authorship

Chapter 1 20

L. Feng (Candidate) Proposed the finite difference method to discretise the time frac-

tional derivative, derived the full discretisation finite element scheme for the equation,

proved the stability and convergence of the scheme, developed the numerical codes in

MATLAB, interpreted the results, and wrote the manuscript.

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

Paper Abstract

In this work, a novel two-dimensional (2D) multi-term time-fractional mixed sub-diffusion

and diffusion-wave equation on convex domains will be considered. Different from the gen-

eral multi-term time-fractional diffusion-wave or sub-diffusion equation, the new equation

not only possesses the diffusion-wave and sub-diffusion terms simultaneously but also has

a special time-space coupled derivative. Although the analytic solution of this kind of

equation can be derived using a multi-level sum of an infinite series and Fox H-functions,

it is extremely complex and difficult to evaluate. Therefore, seeking the numerical so-

lution of the equation is of great importance. In this paper, we will consider the finite

element method for the novel 2D multi-term time fractional mixed diffusion equation.

Firstly, we utilise the mixed L schemes to approximate the time fractional sub-diffusion

term, diffusion-wave term and the coupled time-space derivative, respectively. Secondly,

we establish the variational formulation and use the finite element method to discretise

the equation. Then we adopt linear polynomial basis functions on triangular elements to

derive the matrix form of the numerical scheme. Furthermore, we present the stability and

convergence analysis of the numerical scheme. To show the effectiveness of our method,

three examples are investigated, in which a 2D multi-term time fractional mixed diffusion

equation on a circular domain and a 2D generalised Oldroyd-B fluid in a magnetic field

are analysed.

1.4.8 Chapter 9: Conclusions

In this chapter, we summarise the main contributions of this thesis and draw the conclu-

sions. Finally, we present some recommendations for future research.

1.4.9 Appendix A: FVM for the Riesz space distributed-order diffusion equation

In this chapter, we present the work on a novel finite volume method for the Riesz space

distributed-order diffusion equation. This work has been published in the following paper:

21 Chapter 1

F J. Li, F. Liu, L. Feng, I. Turner, A novel finite volume method for the Riesz space

distributed-order diffusion equation, Computers and Mathematics with Applications

74 (2017) 772-783.

Statement of Joint Authorship

J. Li Proposed the finite volume method to discretise the Riesz space distributed-order

diffusion equation, derived the Crank-Nicolson scheme for the equation, proved the sta-

bility and convergence of the scheme, and wrote the majority of the manuscript.

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

L. Feng (Candidate) Developed the numerical codes in MATLAB, interpreted the nu-

merical results and wrote the numerical parts, and proofread the manuscript.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

Paper Abstract

In recent years, considerable attention has been devoted to distributed-order differential

equations mainly because they appear to be more effective for modelling complex processes

which obey a mixture of power laws or flexible variations in space. In this paper, we

propose a novel finite volume method (FVM) for a distributed-order space-fractional

diffusion equation (FDE). Firstly, we use the mid-point quadrature rule to transform the

space distributed-order diffusion equation into a multi-term fractional equation. Secondly,

the transformed multi-term fractional equation is solved by discretising in space using

the finite volume method and then in time using the Crank-Nicolson scheme. Thirdly, we

prove that the Crank-Nicolson scheme with FVM is unconditionally stable and convergent

with second order accuracy in both time and space. Finally, two numerical examples

are presented to show the effectiveness of the numerical method. These methods and

techniques can also be used to solve other types of fractional partial differential equations.

1.4.10 Appendix B: FVM for the Riesz space distributed-order advection-diffusion

equation

In this chapter, we present the work on a novel finite volume method for the Riesz

space distributed-order advection-diffusion equation. This work has been published in

the following paper:

F J. Li, F. Liu, L. Feng, I. Turner, A novel finite volume method for the Riesz space

distributed-order advection-diffusion equation, Applied Mathematical Modelling 46

(2017) 536-553.

Chapter 1 22

Statement of Joint Authorship

J. Li Proposed the finite volume method to discretise the Riesz space distributed-order

advection-diffusion equation, derived the Crank-Nicolson scheme for the equation, proved

the stability and convergence of the scheme, and wrote the majority of the manuscript.

F. Liu Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript and acted as the corresponding

author.

L. Feng (Candidate) Developed the numerical codes in MATLAB, interpreted the nu-

merical results and wrote the numerical parts, and proofread the manuscript.

I. Turner Directed and guided the work, assisted with the interpretation of results and

the preparation of the paper, proofread the manuscript.

Paper Abstract

In this paper, we investigate the finite volume method (FVM) for a distributed-order

space-fractional advection-diffusion (AD) equation. The mid-point quadrature rule is

used to approximate the distributed-order equation by a multi-term fractional model.

Next, the transformed multi-term fractional equation is solved by discretizing in space by

the finite volume method and in time using the Crank-Nicolson scheme. We use a novel

technique to deal with the convection term, by which the Riesz fractional derivative of

order 0 < γ < 1 is transformed into a fractional integral form. An important contribution

of our work is the use of nodal basis function to derive the discrete form of our model. The

unique solvability of the scheme is also discussed and we prove that the Crank-Nicolson

scheme is unconditionally stable and convergent with second-order accuracy. Finally, we

give some examples to show the effectiveness of the numerical method.

Chapter 2

High-order numerical methods for the Riesz space fractional

advection-dispersion equations

2.1 Introduction

There have been increasing interests in the description of the physical and chemical pro-

cess by means of equations involving fractional derivatives over the last decades. And,

fractional derivatives have been successfully applied into many sciences, such as physics

[230], biology [174], chemistry [274], hydrology [14, 15, 147, 148], and even finance [220].

In groundwater hydrology, the fractional advection-dispersion equation (FADE) is uti-

lized to model the transport of passive tracers carried by fluid flow in a porous medium

[148, 192].

Considerable numerical methods for solving the FADE have been proposed. Kilbas et al.

[128] introduced theory and applications of fractional differential equations. Meerschaert

and Tadjeran [181] developed practical numerical methods to solve the one-dimensional

space FADE with variable coefficients on a finite domain. Liu et al. [148] transformed the

space fractional Fokker-Planck equation into a system of ordinary differential equations

(method of lines), which was then solved using backward differentiation formulas. Momani

and Odibat [192] developed two reliable algorithms, the Adomian decomposition method,

and variational iteration method, to construct numerical solutions of the space-time FADE

in the form of a rapidly convergent series with easily computable components. Zhuang et

al. [300] discussed a variable-order fractional advection-diffusion equation with a nonlinear

source term on a finite domain. Liu et al. [160] proposed an approximation of the Levy-

Feller advection-dispersion process by employing a random walk and finite difference

methods. In addition, other finite difference methods [149], finite element method [68],

finite volume method [100], homotopy perturbation method [96] and spectral method

[34, 292] are also employed to approximate the FADE.

In this paper, we consider the following Riesz space fractional advection-dispersion equa-

tion (RSFADE):

∂u(x, t)

∂t= Kα

∂αu(x, t)

∂|x|α+Kβ

∂βu(x, t)

∂|x|β, 0 < x < L, 0 < t ≤ T, (2.1)

subject to the initial condition:

u(x, 0) = ψ(x), 0 ≤ x ≤ L, (2.2)

23

Chapter 2 24

and the zero Dirichlet boundary conditions:

u(0, t) = 0, u(L, t) = 0, 0 ≤ t ≤ T (2.3)

where 0 < α < 1, 1 < β ≤ 2, Kα ≥ 0 and Kβ > 0 represent the average fluid velocity and

the dispersion coefficient. The Riesz space fractional operators ∂αu∂|x|α and ∂βu

∂|x|βon a finite

domain [0, L] are defined respectively as

∂αu(x, t)

∂|x|α= −cα [0D

αxu(x, t) + xD

αLu(x, t)] ,

∂βu(x, t)

∂|x|β= −cβ

[0D

βxu(x, t) + xD

βLu(x, t)

],

where

cα =1

2 cos πα2, cβ =

1

2 cos πβ2,

and

0Dαxu(x, t) =

1

Γ(1− α)

∂

∂x

∫ x

0(x− ξ)−αu(ξ, t)dξ,

xDαLu(x, t) =

−1

Γ(1− α)

∂

∂x

∫ L

x(ξ − x)−αu(ξ, t)dξ,

0Dβxu(x, t) =

1

Γ(2− β)

∂2

∂x2

∫ x

0(x− ξ)1−βu(ξ, t)dξ,

xDβLu(x, t) =

1

Γ(2− β)

∂2

∂x2

∫ L

x(ξ − x)1−βu(ξ, t)dξ,

where Γ(·) represents the Euler gamma function.

The fractional kinetic equation (2.1) possesses a physical meaning (see [217, 275] for

further details). Physical considerations of a fractional advection-dispersion transport

model restrict 0 < α < 1, 1 < β ≤ 2, and we assume Kα ≥ 0 and Kβ > 0 so that the

flow is from left to right. In the case of α = 1 and β = 2, Eq.(2.1) reduces to the classical

advection-dispersion equation (ADE). In this paper, we only consider the fractional cases:

when Kα = 0, Eq.(2.1) reduces to the Riesz space fractional diffusion equation (RSFDE)

[217] and when Kα 6= 0, the RSFADE is obtained [275].

For the RSFADE (2.1), Anh and Leonenko [3] presented a spectral representation of the

mean-square solution without the non-homogeneous part and for some range of values of

the parameters. Later, Shen et al. [226] derived the fundamental solution of Eq.(2.1) and

discussed the numerical approximation of the Eq.(2.1) using finite difference method with

the first convergence order. Another method based on the spectral approach and the weak

solution formulation was given in Leonenko and Phillips [137]. In addition, Zhang et al.

[280] use the Galerkin finite element method to approximate the RSFADE. Besides, Yang

et al. [259] applied the L1/L2-approximation method, the standard/shifted Grunwald

method, and the matrix transform method to solve the RSFADE. And, Ding et al. [61]

25 Chapter 2

also consider the numerical solution of the RSFADE by using improved matrix transform

method and the (2, 2) Pade approximation. Most of the numerical methods proposed by

these authors are low order or lack stability analysis.

In this paper, based on the weighted and shifted Grunwald difference (WSGD) opera-

tors to approximate the Riesz space fractional derivative, we obtain the second order

approximation of the RSFADE. Furthermore, we propose the finite difference method

for the RSFADE and obtain the Crank-Nicolson scheme. Moreover, we prove that the

Crank-Nicolson scheme is unconditionally stable and convergent with the accuracy of

O(τ2 +h2) and improve the convergence order to O(τ4 +h4) by applying the Richardson

extrapolation method.

The outline of this chapter is as follows. In Section 2.2, the WSGD operators and some

lemmas are given. In Section 2.3, we first present the finite difference method for the

RSFADE, and then derive the Crank-Nicolson scheme. We proceed with the proof of

the stability and convergence of the Crank-Nicolson scheme in Section 2.4. Besides, we

further improve the convergence order by applying the Richardson extrapolation method.

In order to verify the effectiveness of our theoretical analysis, some numerical examples are

carried out and the results are compared with the exact solution in Section 2.5. Finally,

the conclusions are drawn.

2.2 The approximation for the Riemann-Liouville fractional derivative

First, in the interval [a, b], we take the mesh points xi = a + ih, i = 0, 1, · · · ,m, and

tn = nτ , n = 0, 1, · · · , N , where h = (b − a)/M , τ = T/N , i.e., h and τ are the uniform

spatial step size and temporal step size. Now, we give the definition of the Riemann-

Liouville fractional derivative.

Definition 2.2.1 The γ (n−1 < γ < n) order left and right Riemann-Liouville fractional

derivatives of the function v(x) on [a, b], are given by [203]

• left Riemann-Liouville fractional derivative:

aDγxv(x) =

1

Γ(n− γ)

dn

dxn

∫ x

a(x− ξ)n−γ−1v(ξ) dξ,

• right Riemann-Liouville fractional derivative:

xDγb v(x) =

(−1)n

Γ(n− γ)

dn

dxn

∫ b

x(ξ − x)n−γ−1v(ξ) dξ.

Generally, the standard Grunwald-Letnikov difference formula is applied to approximate

the Riemann-Liouville fractional derivative. Meerschaert and Tadjeran [181] showed that

Chapter 2 26

the standard Grunwald-Letnikov difference formula was often unstable for time dependent

problems and they proposed the the shifted Grunwald difference operators

Aγh,pv(x) =1

hγ

∞∑k=0

g(γ)k v(x− (k − p)h),

Bγh,qv(x) =

1

hγ

∞∑k=0

g(γ)k v(x+ (k − q)h),

whose accuracies are first order, i.e.,

Aγh,pv(x) = −∞Dγxv(x) +O(h),

Bγh,qv(x) = xD

γ+∞v(x) +O(h),

where p, q are integers and g(γ)k = (−1)k

(γk

). In fact, the coefficients g

(γ)k are the coefficients

of the power series of the function (1− z)γ ,

(1− z)γ =∞∑k=0

(−1)k(γ

k

)zk =

∞∑k=0

g(γ)k zk,

for all |z| ≤ 1, and they can be evaluated recursively

g(γ)0 = 1, g

(γ)k = (1− γ + 1

k)g

(γ)k−1, k = 1, 2, · · ·

Lemma 2.2.1 Suppose that 0 < α < 1, then the coefficients g(α)k satisfy [149]

g(α)0 = 1, g

(α)1 = −α < 0, g

(α)2 = α(α−1)

2 < 0,

g(α)1 < g

(α)2 < g

(α)3 < · · · < 0,

∞∑k=0

g(α)k = 0,

m∑k=0

g(α)k > 0, m ≥ 1.

Lemma 2.2.2 Suppose that 1 < β ≤ 2, then the coefficients g(β)k satisfy [149, 241]

g(β)0 = 1, g

(β)1 = −β < 0, g

(β)2 = β(β−1)

2 > 0,

1 ≥ g(β)2 ≥ g(β)

3 ≥ · · · ≥ 0,∞∑k=0

g(β)k = 0,

m∑k=0

g(β)k < 0, m ≥ 1.

Inspired by the shifted Grunwald difference operators and multi-step method, Tian et al.

[241] derive the WSGD operators:

LDγh,p,qv(x) =

γ − 2q

2(p− q)Aγh,pv(x) +

2p− γ2(p− q)

Aγh,qv(x),

RDγh,p,qv(x) =

γ − 2q

2(p− q)Bγh,pv(x) +

2p− γ2(p− q)

Bγh,qv(x).

27 Chapter 2

Lemma 2.2.3 Supposing that 1 < γ < 2, let v(x) ∈ L1(R), −∞Dγxv(x) and xD

γ+∞v(x)

and their Fourier transforms belong to L1(R), then the WSGD operators satisfy [241]

LDγh,p,qv(x) = −∞D

γxv(x) +O(h2),

RDγh,p,qv(x) = xD

γ+∞v(x) +O(h2),

uniformly for x ∈ R, where p, q are integers and p 6= q.

Tian et al. prove Lemma 2.2.3 under the additional conditions that 1 < γ < 2. In fact,

their proof also holds for the 0 < γ < 1 case. The proof proceeds the same as [241], hence

we will not repeat it here.

Remark 2.2.1 Considering a well defined function v(x) on the bounded interval [a, b], if

v(a) = 0 or v(b) = 0, the function v(x) can be zero extended for x < a or x > b. And

then the γ order left and right Riemann-Liouville fractional derivatives of v(x) at each

point x can be approximated by the WSGD operators with second order accuracy

aDγxv(x) =

λ1

hγ

[x−ah

]+p∑k=0

g(γ)k v(x− (k − p)h) +

λ2

hγ

[x−ah

]+q∑k=0

g(γ)k v(x− (k − q)h) +O(h2),

xDγb v(x) =

λ1

hγ

[ b−xh

]+p∑k=0

g(γ)k v(x+ (k − p)h) +

λ2

hγ

[ b−xh

]+q∑k=0

g(γ)k v(x+ (k − q)h) +O(h2),

where λ1 = γ−2q2(p−q) and λ2 = 2p−γ

2(p−q) .

When (p, q) = (1, 0), 0 < α < 1 and 1 < β ≤ 2, the discrete approximations for the

Riemann-Liouville fractional derivatives on the domain [0, L] are

0Dαxv(xi) =

1

hα

i+1∑k=0

w(α)k v(xi−k+1) +O(h2), (2.4)

xDαLv(xi) =

1

hα

m−i+1∑k=0

w(α)k v(xi+k−1) +O(h2), (2.5)

0Dβxv(xi) =

1

hβ

i+1∑k=0

w(β)k v(xi−k+1) +O(h2), (2.6)

xDβLv(xi) =

1

hβ

m−i+1∑k=0

w(β)k v(xi+k−1) +O(h2), (2.7)

where

w(α)0 =

α

2g

(α)0 , w

(α)k =

α

2g

(α)k +

2− α2

g(α)k−1, k ≥ 1, (2.8)

w(β)0 =

β

2g

(β)0 , w

(β)k =

β

2g

(β)k +

2− β2

g(β)k−1, k ≥ 1. (2.9)

Chapter 2 28

Now, we discuss the properties of the coefficients w(α)k and w

(β)k .

Lemma 2.2.4 Suppose that 0 < α < 1, then the coefficients w(α)k satisfy

w(α)0 = α

2 > 0, w(α)1 = 2−α−α2

2 > 0, w(α)2 = α(α2+α−4)

4 < 0,

w(α)2 < w

(α)3 < w

(α)4 < · · · < 0,

∞∑k=0

w(α)k = 0,

m∑k=0

w(α)k > 0, m ≥ 1.

Proof. Combining the definition of w(α)k and the property of g

(α)k , it is easy to derive

the value of w(α)0 , w

(α)1 and w

(α)2 . When k ≥ 2, by the definition of w

(α)k

w(α)k =

α

2g

(α)k +

2− α2

g(α)k−1,

we have w(α)k < 0 as g

(α)k < 0 for k ≥ 1 and 0 < α < 1.

Moreover,

w(α)k+1 − w

(α)k =

α

2(g

(α)k+1 − g

(α)k ) +

2− α2

(g(α)k − g(α)

k−1).

Recalling Lemma 2.2.1, g(α)k < g

(α)k+1 when k ≥ 1, we obtain when k ≥ 2

w(α)k+1 − w

(α)k > 0,

i.e.,

w(α)k < w

(α)k+1.

For the sum∞∑k=0

w(α)k , we have

∞∑k=0

w(α)k = w

(α)0 +

∞∑k=1

w(α)k =

α

2g

(α)0 +

∞∑k=1

(α2g

(α)k +

2− α2

g(α)k−1

)=α

2

∞∑k=0

g(α)k +

2− α2

∞∑k=0

g(α)k =

∞∑k=0

g(α)k = 0.

Since w(α)k < 0 when k ≥ 2,

m∑k=0

w(α)k > 0 for m ≥ 1.

Lemma 2.2.5 Suppose that 1 < β ≤ 2, then the coefficients w(β)k satisfy [241]

w(β)0 = β

2 > 0, w(β)1 = 2−β−β2

2 < 0, w(β)2 = β(β2+β−4)

4 ,

1 ≥ w(β)0 ≥ w(β)

3 ≥ w(β)4 ≥ · · · ≥ 0,

∞∑k=0

w(β)k = 0,

m∑k=0

w(β)k < 0, m ≥ 2.

29 Chapter 2

2.3 The finite difference method for the RSFADE

In this section, we utilize the Eqs.(2.4)-(2.7) to approximate the Riesz space fractional

derivative and derive the Crank-Nicolson scheme of the equation. We define tn = nτ ,

n = 0, 1, · · · , N , let Ω = [0, L] be a finite domain, setting Sh be a uniform partition of Ω,

which is given by xi = ih for i = 0, 1, · · · ,m, where τ = T/N and h = L/m are the time

and space steps, respectively. First, we present the semi-discrete form of Eq.(2.1),

u(xi, tn)− u(xi, tn−1)

τ− 1

2

Kα

∂αu(xi, tn)

∂|x|α+Kβ

∂βu(xi, tn)

∂|x|β

=

1

2

Kα

∂αu(xi, tn−1)

∂|x|α+Kβ

∂βu(xi, tn−1)

∂|x|β

(2.10)

Let uni be the approximation solution of u(xi, tn). Substituting (2.4)-(2.7) into (2.10), we

obtain

uni + µα

[ i+1∑k=0

w(α)k uni−k+1 +

m−i+1∑k=0

w(α)k uni+k−1

]

+µβ

[ i+1∑k=0

w(β)k uni−k+1 +

m−i+1∑k=0

w(β)k uni+k−1

]

=un−1i − µα

[ i+1∑k=0

w(α)k un−1

i−k+1 +

m−i+1∑k=0

w(α)k un−1

i+k−1

]

−µβ[ i+1∑k=0

w(β)k un−1

i−k+1 +m−i+1∑k=0

w(β)k un−1

i+k−1

], (2.11)

where µα = τKαcα2hα and µβ =

τKβcβ2hβ

. Denote

A =

w(α)1 w

(α)0 0 · · · 0 0

w(α)2 w

(α)1 w

(α)0 · · · 0 0

w(α)3 w

(α)2 w

(α)1 · · · 0 0

......

.... . .

......

w(α)m−2 w

(α)m−3 w

(α)m−4 · · · w

(α)1 w

(α)0

w(α)m−1 w

(α)m−2 w

(α)m−3 · · · w

(α)2 w

(α)1

, (2.12)

B =

w(β)1 w

(β)0 0 · · · 0 0

w(β)2 w

(β)1 w

(β)0 · · · 0 0

w(β)3 w

(β)2 w

(β)1 · · · 0 0

......

.... . .

......

w(β)m−2 w

(β)m−3 w

(β)m−4 · · · w

(β)1 w

(β)0

w(β)m−1 w

(β)m−2 w

(β)m−3 · · · w

(β)2 w

(β)1

, (2.13)

Chapter 2 30

and Un = [un1 , un2 , · · · , unm−1]T ,

D = µα(A+AT ) + µβ(B +BT ). (2.14)

Thus, Eq.(2.11) can be simplified as

(I +D)Un = (I −D)Un−1. (2.15)

The boundary and initial conditions are discretized as

u0i = ψ(ih), U0 = [u0

1, u02, · · · , u0

m−1]T ,

where i = 1, 2, · · · ,m− 1.

2.4 Theoretical analysis of the finite difference method

2.4.1 Stability

Before giving the proof, we start with some useful lemmas.

Lemma 2.4.1 Let A be an m − 1 order positive define matrix, then for any parameter

θ ≥ 0, the following two inequalities

||(I + θA )−1|| ≤ 1 and ||(I + θA )−1(I − θA )|| ≤ 1

hold [282].

Now, we discuss the property of matrix D.

Theorem 2.4.1 Suppose that 0 < α < 1 and 1 < β ≤ 2, A, B and D are defined as

(2.12)-(2.14), then the coefficients Dij satisfy

|Dii| >m−1∑

j=1,j 6=i|Dij |, i = 1, 2, · · · ,m− 1,

i.e., D is strictly diagonally dominant.

Proof. It is easy to obtain

Dij =

µαw(α)j−i+1 + µβw

(β)j−i+1, j > i+ 1,

µα(w(α)0 + w

(α)2 ) + µβ(w

(β)0 + w

(β)2 ), j = i+ 1,

2µαw(α)1 + 2µβw

(β)1 , j = i,

µα(w(α)0 + w

(α)2 ) + µβ(w

(β)0 + w

(β)2 ), j = i− 1,

µαw(α)i−j+1 + µβw

(β)i−j+1, j < i− 1,

31 Chapter 2

where µα = τKαcα2hα > 0 and µβ =

τKβcβ2hβ

< 0. First, we consider the signs of Dij .

According to Lemma 2.2.4, when k ≥ 3, w(α)k < 0, thus µαw

(α)k < 0. According to Lemma

2.2.5, when k ≥ 3, w(β)k > 0, thus µβw

(β)k < 0. Therefore, Dij < 0 when j > i + 1 or

j < i− 1. For the items Di,i+1 and Di,i−1, we have

w(α)0 + w

(α)2 =

α

2+α(α2 + α− 4)

4=α(α+ 2)(α− 1)

4< 0,

w(β)0 + w

(β)2 =

β

2+β(β2 + β − 4)

4=β(β + 2)(β − 1)

4> 0.

Since µα > 0 and µβ < 0, then

Di,i+1 = Di,i−1 = µα(w(α)0 + w

(α)2 ) + µβ(w

(β)0 + w

(β)2 ) < 0.

According to Lemma 2.2.4 and Lemma 2.2.5, we have w(α)1 > 0 and w

(β)1 < 0, thus

Di,i = 2µαw(α)1 + 2µβw

(β)1 > 0,

as µα > 0 and µβ < 0. Now, for a given i, we consider the sum

m−1∑j=1,j 6=i

|Dij | =i−2∑j=1

|Dij |+m−1∑j=i+2

|Dij |+ |Di,i−1|+ |Di,i+1|

=−i−2∑j=1

(µαw(α)i−j+1 + µβw

(β)i−j+1)−

m−1∑j=i+2

(µαw(α)j−i+1 + µβw

(β)j−i+1)

−2µα(w(α)0 + w

(α)2 )− 2µβ(w

(β)0 + w

(β)2 )

<

i−2∑j=−∞

(µαw(α)i−j+1 + µβw

(β)i−j+1)−

+∞∑j=i+2

(µαw(α)j−i+1 + µβw

(β)j−i+1)

−2µα(w(α)0 + w

(α)2 )− 2µβ(w

(β)0 + w

(β)2 )

=− 2µα

+∞∑k=3

w(α)k − 2µβ

+∞∑k=3

w(β)k − 2µα(w

(α)0 + w

(α)2 )− 2µβ(w

(β)0 + w

(β)2 )

=2µαw(α)1 + 2µβw

(β)1 − 2µα

+∞∑k=0

w(α)k − 2µβ

+∞∑k=0

w(β)k

=2µαw(α)1 + 2µβw

(β)1 = |Di,i|

i.e.,m−1∑

j=1,j 6=i|Dij | < |Dii|.

Thus, the proof is completed.

According to the theorem, it is easy to conclude the following corollaries.

Corollary 2.4.1 The matrix I + D is strictly diagonally dominant as well. Therefore,

I +D is invertible and Eq.(2.15) is solvable.

Chapter 2 32

Corollary 2.4.2 The matrix D is symmetric positive definite.

Proof. In view of (2.14), the symmetry of D is evident. Let λ0 be one eigenvalue of the

matrix D. Then by the Gerschgorin’s circle theorem [109], we have

|λ0 −Dii| ≤ ri =m−1∑

j=1,j 6=i|Dij |

i.e.,

Dii −m−1∑

j=1,j 6=i|Dij | ≤ λ0 ≤ Dii +

m−1∑j=1,j 6=i

|Dij |.

In view of Theorem 2.4.1, we have λ0 > 0, thus D is positive definite, which completes

the proof.

Since D is symmetric positive definite, by Lemma 2.4.1, the following two inequalities

hold.

Corollary 2.4.3

||(I +D)−1|| ≤ 1, and ||(I +D)−1(I −D)|| ≤ 1.

Theorem 2.4.2 The difference scheme (2.15) is unconditionally stable.

Proof. Let Un and un be the numerical and exact solution vectors respectively and

un = [u(x1, tn), u(x2, tn), · · · , u(xm−1, tn)]T . Since the matrix (I + D) is invertible, then

we can obtain the following error equation

E n = ME n−1, (2.16)

where E n = Un − un and M = (I +D)−1(I −D). By (2.16) we have

E n = Mn−1E 0,

Applying Corollary 2.4.3, we obtain

||E n|| ≤ ||Mn−1|| ||E 0|| ≤ ||M ||n−1||E 0|| ≤ ||E 0||,

which means difference scheme (2.15) is unconditionally stable.

2.4.2 Convergence

In the following, we suppose the symbol C is a generic positive constant, which may

take different values at different places. First we give the local truncation error of the

33 Chapter 2

Crank-Nicolson scheme. It is easy to conclude that,

u(xi, tn)− u(xi, tn−1)

τ=

(∂u(x, t)

∂t

)n−1/2

i

+O(τ2).

(Kα

∂αu(x, t)

∂|x|α+Kβ

∂βu(x, t)

∂|x|β

)n−1/2

i

=1

2

(Kα

∂αu(xi, tn)

∂|x|α+Kβ

∂βu(xi, tn)

∂|x|β

)+

1

2

(Kα

∂αu(xi, tn−1)

∂|x|α+Kβ

∂βu(xi, tn−1)

∂|x|β

)+O(τ2).

From Eqs.(2.4)-(2.7), we have

Kα∂αu(xi, tn)

∂|x|α= µα

[ i+1∑k=0

w(α)k uni−k+1 +

m−i+1∑k=0

w(α)k uni+k−1

]+O(h2),

Kβ∂βu(xi, tn)

∂|x|β= µβ

[ i+1∑k=0

w(β)k uni−k+1 +

m−i+1∑k=0

w(β)k uni+k−1

]+O(h2).

Therefore, the local truncation of (2.11) gives

Rni = O(τ3 + τh2).

Theorem 2.4.3 The numerical solution Un unconditionally converges to the exact solu-

tion un as h and τ tend to zero, and

||Un − un|| ≤ C(τ2 + h2).

Proof. Let eni denote the error at grid points (xi, tn). Substituting u(xi, tn) = Uni − eniinto Eq.(2.11) and combining Eqs.(2.4)-(2.7) yields

eni + µα

[ i+1∑k=0

w(α)k eni−k+1 +

m−i+1∑k=0

w(α)k eni+k−1

]

+µβ

[ i+1∑k=0

w(β)k eni−k+1 +

m−i+1∑k=0

w(β)k eni+k−1

]

=en−1i − µα

[ i+1∑k=0

w(α)k en−1

i−k+1 +

m−i+1∑k=0

w(α)k en−1

i+k−1

]

−µβ[ i+1∑k=0

w(β)k en−1

i−k+1 +m−i+1∑k=0

w(β)k en−1

i+k−1

]+O(τ3 + τh2).

Chapter 2 34

Using the conditions (2.2) and (2.3), we obtain the errors e0i = 0 and en0 = enm = 0 for

i = 1, 2, · · · ,m− 1 and j = 0, 1, · · · , N . We can write the system in matrix-vector form

(I +D)En = (I −D)En−1 +O(τ3 + τh2)χ

or

En = MEn−1 + b,

where χ = [1, 1, · · · , 1]T , En = [en1 , en2 , · · · , enm−1]T , D = µα(A + AT ) + µβ(B + BT ),

M = (I + D)−1(I − D) and b = O(τ3 + τh2)(I + D)−1. By iterating and noting that

E0 = 0, we obtain

En = (Mn−1 +Mn−2 + · · ·+ I)b.

Now, from Corollary 2.4.3, we have ||M || < 1 and ||(I + D)−1|| < 1. Then upon taking

norms,

||En|| ≤ (||Mn−1||+ ||Mn−2||+ · · ·+ 1)||b||

≤ (1 + 1 + · · ·+ 1)||b||

≤ nO(τ3 + τh2) = TO(τ2 + h2).

Thus,

||En|| ≤ C(τ2 + h2),

which completes the proof.

2.4.3 Improving the convergence order

Here we use the Richardson extrapolation method (REM) to improve the convergence

order. Suppose that Ihf is the approximation solution of the function of f(x) with an

asymptotic expansion

f = Ihf + C1h2 +O(h3), h→ 0, C1 6= 0.

Then we have

f = Ih/2f + C1(h/2)2 +O(h3).

Eliminating the middle terms C1h2 on the right, we find

f =4Ih/2f − Ihf

3+O(h3), (2.17)

35 Chapter 2

which means the approximation order of f(x) has been improved from O(h2) to O(h3).

Repeatedly, we can improve the approximation order of f(x) from O(h3) to O(h4). Sup-

pose that Ghf is the approximation solution of the function of f(x) with an asymptotic

expansion

f = Ghf + C2h3 +O(h4), h→ 0, C2 6= 0.

Then we have

f = Gh/2f + C2(h/2)3 +O(h4).

Eliminating the middle terms C2h3 on the right, we find

f =8Gh/2f − Ghf

7+O(h4). (2.18)

According to Theorem 2.4.3, we know that the numerical method converges at the rate

of O(τ2 +h2). In order to improve the convergence order, we apply the REM on a coarse

grid τ = h and then on a finer grid of size τ/2 = h/2 and τ/4 = h/4. By applying the

Richardson extrapolation formulae (2.17) and (2.18) consecutively, the convergence order

can be improved from O(τ2 + h2) to O(τ4 + h4).

Remark 2.4.1 Compared to the general method, REM needs at least twice the calculation

of the numerical scheme, which will increase the CPU time. It is also more complex

because it involves the error calculation between the coarse grid and the fine grid.

2.5 Numerical examples

In order to demonstrate the effectiveness of numerical methods, some examples are pre-

sented.

Example 2.5.1 First, we consider the following RSFDE (Kα = 0 ) with a source term∂u(x,t)∂t = ∂βu(x,t)

∂|x|β+ f(x, t), 0 < x < 1, 0 < t ≤ T,

u(x, 0) = x2(1− x)2, 0 ≤ x ≤ 1,

u(0, t) = u(1, t) = 0, 0 ≤ t ≤ T,

where 1 < β ≤ 2,

f(x, t) = −x2(1− x)2e−t +e−t

2 cos βπ2

24

Γ(5− β)[x4−β + (1− x)4−β]

− 12

Γ(4− β)[x3−β + (1− x)3−β] +

2

Γ(3− β)[x2−β + (1− x)2−β]

,

and the exact solution is u(x, t) = x2(1− x)2e−t.

The related numerical results are given in Table 2.1. It shows the error and convergenceorder of the Crank-Nicolson scheme at t = 1 with τ = h, where β is corresponding to

Chapter 2 36

three distinct values: β = 1.2, β = 1.5, β = 1.8. It can be seen that the numerical resultsare in excellent agreement with the exact solution.

Table 2.1: The error and convergence order of the Crank-Nicolson scheme of the RSFDEfor different β at t = 1 with τ = h

τ = hβ = 1.2 β = 1.5 β = 1.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/8 1.6781E-03 – 1.8350E-03 – 1.7827E-03 –1/16 3.9728E-04 2.08 4.3608E-04 2.07 4.3250E-04 2.041/32 9.4291E-05 2.07 1.0349E-04 2.08 1.0482E-04 2.041/64 2.2442E-05 2.07 2.4538E-05 2.08 2.5351E-05 2.051/128 5.3679E-06 2.06 5.8261E-06 2.07 6.1256E-06 2.05

Example 2.5.2 Then, we consider the following RSFADE with a source term:∂u(x,t)∂t = Kα

∂αu(x,t)∂|x|α +Kβ

∂βu(x,t)∂|x|β + f(x, t), 0 < x < 1, 0 < t ≤ T,

u(x, 0) = 0, 0 ≤ x ≤ 1,

u(0, t) = u(1, t) = 0, 0 ≤ t ≤ T,

where 0 < α < 1, 1 < β ≤ 2, l(x, p) = xp + (1− x)p and

f(x, t) =Kαt

βeαt

2 cos(απ/2)

Γ(7)l(x, 6− α)

Γ(7− α)− 6Γ(8)l(x, 7− α)

Γ(8− α)+

15Γ(9)l(x, 8− α)

Γ(9− α)

− 20Γ(10)l(x, 9− α)

Γ(10− α)+

15Γ(11)l(x, 10− α)

Γ(11− α)− 6Γ(12)l(x, 11− α)

Γ(12− α)

+Γ(13)l(x, 12− α)

Γ(13− α)

+

Kβtβeαt

2 cos(βπ/2)

Γ(7)l(x, 6− β)

Γ(7− β)− 6Γ(8)l(x, 7− β)

Γ(8− β)

+15Γ(9)l(x, 8− β)

Γ(9− β)− 20Γ(10)l(x, 9− β)

Γ(10− β)+

15Γ(11)l(x, 10− β)

Γ(11− β)

− 6Γ(12)l(x, 11− β)

Γ(12− β)+

Γ(13)l(x, 12− β)

Γ(13− β)

+ tβ−1eαt(β + αt)x6(1− x)6

and the exact solution is u(x, t) = tβeαtx6(1− x)6.

Here, we take Kα = Kβ = 2, α = 0.1, 0.5, 0.9 and β = 1.2, 1.5, 1.8. The error and

convergence order of the Crank-Nicolson scheme of the RSFADE at t = 1 with τ = h is

shown in Table 2.2. As expected, the convergence order can be reached second order in

both time and space direction. By applying the REM (2.17) and (2.18) consecutively, we

get the improved error and convergence order which is shown in Table 2.3. It can be seen

that the convergence order has been improved from second order to fourth order.

37 Chapter 2

Table 2.2: The error and convergence order of the Crank-Nicolson scheme of the RSFADEfor different α and β at t = 1 with τ = h

α = 0.1, τ = hβ = 1.2 β = 1.5 β = 1.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/8 2.8322E-05 – 2.8689E-05 – 2.4212E-05 –1/16 7.1828E-06 1.98 7.0817E-06 2.02 5.7905E-06 2.061/32 1.8334E-06 1.97 1.7860E-06 1.99 1.4386E-06 2.011/64 4.6575E-07 1.98 4.5086E-07 1.99 3.5998E-07 2.001/128 1.1756E-07 1.99 1.1343E-07 1.99 9.0132E-08 2.00

α = 0.5, τ = hβ = 1.2 β = 1.5 β = 1.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/8 4.2703E-05 – 4.3144E-05 – 3.6635E-05 –1/16 1.0824E-05 1.98 1.0665E-05 2.02 8.7850E-06 2.061/32 2.7622E-06 1.97 2.6913E-06 1.99 2.1851E-06 2.011/64 7.0166E-07 1.98 6.7960E-07 1.99 5.4714E-07 2.001/128 1.7712E-07 1.99 1.7101E-07 1.99 1.3704E-07 2.00

α = 0.9, τ = hβ = 1.2 β = 1.5 β = 1.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/8 6.6633E-05 – 6.6253E-05 – 5.6545E-05 –1/16 1.6852E-05 1.98 1.6414E-05 2.01 1.3633E-05 2.051/32 4.3006E-06 1.97 4.1491E-06 1.98 3.4009E-06 2.001/64 1.0929E-06 1.98 1.0488E-06 1.98 8.5302E-07 2.001/128 2.7596E-07 1.99 2.6409E-07 1.99 2.1385E-07 2.00

Table 2.3: The error and convergence order of the RSFADE by applying the REM

β = 1.8, τ = hα = 0.1 α = 0.5 α = 0.9

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/8 2.5866E-08 – 3.9130E-08 – 6.2601E-08 –1/16 1.7240E-09 3.91 2.6351E-09 3.89 4.3563E-09 3.851/32 1.1610E-10 3.89 1.7772E-10 3.89 2.9783E-10 3.871/64 7.6467E-12 3.92 1.1768E-11 3.92 1.9906E-11 3.90

Chapter 2 38

Example 2.5.3 Now, we consider the following RSFADE:∂u(x,t)∂t = Kα

∂αu(x,t)∂|x|α +Kβ

∂βu(x,t)

∂|x|β, 0 < x < π, 0 < t ≤ T,

u(x, 0) = x2(π − x), 0 ≤ x ≤ π,

u(0, t) = u(π, t) = 0, 0 ≤ t ≤ T,

where 0 < α < 1, 1 < β ≤ 2. According to [259], the analytical solution is given by

u(x, t) =

∞∑n=1

[ 8

n3(−1)n+1 − 4

n3

]sin(nx) exp

(− [Kα(n2)α/2 +Kβ(n2)β/2]t

).

Here, we take Kα = Kβ = 0.15. In Figure 2.1, we present the comparison of the numerical

solution with the analytical solution at T = 0.4 with fixed α = 0.4, β = 1.8, h =

τ = 1/100. In Figure 2.2, we present the behavior of the RSFADE for different α =

0.1, 0.3, 0.5, 0.7, 0.9 at T = 10.0 with fixed β = 1.7, h = τ = 1/100. In Figure 2.3, we

present the behavior of the RSFADE for different β = 1.2, 1.4, 1.6, 1.8, 2.0 at T = 10.0

with fixed α = 0.3, h = τ = 1/100. In Figure 2.4, we present the behavior of the RSFADE

at different time T = 1.0, 2.0, 4.0, 8.0 with fixed α = 0.4, β = 1.6, h = τ = 1/100.

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

u(x,

t)

NumericalAnalytical

Figure 2.1: The comparison of the numerical solution and analytical solution at T = 0.4with fixed α = 0.4, β = 1.8, h = τ = 1/100

39 Chapter 2

0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

u(x,

t)

α=0.1α=0.3α=0.5α=0.7α=0.9

Figure 2.2: The numerical approximation of u(x, t) for α = 0.1, 0.3, 0.5, 0.7, 0.9 atT = 10.0 with fixed β = 1.7, h = τ = 1/100

0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

u(x,

t)

β=1.2β=1.4β=1.6β=1.8β=2.0

Figure 2.3: The numerical approximation of u(x, t) for β = 1.2, 1.4, 1.6, 1.8, 2.0 atT = 10.0 with fixed α = 0.3, h = τ = 1/100

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

x

u(x,

t)

t=1.0t=2.0t=4.0t=8.0

Figure 2.4: The numerical approximation of u(x, t) for α = 0.4, β = 1.6 at t = 1.0, 2.0,4.0, 8.0 with h = τ = 1/100

Chapter 2 40

2.6 Conclusions

In this chapter, we developed and demonstrated a second-order finite difference method

for solving a class of the RSFADE. Firstly, we extended the WSGD operators to ap-

proximate the Riesz fractional derivative of order α ∈ (0, 1) and β ∈ (1, 2], respectively.

Subsequently, we discussed the Crank-Nicolson scheme of the RSFADE and proved that

the Crank-Nicolson scheme is unconditionally stable and convergent with the accuracy

of O(τ2 + h2). Furthermore, we applied the REM to improve the convergence order to

O(τ4+h4). Finally, some numerical results for the fractional finite difference method were

given to show the stability, consistency, and convergence of our computational approach.

This technique can be extended to two-dimensional or three-dimensional RSFADE prob-

lems.

Chapter 3

A fast second-order accurate method for a two-sided space-fractional

diffusion equation with variable coefficients

3.1 Introduction

In the past decades, fractional derivatives have been successfully applied in physics [230],

biology [174], chemistry [274], hydrology [14, 15, 148], and finance [220]. Some theory

and applications of fractional differential equations were introduced in [60, 128, 166, 186,

203, 218]. Considerable numerical methods for solving the FDE have been proposed.

Meerschaert and Tadjeran [181] developed practical numerical methods to solve the one-

dimensional space FDE with variable coefficients on a finite domain. Liu et al. [148]

transformed the space fractional Fokker-Planck equation into a system of ordinary differ-

ential equations (method of lines), which was then solved using backward differentiation

formulas. Liu et al. [149] also discussed the stability and convergence of the difference

methods for the space-time fractional advection-diffusion equation. Momani and Odi-

bat [192] developed two reliable algorithms, the Adomian decomposition method, and

variational iteration method, to construct numerical solutions of the space-time FDE in

the form of a rapidly convergent series with easily computable components. Zhuang et

al. [300] discussed a variable-order fractional advection-diffusion equation with a non-

linear source term on a finite domain. In addition, high-other finite difference methods

[45, 62, 78, 213, 267], finite element method [68, 79, 166, 290], finite volume method

[77, 100, 155] and spectral method [16, 145, 277, 297] are also employed to approximate

the FDE. In terms of computation, Wang et al. [248, 249] developed a fast finite differ-

ence method for FDE, which require less storage and computation cost while retaining

the same accuracy and approximation property. Wang et al. [250] also constructed a

fast conjugate gradient squared method (FCGS) by decomposing the coefficient matrix

into a combination of sparse and Toeplitz like matrices. Moroney and Yang [194] pre-

sented a fast Poisson preconditioner for a Jacobian-free Newton-Krylov method to solve

the nonlinear space-fractional diffusion equations. Chen et al. [48] derived a fast semi-

implicit difference method for a nonlinear two-sided space-fractional diffusion equation

with variable diffusivity coefficients.

However, less focuses are on the variable coefficients FDE in conservative form. The

diffusion coefficient is generally space or time dependent in practical problems. Hence,

we aim at deriving a more general fractional diffusion model with variable coefficients in

conservative form. According to the principle of conservation of mass, the equation of

41

Chapter 3 42

continuity in 1D form is given by

∂u(x, t)

∂t+∂q(x, t)

∂x= 0, (3.1)

in which u(x, t) is the distribution function of the diffusing quantity and q(x, t) denotes

the diffusion flux. Suppose that the flux term possesses the following form

q(x, t) = −(K1(x)

∂

∂x

∫ x

ak+(x, ξ)u(ξ, t) dξ +K2(x)

∂

∂x

∫ b

xk−(x, ξ)u(ξ, t) dξ

), (3.2)

on the closed interval [a, b], where k+(x, ξ), k−(x, ξ) are the kernel functions, K1(x) and

K2(x) are the nonnegative diffusion coefficients. We can regard Fick’s first law as a special

case of Eq.(3.2) by taking k+(x, ξ) = k−(x, ξ) = δ(x− ξ). We consider the most common

case k+(x, ξ) = (x−ξ)−αΓ(1−α) , for a ≤ ξ ≤ x,

k−(x, ξ) = (ξ−x)−α

Γ(1−α) , for x ≤ ξ ≤ b,

where 0 < α < 1. Combining Eqs.(3.1)-(3.2), we obtain a conservative form of the

two-sided space fractional diffusion equation with variable coefficients:

∂u(x, t)

∂t=

∂

∂x

(K1(x)aD

αxu(x, t)−K2(x)xD

αb u(x, t)

),

where the operators aDαx , xD

αb are the left and right Riemann-Liouville fractional deriva-

tives (see [203]) defined as

aDαxu(x, t) =

1

Γ(n− α)

dn

dxn

∫ x

a(x− ξ)n−α−1u(ξ, t) dξ,

xDαb u(x, t) =

(−1)n

Γ(n− α)

dn

dxn

∫ b

x(ξ − x)n−α−1u(ξ, t) dξ,

respectively, for n− 1 ≤ α < n, where n is an integer.

In this paper, we consider the following two-sided space FDE with variable coefficients.

∂u(x, t)

∂t=

∂

∂x

[K−(x)aD

αxu(x, t)−K+(x)xD

αb u(x, t)

]+ f(x, t), (3.3)

(x, t) ∈ (a, b)× (0, T ],

subject to the initial condition

u(x, 0) = ϕ(x), a ≤ x ≤ b, (3.4)

and the boundary conditions

u(a, t) = 0, u(b, t) = 0, 0 ≤ t ≤ T, (3.5)

43 Chapter 3

where 0 < α < 1, f(x, t) is a source term. The left and right Riemann-Liouville fractional

derivatives on the finite domain [a, b] are defined as

aDαxu(x, t) =

1

Γ(1− α)

∂

∂x

∫ x

a(x− ξ)−αu(ξ, t)dξ,

xDαb u(x, t) =

−1

Γ(1− α)

∂

∂x

∫ b

x(ξ − x)−αu(ξ, t)dξ.

In this paper, based on a second-order numerical scheme to approximate the Riemann-

Liouville fractional derivative, we obtain the second order approximation of the FDE.

Furthermore, we derive the Crank-Nicolson scheme and prove the scheme is uncondition-

ally stable and convergent with the accuracy of O(τ2 +h2). According to the structure of

the coefficient matrix, we decompose it and develop a fast bi-conjugate gradient stabilized

(FBi-CGSTAB) method to solve the Crank-Nicolson scheme.

The outline of this chapter is as follows. In Section 3.2, the second-order scheme and some

lemmas are given. In Section 3.3, we present the finite difference method for the FDE

and derive the Crank-Nicolson scheme. We proceed with the proof of the stability and

convergence of the Crank-Nicolson scheme in Section 3.4 and develop a FBi-CGSTAB

method in Section 3.5. In order to verify the effectiveness of our theoretical analysis,

several numerical examples are carried out and the results are compared with the exact

solution in Section 3.6. Finally, the conclusions are drawn.

3.2 Preliminary knowledge

First, in the interval [a, b], we take the mesh points xi = a + ih, i = 0, 1, · · · ,m, and

tn = nτ , n = 0, 1, · · · , N , where h = (b − a)/m, τ = T/N , i.e., h and τ are the uniform

spatial step size and temporal step size. In the following, we suppose the symbol C is a

generic positive constant, which may take different values at different places. Now, we

give some useful lemmas.

Lemma 3.2.1 If the function v(x) ∈ C2[a, b], then we have

v(η) =(xi+1 − η)v(xi) + (η − xi)v(xi+1)

h− 1

2(η − xi)(xi+1 − η)v′′(ς), (3.6)

where xi ≤ ς ≤ xi+1, i = 0, 1, · · · , m− 1.

Proof. By the Taylor expansion, we expand the v(xi) and v(xi+1) at x = η to obtain

v(xi) = v(η) + (xi − η)v′(η) +1

2(xi − η)2v′′(ςi1), (3.7)

v(xi+1) = v(η) + (xi+1 − η)v′(η) +1

2(xi+1 − η)2v′′(ςi2), (3.8)

Chapter 3 44

where xi ≤ ςi1 ≤ η ≤ ςi2 ≤ xi+1. From (3.7) and (3.8), we have

(xi+1 − η)v(xi) + (η − xi)v(xi+1)

h

=v(η) +1

2(η − xi)(xi+1 − η)

[v′′(ςi2)

xi+1 − ηh

+ v′′(ςi1)η − xih

].

Since v(x) ∈ C2[a, b], then there exists a ς ∈ [ςi1, ςi2] such that

v′′(ς) = v′′(ςi2)xi+1 − η

h+ v′′(ςi1)

η − xih

.

Thus, Eq.(3.6) holds.

Lemma 3.2.2 Suppose that 0 < α < 1, v(x) ∈ C2[a, b]. By the definition of Riemann-

Liouville fractional derivative, we have

aDα−1x v(xi) =

1

Γ(1− α)

∫ xi

a

v(η)

(xi − η)αdη

=1

Γ(1− α)

i−1∑k=0

∫ xk+1

xk

(xk+1−η)v(xk)+(η−xk)v(xk+1)h

(xi − η)αdη +R−i

=1

Γ(1− α)h

i−1∑k=0

[A−i,kv(xk) + B−i,kv(xk+1)

]+R−i ,

where

A−i,k =

∫ xk+1

xk

(xk+1 − η)

(xi − η)αdη

=

∫ xk+1

xk

(xk+1 − xi) + (xi − η)

(xi − η)αdη

=(i− k − 1)

1− α[(i− k − 1)1−α − (i− k)1−α] · h2−α

− 1

2− α[(i− k − 1)2−α − (i− k)2−α] · h2−α,

B−i,k =

∫ xk+1

xk

(η − xk)(xi − η)α

dη

=

∫ xk+1

xk

(η − xi) + (xi − xk)(xi − η)α

dη

=(i− k)

1− α[(i− k)1−α − (i− k − 1)1−α] · h2−α

+1

2− α[(i− k − 1)2−α − (i− k)2−α] · h2−α,

and

R−i = − 1

2Γ(1− α)

i−1∑k=0

∫ xk+1

xk

(η − xk)(xk+1 − η)v′′(ς−k )

(xi − η)αdη.

45 Chapter 3

It is easy to conclude that

|R−i | ≤h2

2Γ(1− α)maxx∈[a,b]

|v′′(x)|i−1∑k=0

∫ xk+1

xk

1

(xi − η)αdη ≤ Ch2.

Lemma 3.2.3 Suppose that 0 < α < 1, v(x) ∈ C2[a, b]. By the definition of Riemann-

Liouville fractional derivative, we have

xDα−1b v(xi) =

1

Γ(1− α)

∫ b

xi

v(η)

(η − xi)αdη

=1

Γ(1− α)

m−1∑k=i

∫ xk+1

xk

(xk+1−η)v(xk)+(η−xk)v(xk+1)h

(η − xi)αdη +R+

i

=1

Γ(1− α)h

m−1∑k=i

[A+i,kv(xk) + B+

i,kv(xk+1)]

+R+i ,

where

A+i,k =

∫ xk+1

xk

(xk+1 − η)

(η − xi)αdη

=

∫ xk+1

xk

(xk+1 − xi) + (xi − η)

(η − xi)αdη

=(k − i+ 1)

1− α[(k − i+ 1)1−α − (k − i)1−α] · h2−α

− 1

2− α[(k − i+ 1)2−α − (k − i)2−α] · h2−α,

B+i,k =

∫ xk+1

xk

(η − xk)(η − xi)α

dη

=

∫ xk+1

xk

(η − xi) + (xi − xk)(η − xi)α

dη

=(i− k)

1− α[(k − i+ 1)1−α − (k − i)1−α] · h2−α

+1

2− α[(k − i+ 1)2−α − (k − i)2−α] · h2−α,

and

R+i = − 1

2Γ(1− α)

m−1∑k=i

∫ xk+1

xk

(η − xk)(xk+1 − η)v′′(ς+k )

(η − xi)αdη.

It is easy to conclude that

|R+i | ≤

h2

2Γ(1− α)maxx∈[a,b]

|v′′(x)|m−1∑k=i

∫ xk+1

xk

1

(η − xi)αdη ≤ Ch2.

Lemma 3.2.4 If the function v(x) ∈ C3[a, b], then

v′(x) =v(x+ h

2 )− v(x− h2 )

h− h2

24v′′′(ξ),

Chapter 3 46

where x− h2 ≤ ξ ≤ x+ h

2 .

Remark 3.2.1 In Eq.(3.3), if K−(x) = βK(x), K+(x) = (1 − β)K(x), then Eq.(3.3)

can change into

∂u(x, t)

∂t=

∂

∂x

K(x)

[βaD

αxu(x, t)− (1− β)xD

αb u(x, t)

]+ f(x, t).

Remark 3.2.2 In Eq.(3.3), if K−(x) = K+(x) = − 12 cos((α+1)π/2) , then Eq.(3.3) can

change into

∂u(x, t)

∂t=∂1+αu(x, t)

∂|x|1+α+ f(x, t).

3.3 The finite difference method for the FDE

In this section, we utilize the finite difference method to approximate Eq.(3.3) and derive

the Crank-Nicolson scheme. First, we consider the discretization of ∂∂x

[K−(x)aD

αxu(x, t)

]and ∂

∂x

[K+(x)xD

αb u(x, t)

]at node (xi, tn). Applying Lemma 3.2.4, we have

∂

∂x

[K−(x)aD

αxu(x, t)

]∣∣∣(xi,tn)

=K−(xi+1/2)

h

[aD

αxu(x, tn)

]∣∣∣xi+1/2

−K−(xi−1/2)

h

[aD

αxu(x, tn)

]∣∣∣xi−1/2

−h2

24

∂3

∂x3

[K−(x)aD

αxu(x, tn)

]∣∣∣x=ζ−i

, ∂

∂x

[K+(x)xD

αb u(x, t)

]∣∣∣(xi,tn)

=K+(xi+1/2)

h

[xD

αb u(x, tn)

]∣∣∣xi+1/2

−K+(xi−1/2)

h

[xD

αb u(x, tn)

]∣∣∣xi−1/2

−h2

24

∂3

∂x3

[K+(x)xD

αb u(x, tn)

]∣∣∣x=ζ+i

,

where xi−1/2 ≤ ζ−i , ζ+i ≤ xi+1/2. By the definition of the Riemann-Liouville fractional

derivative, we have

aDαxu(x, tn) =

∂

∂xaD

α−1x u(x, tn), xD

αb u(x, tn) = − ∂

∂xxD

α−1b u(x, tn).

47 Chapter 3

Then, we can obtain

∂

∂x

[K−(x)aD

αxu(x, t)

]∣∣∣(xi,tn)

=K−(xi+1/2)

h

[ ∂∂x

aDα−1x u(xi+1/2, tn)

]−K−(xi−1/2)

h

[ ∂∂x

aDα−1x u(xi−1/2, tn)

]+O(h2), (3.9) ∂

∂x

[K+(x)xD

αb u(x, t)

]∣∣∣(xi,tn)

= −K+(xi+1/2)

h

[ ∂∂x

xDα−1b u(xi+1/2, tn)

]+K+(xi−1/2)

h

[ ∂∂x

xDα−1b u(xi−1/2, tn)

]+O(h2). (3.10)

For the items ∂∂xaD

α−1x u(xi+1/2, tn), ∂

∂xaDα−1x u(xi−1/2, tn), ∂

∂xxDα−1b u(xi+1/2, tn) and

∂∂xxD

α−1b u(xi−1/2, tn), we use Lemma 3.2.4 again to get

∂

∂xaD

α−1x u(xi+1/2, tn) =

1

h[aD

α−1x u(xi+1, tn)− aD

α−1x u(xi, tn)] +O(h2),

∂

∂xaD

α−1x u(xi−1/2, tn) =

1

h[aD

α−1x u(xi, tn)− aD

α−1x u(xi−1, tn)] +O(h2),

∂

∂xxD

α−1b u(xi+1/2, tn) =

1

h[xD

α−1b u(xi+1, tn)− xD

α−1b u(xi, tn)] +O(h2),

∂

∂xxD

α−1b u(xi−1/2, tn) =

1

h[xD

α−1b u(xi, tn)− xD

α−1b u(xi−1, tn)] +O(h2).

Then Eq.(3.9) can be written as ∂

∂x

[K−(x)aD

αxu(x, t)

]∣∣∣(xi,tn)

=K−(xi+1/2)

h2

[aD

α−1x u(xi+1, tn)− aD

α−1x u(xi, tn)

]−K−(xi−1/2)

h2

[aD

α−1x u(xi, tn)− aD

α−1x u(xi−1, tn)

]+O(h2). (3.11)

Using Lemma 3.2.2, we have

I−1 , aDα−1x u(xi+1, tn)− aD

α−1x u(xi, tn)

=h1−α

Γ(3− α)

i−1∑k=1

u(xk, tn)Ci−k−1 + P−i,n +K−i,n, (3.12)

where

Ck = 3(k + 1)2−α + (k + 3)2−α − 3(k + 2)2−α − k2−α, k ≥ 0, (3.13)

P−i,n ,h1−α(22−α − 3)u(xi, tn)

Γ(3− α)+h1−αu(xi+1, tn)

Γ(3− α)(3.14)

and

|K−i,n| ≤ 2Ch2 maxx∈[a,b]

|uxx(x, tn)|.

Chapter 3 48

Similarly, we obtain

I−2 , aDα−1x u(xi, tn)− aD

α−1x u(xi−1, tn)

=h1−α

Γ(3− α)

i−2∑k=1

u(xk, tn)Ci−k−2 + P−i−1,n +K−i−1,n, (3.15)

and

|K−i−1,n| ≤ 2Ch2 maxx∈[a,b]

|uxx(x, tn)|.

In a similar fashion, Eq.(3.10) can be written as ∂

∂x

[K+(x)xD

αb u(x, t)

]∣∣∣(xi,tn)

=−K+(xi+1/2)

h2

[xD

α−1b u(xi+1, tn)− xD

α−1b u(xi, tn)

]+K+(xi−1/2)

h2

[xD

α−1b u(xi, tn)− xD

α−1b u(xi−1, tn)

]+O(h2). (3.16)

Using Lemma 3.2.3, we have

I+1 , xD

α−1b u(xi+1, tn)− xD

α−1b u(xi, tn)

= − h1−α

Γ(3− α)

m−1∑k=i+2

Ck−i−2u(xk, tn) + P+i,n +K+

i,n, (3.17)

where

P+i,n , −

h1−αu(xi, tn)

Γ(3− α)− h1−α(22−α − 3)u(xi+1, tn)

Γ(3− α)(3.18)

and

|K+i,n| ≤ 2Ch2 max

x∈[a,b]|uxx(x, tn)|.

Similarly, we obtain

I+2 , xD

α−1b u(xi, tn)− xD

α−1b u(xi−1, tn)

= − h1−α

Γ(3− α)

m−1∑k=i+2

Ck−i−1u(xk, tn) + P+i−1,n +K+

i−1,n, (3.19)

and

|K+i−1,n| ≤ 2Ch2 max

x∈[a,b]|uxx(x, tn)|.

49 Chapter 3

Combining Eqs.(3.11)-(3.15), we find ∂

∂x

[K−(x)aD

αxu(x, t)

]∣∣∣(xi,tn)

=

i−2∑k=1

V −i,ku(xk, tn) +G−i u(xi−1, tn) +B−i u(xi, tn) +D−i u(xi+1, tn) +O(h2),

where

V −i,k =1

h1+αΓ(3− α)

[K−i+1/2Ci−k−1 −K−i−1/2Ci−k−2

],

G−i =1

h1+αΓ(3− α)

[K−i+1/2C0 −K−i−1/2(22−α − 3)

],

B−i =1

h1+αΓ(3− α)

[K−i+1/2(22−α − 3)−K−i−1/2

],

D−i =1

h1+αΓ(3− α)K−i+1/2.

In a similar fashion, combining Eqs.(3.16)-(3.19), we obtain ∂

∂x

[K+(x)xD

αb u(x, t)

]∣∣∣(xi,tn)

=

m−1∑k=i+2

V +i,ku(xk, tn) +G+

i u(xi−1, tn) +B+i u(xi, tn) +D+

i u(xi+1, tn) +O(h2),

where

V +i,k =

1

h1+αΓ(3− α)

[K+i+1/2Ck−i−2 −K+

i−1/2Ck−i−1

],

G+i =

−1

h1+αΓ(3− α)K+i+1/2,

B+i =

1

h1+αΓ(3− α)

[K+i+1/2 −K

+i−1/2(22−α − 3)

],

D+i =

1

h1+αΓ(3− α)

[K+i+1/2(22−α − 3)−K+

i−1/2C0

].

Therefore,

∂

∂x

[K−(x)aD

αxu(x, t)

]∣∣∣(xi,tn)

− ∂

∂x

[K+(x)xD

αb u(x, t)

]∣∣∣(xi,tn)

=

i−2∑k=1

V −i,ku(xk, tn) +

m−1∑k=i+2

V +i,ku(xk, tn) +Giu(xi−1, tn) +Biu(xi, tn) +Diu(xi+1, tn),

where Gi = G−i −G+i , Bi = B−i −B

+i , Di = D−i −D

+i . Now we consider the discretization

of the time. It is easy to conclude that,(∂u(x, t)

∂t

)∣∣∣∣(xi,tn−1/2)

=u(xi, tn)− u(xi, tn−1)

τ+O(τ2),

Chapter 3 50

and ∂

∂x

[K−(x)aD

αxu(x, t)−K+(x)xD

αb u(x, t)

]+ f(x, t)

|(xi,tn−1/2)

=1

2

∂

∂x

[K−(x)aD

αxu(x, t)−K+(x)xD

αb u(x, t)

]+ f(x, t)

|(xi,tn)

+1

2

∂

∂x

[K−(x)aD

αxu(x, t)−K+(x)xD

αb u(x, t)

]+ f(x, t)

|(xi,tn−1) +O(τ2).

Therefore, the Crank-Nicolson scheme of the fractional diffusion Eq.(3.3) at (xi, tn−1/2)

gives

u(xi, tn)− u(xi, tn−1)

τ=

Ωi,n + Ωi,n−1

2+ f(xi, tn−1/2) +O(τ2 + h2), (3.20)

where

Ωi,n =i−2∑k=1

V −i,ku(xk, tn) +m−1∑k=i+2

V +i,ku(xk, tn) +Giu(xi−1, tn) +Biu(xi, tn) +Diu(xi+1, tn)

and

f(xi, tn−1/2) =1

2

[f(xi, tn) + f(xi, tn−1)

].

Let uni be the approximation solution of u(xi, tn) and fn−1/2i = f(xi, tn−1/2), then we

obtain the difference scheme of (3.20)

uni −τ

2Ωni = un−1

i +τ

2Ωn−1i + τf

n−1/2i . (3.21)

The boundary and initial conditions are discretized as

u0i = ϕ(xi), un0 = 0, unm = 0. (3.22)

Define Un = [ un1 , un2 , · · · , unm−1], Fn =

(fn−1/21 , f

n−1/22 , · · · , fn−1/2

m−2 , fn−1/2m−1

)T,

Q = −τ2

B1 D1 V +1,3 · · · V +

1,m−2 V +1,m−1

G2 B2 D2 · · · V +2,m−2 V +

2,m−1

V −3,1 G3 B3 · · · V +3,m−2 V +

3,m−1...

......

. . ....

...

V −m−2,1 V −m−2,2 V −m−2,3 · · · Bm−2 Dm−2

V −m−1,1 V −m−1,2 V −m−1,3 · · · Gm−1 Bm−1

,

then we can rewrite (3.21) in matrix form

(I +Q)Un = (I −Q)Un−1 + τFn. (3.23)

51 Chapter 3

3.4 Theoretical analysis of the finite difference method

3.4.1 Stability

Here, we consider the stability of the Crank-Nicolson scheme (3.23). Before giving the

proof, we start with some useful lemmas.

Lemma 3.4.1 Suppose that 1 < β1 < 2, we define the function p(x), x ∈ (0,+∞) as

p(x) = 2(x+ 1)β1 − xβ1 − (x+ 2)β1 ,

then

p ′(x) = β1

[2(x+ 1)β1−1 − xβ1−1 − (x+ 2)β1−1

]> 0, (3.24)

and

p ′′(x) = β1(β1 − 1)[2(x+ 1)β1−2 − xβ1−2 − (x+ 2)β1−2

]< 0. (3.25)

Proof. Let w(x) = (x+ 1)r − xr, x ∈ (0,+∞), r ∈ (−1, 0) ∪ (0, 1), and

g(x) = w(x)− w(x+ 1) = 2(x+ 1)r − xr − (x+ 2)r.

It is easy to obtain that

w′(x) = r[(x+ 1)r−1 − xr−1].

We can observe that, when 0 < r < 1, w′(x) < 0, which means w(x) is decreasing

monotonically with x increases, then w(x) > w(x + 1). Hence, w(x) − w(x + 1) > 0,

namely, g(x) > 0 for all x ∈ (0,+∞) when 0 < r < 1. Similarly, we can obtain g(x) < 0

for all x ∈ (0,+∞) when −1 < r < 0. Since 1 < β1 < 2, then β1 − 1 ∈ (0, 1) and

β1 − 2 ∈ (−1, 0). According to the discussion above, it is easy to obtain that Eq.(3.24)

and Eq.(3.25) hold.

Lemma 3.4.2 Assume that 0 < α < 1, we define C(k), k = 0, 1, 2, · · · as

C(k) = 3(k + 1)2−α + (k + 3)2−α − 3(k + 2)2−α − k2−α,

then

C(k) < 0, (3.26)

C(k) is increasing monotonically with k increases, i.e.

C(k) < C(k + 1), (3.27)

Chapter 3 52

and

limk→+∞

C(k) = 0. (3.28)

Proof. Since 0 < α < 1, it is easy to verify

C(0) = 3 + 32−α − 3 · 22−α < 0.

We define the function C(x), x ∈ (0,+∞) as

C(x) = 3(x+ 1)2−α + (x+ 3)2−α − 3(x+ 2)2−α − x2−α.

Let β1 = 2− α, it is easy to check that

C(x) = p(x)− p(x+ 1),

where p(x) is defined in Lemma 3.4.1. Thanks to Eq.(3.24), we obtain p(x)−p(x+1) < 0,

i.e., for all x ∈ (0,+∞)

C(x) < 0.

Hence, Eq.(3.26) holds. By using the Taylor expansion and Eq.(3.25), we have

p(x+ 2) = p(x+ 1) + p ′(x+ 1) +1

2p ′′(ξ) < p(x+ 1) + p ′(x+ 1), ξ ∈ (x+ 1, x+ 2),

p(x) = p(x+ 1)− p ′(x+ 1) +1

2p ′′(η) < p(x+ 1)− p ′(x+ 1), η ∈ (x, x+ 1).

To sum each side of the inequalities respectively, we obtain

p(x) + p(x+ 2) < 2p(x+ 1),

which means

p(x)− p(x+ 1) < p(x+ 1)− p(x+ 2),

i.e,

C(x) < C(x+ 1).

Thus, Eq.(3.27) holds. Now we consider the following limit, x > 0, 1 < β2 < 2

limx→+∞

(3(x+ 1)β2 + (x+ 3)β2 − 3(x+ 2)β2 − xβ2

)= limx→+∞

3(1 + 1x)β2 + (1 + 3

x)β2 − 3(1 + 2x)β2 − 1

( 1x)β2

= limt→0+

3(1 + t)β2 + (1 + 3t)β2 − 3(1 + 2t)β2 − 1

tβ2.

53 Chapter 3

By using the Taylor expansion, we obtain

3(1 + t)β2 + (1 + 3t)β2 − 3(1 + 2t)β2 − 1

=3[1 + β2t+β2(β2 − 1)

2t2 + o(t2)] + 1 + 3β2t+

β2(β2 − 1)

2(3t)2 + o(t2)

−3[1 + 2β2t+β2(β2 − 1)

2(2t)2 + o(t2)]− 1 = o(t2).

Then,

limx→+∞

(3(x+ 1)β2 + (x+ 3)β2 − 3(x+ 2)β2 − xβ2

)= limt→0+

o(t2)

tβ2= lim

t→0+

o(t2)

t2· t2−β2 = 0.

As 1 < 2− α < 2, therefore, Eq.(3.28) holds.

Corollary 3.4.1 It holds that

+∞∑k=0

C(k) = 2− 22−α.

Now we consider the property of matrix Q.

Theorem 3.4.1 Suppose that 0 < α < 1, K−(x) ≥ 0 is decreasing monotonically with

respect to x and K+(x) ≥ 0 is increasing monotonically with respect to x on [a, b]. When

6 + 32−α − 24−α ≥ 0 (or α ≥ 0.5546), the coefficients Qij satisfy

|Qii| >m−1∑

j=1,j 6=i|Qij |, i = 1, 2, · · · ,m− 1. (3.29)

i.e., Q is strictly diagonally dominant.

Proof. It is easy to obtain

Qij = k0

K+i−1/2Cj−i−1 −K+

i+1/2Cj−i−2, j > i+ 1,

K−i+1/2 −K+i+1/2(22−α − 3) +K+

i−1/2C0, j = i+ 1,

K−i+1/2(22−α − 3)−K−i−1/2 −K+i+1/2 +K+

i−1/2(22−α − 3), j = i,

K−i+1/2C0 −K−i−1/2(22−α − 3) +K+i−1/2, j = i− 1,

K−i+1/2Ci−j−1 −K−i−1/2Ci−j−2, j < i− 1,

(3.30)

where k0 = − τ2h1+αΓ(3−α)

< 0. As 0 < α < 1, then 22−α − 3 < 1, 0 < K−i+1/2 < K−i−1/2

and 0 < K+i−1/2 < K+

i+1/2, so

(22−α − 3)K−i+1/2 < K−i−1/2, (22−α − 3)K+i−1/2 < K+

i+1/2,

Chapter 3 54

hence, Qii > 0. For the item Qi,i−1,

K−i+1/2C0 −K−i−1/2(22−α − 3) = (K−i+1/2 −K−i−1/2)C0 +K−i−1/2(6 + 32−α − 24−α).

Since k0 < 0, K−i+1/2 − K−i−1/2 < 0, C0 < 0, K−i−1/2 > 0 and 6 + 32−α − 24−α ≥ 0,

K+i−1/2 > 0, then Qi,i−1 < 0. Similarly, we can obtain Qi,i+1 < 0. When j < i− 1,

Qij = k0

[K−i+1/2Ci−j−1 −K−i−1/2Ci−j−2

]= k0

[K−i+1/2(Ci−j−1 − Ci−j−2) + (K−i+1/2 −K

−i−1/2)Ci−j−2

].

According to Lemma 3.4.2, Ci−j−1 − Ci−j−2 > 0 and Ci−j−2 < 0, then it is easy to obtain

Qij < 0. Similarly, when j > i+ 1, we can derive Qij < 0 as well. Now, for a given i, we

consider the sum

m−1∑j=1,j 6=i

|Qij | =i−2∑j=1

|Qij |+m−1∑j=i+2

|Qij |+ |Qi,i−1|+ |Qi,i+1|

=− k0

i−2∑j=1

[K−i+1/2Ci−j−1 −K−i−1/2Ci−j−2

]+

m−1∑j=i+2

[K+i−1/2Cj−i−1 −K+

i+1/2Cj−i−2

]+K−i+1/2C0 −K−i−1/2(22−α − 3) +K+

i−1/2 +K−i+1/2 −K+i+1/2(22−α − 3) +K+

i−1/2C0

=− k0

K−i+1/2

i−2∑j=1

Cj −K−i−1/2

i−3∑j=0

Cj +K+i−1/2

m−i−2∑j=1

Cj −K+i+1/2

m−i−3∑j=0

Cj

+K−i+1/2[1 + C0]−K−i−1/2(22−α − 3) +K+i−1/2[1 + C0]−K+

i+1/2(22−α − 3)

=− k0

(K−i+1/2 −K

−i−1/2)

i−2∑j=0

Cj +K−i+1/2 +K−i−1/2[Ci−2 − 22−α + 3)]

+(K+i−1/2 −K

+i+1/2)

m−i−2∑j=0

Cj +K+i−1/2 +K+

i+1/2[Cm−i−2 − 22−α + 3)]

<− k0

(K−i+1/2 −K

−i−1/2)

+∞∑j=0

Cj +K−i+1/2 +K−i−1/2(3− 22−α)

+(K+i−1/2 −K

+i+1/2)

+∞∑j=0

Cj +K+i−1/2 +K+

i+1/2(3− 22−α)

=− k0

(K−i+1/2 −K

−i−1/2)(2− 22−α) +K−i+1/2 +K−i−1/2(3− 22−α)

+(K+i−1/2 −K

+i+1/2)(2− 22−α) +K+

i−1/2 +K+i+1/2(3− 22−α)

=− k0

K−i+1/2(22−α − 3)−K−i−1/2 −K

+i+1/2 +K+

i−1/2(22−α − 3)

=Qi,i = |Qi,i|

55 Chapter 3

i.e.,m−1∑

j=1,j 6=i|Qij | < |Qii|.

Thus, the proof is completed.

Remark 3.4.1 The condition 0.5546 ≤ α < 1 is only a sufficient condition to guarantee

that matrix Q is strictly diagonally dominant, it does not mean that when 0 < α < 0.5546,

matrix Q is not strictly diagonally dominant, which can be checked by the subsequent

numerical examples.

Remark 3.4.2 It is easy to verify that when K−(x) = K+(x) = Constant, for any

0 < α < 1, matrix Q is strictly diagonally dominant.

Corollary 3.4.2 Let λ = c+ d i be any eigenvalue of the matrix Q, then the real part of

λ satisfies

c > 0. (3.31)

Proof. By the Gershgorin’s circle theorem (see [129]), we have

(Qii − c)2 + (d i)2 ≤ r2i ,

therefore,

|Qii − c− d i| ≤ ri =m−1∑

j=1,j 6=i|Qij |.

Then

(Qii − c)2 + (d i)2 ≤ r2i ,

therefore,

|Qii − c| ≤ ri.

Using (3.29) and Qii > 0, it is easy to derive (3.31) must hold.

Corollary 3.4.3 The matrix I + Q is strictly diagonally dominant as well. Therefore,

I +Q is invertible and Eq.(3.23) is solvable.

Theorem 3.4.2 The difference scheme (3.23) is unconditionally stable.

Chapter 3 56

Proof. Since the eigenvalues λ of matrix Q satisfy Corollary 3.4.2, then the eigenvalues

of the matrix (I +Q)−1(I −Q) satisfy∣∣∣∣1− λ1 + λ

∣∣∣∣ =

∣∣∣∣1− c− d i

1 + c+ d i

∣∣∣∣ < 1.

Hence, the spectral radius of the matrix (I+Q)−1(I−Q) is less than one. Therefore, the

difference scheme (3.23) is unconditionally stable.

3.4.2 Convergence

By Eq.(3.20), we notice that the local truncation error of the Crank-Nicolson scheme

gives,

Rni = O(τ3 + τh2).

Theorem 3.4.3 Let un be the exact solution of the problem (3.3)-(3.5). Then the nu-

merical solution Un unconditionally converges to the exact solution un as h and τ tend

to zero, and

||Un − un|| ≤ C(τ2 + h2).

Proof. Let eni denote the error at grid points (xi, tn) and eni = Uni −u(xi, tn). Substituting

u(xi, tn) = Uni − eni into Eq.(3.20) and combining Eq.(3.21) yields

eni −τ

2Θni = en−1

i +τ

2Θn−1i +O(τ3 + τh2),

where

Θni =

i−2∑k=1

enkV−i,k +

m−1∑k=i+2

enkV+i,k +Gie

ni−1 +Bie

ni +Die

ni+1.

Using the conditions (3.4), (3.5) and (3.22), we obtain the errors e0i = 0 and en0 = enm = 0

for i = 1, 2, · · · ,m − 1 and n = 0, 1, · · · , N . We can write the system in matrix-vector

form

(I +Q)En = (I −Q)En−1 +O(τ3 + τh2)χ

or

En = MEn−1 + b

57 Chapter 3

where χ = [1, 1, · · · , 1]T , En = (en1 , en2 , · · · , enm−1)T , M = (I + Q)−1(I − Q) and b =

O(τ3 + τh2)(I +Q)−1. By iterating and noting that E0 = 0, we obtain

En = (Mn−1 +Mn−2 + · · ·+ I)b.

Now, from Corollary 3.4.2, Corollary 3.4.3 and Theorem 3.4.2, we have ρ((I +Q)−1) < 1

and ρ(M) < 1. Therefore, we can choose a vector norm and induced matrix norm || · ||such that ||M || < 1 and ||(I +Q)−1|| < 1. Then upon taking norms,

||En|| ≤ (||Mn−1||+ ||Mn−2||+ · · ·+ 1)||b||

≤ (1 + 1 + · · ·+ 1)||b||

≤ nO(τ3 + τh2) = TO(τ2 + h2).

Thus,

||En|| ≤ C(τ2 + h2),

which completes the proof.

3.5 A fast iterative algorithm

It is can be seen from Eq.(3.23) that the Crank-Nicolson scheme generates a full coefficient

matrix. Consequently, a direct solver for Eq.(3.23) (for example, Gauss elimination)

requires the computational effort of O(m3) per time step and storage of O(m2). In this

section, we develop a fast and effective iterative method for Eq.(3.23), which possesses a

significantly reduced storage requirement of O(m) and computational cost O(m logm),

while retaining the same accuracy as the Gauss elimination.

3.5.1 Efficient storage of matrix Q

Define q0 = 1, q1 = 22−α − 3, qi = Ci−2, i ≥ 2. Recalling Eq.(3.30), we can decompose

matrix Q as

Q = diag(KL1)QL1 + diag(KR1)QTL1− (diag(KL2)QL2 + diag(KR2)QTL2

),

where

KL1 = k0

[K−3/2,K

−5/2, · · · ,K

−m−1/2

]T, KR1 = k0

[K+

1/2,K+3/2, · · · ,K

+m−3/2

]T,

KL2 = k0

[K−1/2,K

−3/2, · · · ,K

−m−3/2

]T, KR2 = k0

[K+

3/2,K+5/2, · · · ,K

+m−1/2

]T,

Chapter 3 58

and

QL1 =

q1 q0 · · · 0

q2 q1. . .

......

. . .. . . q0

qm−1 qm−2 · · · q1

, QL2 =

q0 0 · · · 0

q1 q0. . .

......

. . .. . . 0

qm−2 qm−3 · · · q0

,both of which are Toeplitz matrices. Recall that a Toeplitz matrix is a matrix in which

each descending diagonal from left to right is constant. Thus, we only need to store q =

[q0, q1, · · · , qm−1]T , KL = k0

[K−1/2,K

−3/2, · · · ,K

−m−1/2

]Tand KR = k0

[K+

1/2,K+3/2, · · · ,

K+m−1/2

]T, which have 3×m parameters instead of the (m− 1)2 parameters. Therefore,

the total memory requirement for Q has significantly reduced from O(m2) to O(m).

3.5.2 A fast bi-conjugate gradient stabilized method

Generally, an iterative method is preferred to solve large linear systems as iterative meth-

ods can reduce the computational cost from O(m3) to O(m2). We utilize the bi-conjugate

gradient stabilized method (Bi-CGSTAB) to the non-symmetric linear system (3.23) to

avoid the irregular convergence patterns that may arise when using the conjugate gradient

squared method (CGS) (see [10, 246]). As all the other operations require only O(m) of

Algorithm 1 The Bi-CGSTAB algorithm

1: In each time level tn, x0 = Un−1, b = (I −Q)Un−1 + τFn;2: Compute r0 = b − (I + Q)x0, r0 is an arbitrary vector, such that (r0, r0) 6= 0, e.g.,r0 = r0;

3: ρ0 = α0 = ω0 = 1, v0 = p0 = 0;4: for i = 1, 2, 3, · · · , do5: ρi = (r0, ri−1); β = (ρi/ρi−1)(αi−1/ωi−1);6: pi = ri−1 + β(pi−1 − ωi−1vi−1); vi = (I +Q)pi, αi = ρi/(r0, vi);7: s = ri−1 − αivi, t = (I +Q)s; ωi = (t, s)/(t, t); xi = xi−1 + αpi + ωis;8: if xi is accurate enough then quit;9: ri = s− ωit;

10: end for11: Un = xi

computational work, the major computational cost of the entire algorithm is the matrix-

vector multiplication QUn−1 of O(m2). Thus, we have to compute the matrix-vector

multiplication in an effective way to reduce the computational complexity.

Theorem 3.5.1 The matrix-vector multiplication QU for any vector U ∈ Rm can be

evaluated in O(m logm) operations.

Proof. See [248, 250].

Then we use the following O(m logm) algorithm to evaluate the matrix-vector multipli-

cation QU , based on the decomposition of matrix Q.

59 Chapter 3

Algorithm 2 Fast O(m logm) algorithm for the evaluation of QU

1: Introduce a (2m− 2)× (2m− 2) circulant matrix [248] and one 2m− 2 vector

Q2m−2,L1 =

[QL1 TL1

TL1 QL1

], U2m−2 =

[U0

];

2: Evaluate the discrete Fourier transform of U2m−2, i.e., w2m−2 = F2m−2U2m−2;3: Evaluate the discrete Fourier transform of q2m−2,L1 , i.e., v2m−2,L1 = F2m−2q2m−2,L1 ,

where q2m−2,L1 is the first column vector of Q2m−2,L1 ;4: Evaluate the Hadamard products z2m−2,L1 = w2m−2 ∗ v2m−2,L1 ;5: Evaluate y2m−2,L1 = F−1

2m−2z2m−2,L1 via inverse fast Fourier transform (FFT), yields

y2m−2,L1 =

[yL1

y′L1

]=

[QL1UTL1U

];

6: Evaluate the Hadamard products UL1 = KL1 ∗ yL1 ;7: Repeat Step 1 to Step 6 to evaluate UR1 , UL2 , UR2 and obtain QU = UL1 + UL2 −

(UL2 + UR2).

3.6 Numerical examples

In order to demonstrate the effectiveness of the finite difference method, some example

are presented. All numerical computations were carried out using MATLAB R2008a on

a ThinkPad SL410 laptop with configuration: Intel(R) Core(TM)2 T6570, 2.10GHz and

2G RAM.

Example 3.6.1 At first, we consider the following two-sided space fractional diffusion

equation (TSSFDE)∂u(x,t)∂t = ∂

∂x

[K−(x)aD

αxu(x, t)−K+(x)xD

αb u(x, t)

]+ f(x, t), (x, t) ∈ (0, 1)× (0, T ]

u(x, 0) = x2(1− x)2, 0 ≤ x ≤ 1,

u(0, t) = 0, u(1, t) = 0, 0 ≤ t ≤ T,

where 0 < α < 1,

f(x, t) = −e−t[x2(1− x)2 +

∂

∂xK−(x)P (x, α) +K−(x)P (x, 1 + α)

− ∂

∂xK+(x)P (1− x, α) +K+(x)P (1− x, 1 + α)

],

P (x, α) =Γ(5)

Γ(5− α)x4−α − 2Γ(4)

Γ(4− α)x3−α +

Γ(3)

Γ(3− α)x2−α

and the exact solution is u(x, t) = x2(1− x)2e−t.

Firstly, we take K−(x) = 2 − x, K+(x) = 2 + x the related numerical results are givenin Table 3.1. It describes the L2 error and convergence order of the Crank-Nicolsonscheme at t = 1 with τ = h, corresponding to six distinct values. It can be seen that nomatter α > 0.5546 or α < 0.5546 the numerical results are all in excellent agreement withthe exact solution. Then, we take three different pairs of K−(x) and K+(x), K−1 (x) =

Chapter 3 60

2−x2, K+1 (x) = 2+x2, K−2 (x) = 3−ex, K+

2 (x) = 3+ex, K−3 (x) = 12+x , K

+3 (x) = 1

2−x , therelated numerical results are given in Table 3.2. It shows the error and convergence orderof the Crank-Nicolson scheme at t = 1 with τ = h for the different K−(x) and K+(x),corresponding to three distinct values of α = 0.2, α = 0.5, α = 0.8. We can noticethat the numerical results are still in excellent agreement with the exact solution, whichfurther demonstrate the effectiveness of our numerical method. What follows are thenumerical results of comparison between the regular Bi-CGSTAB, fast Bi-CGSTAB andGauss elimination. In the process of regular Bi-CGSTAB and fast Bi-CGSTAB methods,we set ||xi−xi−1||∞ < 10−10 as the stopping criterion and the maximum iteration numberis 103. Table 3.3 illustrates the error and convergence property of three algorithms att = 1 with τ = h and α = 0.5 for K−(x) = 2 − x, K+(x) = 2 + x. The results arevery encouraging and show that the FBi-CGSTAB method is efficient and reliable as thethree algorithms approximate the exact solution with the same accuracy, despite the factthat the FBi-CGSTAB method has significantly reduced the storage and computationalcost. Table 3.4 displays the consumed CPU time of three algorithms at t = 1 with τ = hand α = 0.5 for K−(x) = 2 − x, K+(x) = 2 + x. Comparing to the Gauss elimination,the regular Bi-CGSTAB and fast Bi-CGSTAB methods have significantly reduced thecomputational time. Although the consumed time of fast Bi-CGSTAB method is slightlymore than the regular Bi-CGSTAB method at the beginning, the advantage of fast Bi-CGSTAB method becomes increasingly apparent with τ and h decrease, which coincideswith the theoretical analysis. Another advantage of Bi-CGSTAB method to be mentionedis that the average iteration number never increases significantly with τ and h decrease.Here, the average iteration number is approximately 30 regardless of the model scale.

Table 3.1: The error and convergence order of the Crank-Nicolson scheme of the TSSFDEfor different α at t = 1 with τ = h

τ = hα = 0.1 α = 0.3 α = 0.5

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 2.7307E-04 – 2.3763E-04 – 2.3083E-04 –1/32 6.4691E-05 2.08 2.3763E-04 2.03 5.6622E-05 2.031/64 1.5698E-05 2.04 1.4381E-05 2.02 1.3970E-05 2.021/128 3.8713E-06 2.02 3.5796E-06 2.01 3.4657E-06 2.011/256 9.6319E-07 2.01 8.9456E-07 2.00 8.6321E-07 2.01

τ = hα = 0.6 α = 0.75 α = 0.9

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 2.3453E-04 – 2.4721E-04 – 2.6290E-04 –1/32 5.7563E-05 2.03 6.0844E-05 2.02 6.5264E-05 2.011/64 1.4166E-05 2.02 1.4986E-05 2.02 1.6214E-05 2.011/128 3.4998E-06 2.02 3.6950E-06 2.02 4.0287E-06 2.011/256 8.6777E-07 2.01 9.1232E-07 2.02 1.0010E-06 2.01

61 Chapter 3

Table 3.2: The error and convergence order of the Crank-Nicolson scheme of the TSSFDEfor different K−(x), K+(x) and α at t = 1 with τ = h

K−1 (x),K+1 (x)

α = 0.2 α = 0.5 α = 0.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 2.4916E-04 – 2.2805E-04 – 2.5183E-04 –1/32 6.0815E-05 2.03 5.6137E-05 2.02 6.2122E-05 2.021/64 1.4991E-05 2.02 1.3878E-05 2.02 1.5334E-05 2.021/128 3.7256E-06 2.01 3.4474E-06 2.01 3.7869E-06 2.021/256 9.3037E-07 2.00 8.5950E-07 2.00 9.3583E-07 2.02

K−2 (x),K+2 (x)

α = 0.2 α = 0.5 α = 0.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 3.1562E-04 – 2.6807E-04 – 2.5811E-04 –1/32 7.4328E-05 2.09 6.5933E-05 2.02 6.4436E-05 2.001/64 1.7649E-05 2.07 1.6131E-05 2.03 1.6070E-05 2.001/128 4.2268E-06 2.07 3.9499E-06 2.03 4.0024E-06 2.011/256 1.0198E-06 2.05 9.6909E-07 2.03 9.9616E-07 2.01

K−3 (x),K+3 (x)

α = 0.2 α = 0.5 α = 0.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 2.6409E-04 – 2.6173E-04 – 2.8029E-04 –1/32 6.4156E-05 2.04 6.3402E-05 2.05 6.8770E-05 2.031/64 1.5861E-05 2.02 1.5493E-05 2.03 1.6906E-05 2.021/128 3.9585E-06 2.00 3.8110E-06 2.02 4.1606E-06 2.021/256 9.9217E-07 2.00 9.4205E-07 2.02 1.0248E-06 2.02

Table 3.3: Comparison of the convergence property of FBi-CGSTAB versus Gauss elim-ination and Bi-CGSTAB for K−(x) = 2 − x, K+(x) = 2 + x at t = 1 with τ = h andα = 0.5

τ = hGauss elimination Bi-CGSTAB FBi-CGSTAB

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/64 1.3970E-05 – 1.3970E-05 – 1.3970E-05 –1/128 3.4657E-06 2.01 3.4656E-06 2.01 3.4656E-06 2.011/256 8.6321E-07 2.01 8.6300E-07 2.01 8.6319E-07 2.011/512 2.1555E-07 2.00 2.1545E-07 2.00 2.1494E-07 2.011/1024 5.3917E-08 2.00 5.3286E-08 2.02 5.3732E-08 2.00

Chapter 3 62

Table 3.4: Comparison of the consumed CPU time of FBi-CGSTAB versus Gauss elim-ination and Bi-CGSTAB for K−(x) = 2 − x, K+(x) = 2 + x at t = 1 with τ = h andα = 0.5

τ = hGauss elimination Bi-CGSTAB FBi-CGSTAB

CPU time CPU time CPU time

1/128 7.64s 4.54s 5.63s1/256 1min11.2s 17.44s 18.29s1/512 19min33s 1min22s 1min29s1/1024 5h57min39s 7min26s 6min7s1/2048 >100h 39min39s 26min43s

Example 3.6.2 Now, we consider another type of TSSFDE∂u(x,t)∂t = ∂

∂x

K(x)

[βaD

αxu(x, t)− (1− β)xD

αb u(x, t)

]+ f(x, t), (x, t) ∈ (0, 1)× (0, T ]

u(x, 0) = x2(1− x)2, 0 ≤ x ≤ 1,

u(0, t) = 0, u(1, t) = 0, 0 ≤ t ≤ T,

where 0 < α < 1, 0 < β < 1,

f(x, t) = −e−t[x2(1− x)2 + βK ′(x)P (x, α) + βK(x)P (x, 1 + α)

− (1− β)K ′(x)P (1− x, α) + (1− β)K(x)P (1− x, 1 + α)],

P (x, α) =Γ(5)

Γ(5− α)x4−α − 2Γ(4)

Γ(4− α)x3−α +

Γ(3)

Γ(3− α)x2−α

and the exact solution is u(x, t) = x2(1− x)2e−t.

Here we take K(x) = x2. The related numerical results are given in Table 3.5. It describes

the error and convergence order of the Crank-Nicolson scheme at t = 1 with τ = h for

different β and α.

Example 3.6.3 Finally, we consider the following Riesz space fractional diffusion equa-

tion (RSFDE)∂u(x,t)∂t = ∂1+αu(x,t)

∂|x|1+α + f(x, t), (x, t) ∈ (0, 1)× (0, T ]

u(x, 0) = x2(1− x)2, 0 ≤ x ≤ 1,

u(0, t) = 0, u(1, t) = 0, 0 ≤ t ≤ T,

where 0 < α < 1,

f(x, t) = −e−tx2(1− x)2 +e−t

2 cos (1+α)π2

[P (x, 1 + α) + P (1− x, 1 + α)

],

P (x, α) =Γ(5)

Γ(5− α)x4−α − 2Γ(4)

Γ(4− α)x3−α +

Γ(3)

Γ(3− α)x2−α

63 Chapter 3

Table 3.5: The error and convergence order of the Crank-Nicolson scheme of TSSFDE fordifferent β and α at t = 1 with τ = h

β = 0.3α = 0.2 α = 0.5 α = 0.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 1.9224E-04 – 1.9667E-04 – 2.2955E-04 –1/32 4.6285E-05 2.05 4.8975E-05 2.01 5.6832E-05 2.011/64 1.1396E-05 2.02 1.2286E-05 2.00 1.4284E-05 1.991/128 2.8351E-06 2.01 3.0860E-06 1.99 3.5815E-06 2.001/256 7.0839E-07 2.00 7.7512E-07 1.99 8.9410E-07 2.00

β = 0.5α = 0.2 α = 0.5 α = 0.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 1.4166E-04 – 2.4418E-04 – 2.7135E-04 –1/32 3.5555E-05 1.99 5.9068E-05 2.05 6.8227E-05 1.991/64 9.0331E-06 1.98 1.4346E-05 2.04 1.7192E-05 1.991/128 2.3075E-06 1.97 3.4962E-06 2.04 4.3030E-06 2.001/256 5.8963E-07 1.97 8.5527E-07 2.03 1.0709E-06 2.01

β = 0.7α = 0.2 α = 0.5 α = 0.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 4.3589E-04 – 3.7937E-04 – 3.2791E-04 –1/32 1.0412E-04 2.07 9.2419E-05 2.04 8.4150E-05 1.961/64 2.4782E-05 2.07 2.2360E-05 2.05 2.1439E-05 1.971/128 5.8975E-06 2.07 5.3904E-06 2.05 5.4107E-06 1.991/256 1.4052E-06 2.07 1.2976E-06 2.05 1.3565E-06 2.00

and the exact solution is u(x, t) = x2(1− x)2e−t.

Table 3.6 shows the error and convergence order of the Crank-Nicolson scheme at t = 1with τ = h for α. The results are in excellent agreement with the exact solution, whichverified the analysis of Remark 3.4.2. According to Examples 3.6.2 and 3.6.3, we canfurther conclude that our numerical method is effective and widely applicable.

Table 3.6: The error and convergence order of the Crank-Nicolson scheme of the RSFDEfor different α

τ = hα = 0.2 α = 0.5 α = 0.8

||E(h, τ)|| Order ||E(h, τ)|| Order ||E(h, τ)|| Order

1/16 2.4964E-04 – 2.8341E-04 – 3.0589E-04 –1/32 5.9664E-05 2.06 6.7700E-05 2.07 7.5738E-05 2.011/64 1.4511E-05 2.04 1.6255E-05 2.06 1.8723E-05 2.021/128 3.5765E-06 2.02 3.9262E-06 2.05 4.6213E-06 2.021/256 8.8913E-07 2.01 9.5441E-07 2.04 1.1393E-06 2.02

Chapter 3 64

3.7 Conclusions

In this chapter, we have developed and demonstrated a second order finite difference

method for solving a class of two-sided space fractional diffusion equation with variable co-

efficients. Firstly, based on a second-order scheme, applying the finite difference method,

we derived the Crank-Nicolson scheme of the problem and rewrote the scheme as a matrix

form. Subsequently, we proved that the scheme is unconditionally stable and convergent

with the accuracy of O(τ2 + h2). Moreover, we developed a fast bi-conjugate stabilized

method which needs less storage and computational cost while retaining the same accura-

cy as Gauss elimination. Finally, some numerical results were given to show the stability,

consistency, and convergence of our computational approach. This technique could be

extended to two-dimensional or three-dimensional problems with complex regions. In the

future, we would like to investigate finite difference method for the two-sided FDE in high

dimensions.

Chapter 4

Unstructured mesh finite difference/finite element method for the

2D time-space Riesz fractional diffusion equation on irregular convex

domains

4.1 Introduction

Over the past decades, fractional derivatives have been widely used in physics [230, 275],

biology [158, 174], chemistry [274], hydrology [14, 148], finance [220] and the related theory

has been expanding at a fast rate [60, 128, 203]. A considerable number of computational

models emerged that are based on applying the finite element method (FEM) to frac-

tional diffusion equations (FDE). This work dates back to Roop and Ervin [68, 216], who

constructed appropriate fractional derivative spaces and presented a theoretical frame-

work for the Galerkin finite element approximation to the fractional advection-dispersion

equation. Deng [58] developed the FEM for the numerical resolution of the space and

time fractional Fokker-Planck equation. Zhang et al. [280] discussed the Galerkin finite

element approximation of symmetric space-fractional partial differential equations. Li et

al. [138] studied the Galerkin FEM for time-space fractional order nonlinear subdiffusion

and superdiffusion equations. Liu et al. [162] investigated the finite element approx-

imation for a modified anomalous subdiffusion equation. Zeng et al. [276] developed

finite difference and finite element approaches for the time-fractional subdiffusion equa-

tion with Dirichlet boundary conditions. Jin et al. [116] studied the Galerkin FEM and

lumped mass Galerkin FEM for the initial boundary value problem of a homogeneous

time-fractional diffusion equation. Jin et al. also [117] considered the Galerkin FEM

for the initial/boundary value problem involving multiple time-fractional derivatives on

a bounded convex polyhedral domain. Bu et al. [20] considered the Galerkin FEM for

two-dimensional Riesz space FDE. Liu et al. [164] presented a mixed FEM for a time-

fractional fourth-order partial differential equation. Feng et al. [79] considered the FEM

with a second-order time scheme for a space-time FDE. Zhuang et al. [303] introduced the

Galerkin FEM and error analysis for the fractional cable equation. Recently, Zhao et al.

[291] established the nonconforming FEM for two-dimensional multi-term time fractional

subdiffusion equations. Jin et al. [119] developed variational formulations of a Petrov-

Galerkin FEM type for one-dimensional fractional boundary value problems. Yang et

al. [262] considered the FEM for nonlinear Riesz space fractional diffusion equations on

irregular domains. Fan et al. [72] discussed the FEM for the two-dimensional time-space

fractional wave equation on irregular domains.

65

Chapter 4 66

In fact, many problems from science and engineering involve mathematical models that

must be computed on irregular domains and therefore seeking effective numerical meth-

ods to solve FDE on such domains is important. Although existing numerical methods

for FDE are numerous, most of them are limited to regular domains and uniform meshes.

Research involving unstructured meshes [72, 210] and irregular domains [157, 261, 262] is

more sparse. For the classical diffusion equation with integer order derivatives, there is

some theory and research involving unstructured meshes with the finite volume element

method (see [29, 36, 236] and references therein). For the fractional case, recently, Karaa

et al. [120] proposed a finite volume element method with unstructured mesh for approx-

imating the anomalous subdiffusion equations with the temporal fractional derivative.

Le et al. [134] studied the finite element approximation for a time-fractional diffusion

problem on a domain with a re-entrant corner. Fan et al. [72] discussed the unstructured

mesh finite element method for the time-space fractional wave equation. They used the

L2 scheme to approximate the temporal derivative with low accuracy and extended the

properties of the fractional derivative space to the two-dimensional convex domain case

without detailed proof. Here, we will consider a finite element method suitable for imple-

mentation with an unstructured mesh for the time-space fractional diffusion equation.

In this paper, we consider the following two-dimensional time-space Riesz fractional dif-

fusion equation (2D-TSRFDE) on an irregular convex domain:

C0 D

γt u(x, y, t) = K1

∂2αu(x, y, t)

∂|x|2α+K2

∂2βu(x, y, t)

∂|y|2β+f(x, y, t), (x, y, t) ∈ Ω×(0, T ], (4.1)

with the initial condition

u(x, y, 0) = φ(x, y), (x, y) ∈ Ω, (4.2)

and boundary condition

u(x, y, t) = 0, (x, y, t) ∈ ∂Ω× [0, T ], (4.3)

where 0 < γ < 1, 12 < α, β ≤ 1, K1 > 0, K2 > 0, f(x, y, t) and φ(x, y) are two known

smooth functions. The irregular convex domain Ω is defined as (see Figure 4.1): Ω =

(x, y)| c(y) < x < r(y), a1 < y < b1 or Ω = (x, y)| g(x) < y < m(x), c1 < x < d1,where a1 = min

(x,y)∈Ωg(x), b1 = max

(x,y)∈Ωm(x), c1 = min

(x,y)∈Ωc(y), and d1 = max

(x,y)∈Ωr(y). Here

the irregular convex domain means the convex domain in with a curved boundary, which

is compared with the regular convex domain such as a square or rectangle. In Eq.(4.1),

the Caputo fractional derivative C0 D

γt u is defined as [128]

C0 D

γt u(x, y, t) =

1

Γ(1− γ)

∫ t

0(t− s)−γ ∂u(x, y, s)

∂sds.

As Ω is an irregular convex domain, the boundary value in the space fractional derivative is

no longer a fixed constant, which is distinct from the common Riemann-Liouville fractional

67 Chapter 4

c(y) r(y)

b1

a1

y

xO

g(x)

m(x)

c1 d1

y

xO

Figure 4.1: The boundaries of convex domain Ω

derivative definition [203]

aD2αx u(x, y, t) =

1

Γ(n− 2α)

∂n

∂xn

∫ x

a(x− s)n−2α−1u(s, y, t) ds,

xD2αb u(x, y, t) =

(−1)n

Γ(n− 2α)

∂n

∂xn

∫ b

x(s− x)n−2α−1u(s, y, t) ds.

Here, we define the Riesz fractional derivatives ∂2αu∂|x|2α and ∂2βu

∂|y|2βas

∂2αu(x, y, t)

∂|x|2α= − 1

2 cos(απ)

(c(y)D

2αx u(x, y, t) + xD

2αr(y)u(x, y, t)

),

∂2βu(x, y, t)

∂|y|2β= − 1

2 cos(βπ)

(g(x)D

2βy u(x, y, t) + yD

2βm(x)u(x, y, t)

),

and the Riemann-Liouville fractional derivative with varying boundary c(y)D2αx u(x, y, t),

xD2αr(y)u(x, y, t), g(x)D

2βy u(x, y, t) and yD

2βm(x)u(x, y, t) (n− 1 < 2α, 2β < n) are given by

c(y)D2αx u(x, y, t) :=

1

Γ(n− 2α)

∂n

∂xn

∫ x

c(y)(x− s)n−2α−1u(s, y, t) ds,

xD2αr(y)u(x, y, t) :=

(−1)n

Γ(n− 2α)

∂n

∂xn

∫ r(y)

x(s− x)n−2α−1u(s, y, t) ds,

g(x)D2βy u(x, y, t) :=

1

Γ(n− 2β)

∂n

∂yn

∫ y

g(x)(y − s)n−2β−1u(x, s, t) ds,

yD2βm(x)u(x, y, t) :=

(−1)n

Γ(n− 2β)

∂n

∂yn

∫ m(x)

y(s− y)n−2β−1u(x, s, t) ds.

The major contribution of this paper is as follows. Firstly, we establish and prove some

new definitions and lemmas for convex domains, which extends the properties of the

fractional derivative space from the one-dimensional case to the two-dimensional convex

domain case. We believe this is a new contribution to the literature that was not discussed

in [21, 72]. Secondly, we propose a novel technique utilizing FEM and unstructured trian-

gular meshes to deal with the space fractional derivative on an irregular convex domain,

which we believe is very flexible because our considered solution domain can be arbitrarily

Chapter 4 68

convex and we need fewer grid nodes to generate the meshes. For general convex domains,

such as a human brain-like domain, the software Gmsh [95] can be used to partition and

generate regular unstructured triangular meshes (see Figure 4.2(b)). Due to the irregular

domain, the treatment of the space fractional derivative is not straightforward. Therefore,

we reduce the calculation from the whole domain Ω to every single triangular element

and deal with it approximately by the Gauss quadrature technique, which is elaborated

in Section 4.3. In [72], Fan et al. considered the time-space fractional wave equation,

of which the order of time fractional derivative is 1 < γ < 2 and discretised the time

fractional derivative using the L2 scheme, which requires discretisation of two time levels.

Different to [72], we consider the application of the FEM to time-space fractional diffusion

equations, of which the order of time fractional derivative is 0 < γ < 1. The time frac-

tional derivative is discretised using the L1 scheme, which only requires discretisation of

the time fractional derivative at one temporal level and is more computationally efficient.

Furthermore, we present a second order numerical scheme for the temporal fractional

derivative based on the finite difference. Moreover, we extend the method to non-convex

domains and solve a problem on a multiply-connected domain, respectively. Another im-

portant contribution of our work is the extension of the theory to allow the solution of

the 2D-TSRFDE on a multiply-connected domain and we illustrate the numerical results

for a particular case.

In order to demonstrate the applicability of our computational approach, we solve the two-

dimensional coupled fractional Bloch-Torrey equation on a human brain-like domain nu-

merically and analyse the impact of the space fractional index on the diffusion behaviour.

In clinical settings, diffusion-weighted imaging (DWI) is increasingly utilised to study

heterogeneous water diffusion in the human brain, which is thought to be anomalous.

In [13], the researchers used the stretched exponential model to investigate the diffusion

behaviour in complex biological tissues, such as human brain gray matter, glioblastoma

tissues, and erythrocyte ghosts. In [175], Magin et al. showed that the model was a

fundamental extension of the classical Bloch-Torrey equation through the application of

the operators of fractional calculus. The time and space fractional Bloch-Torrey equation

has the following form [175, 272]:

ωα−1C0 D

αt Mxy(r, t)

=Dµ2(β−1)

(∂2βMxy(r, t)

∂|x|2β+∂2βMxy(r, t)

∂|y|2β+∂2βMxy(r, t)

∂|z|2β

)+ λ(t)Mxy(r, t), (4.4)

where λ(t) = −iγ(r · G(t)) , G(t) is the magnetic field gradient, γ and D are the gy-

romagnetic ratio and the diffusion coefficient respectively, ω and µ are time and space

constants, r = (x, y, z), i =√−1 and Mxy(r, t) = Mx(r, t) + iMy(r, t). Therefore, models

using fractional-order calculus have emerged as promising tools to analyse diffusion im-

ages of the human brain to provide new insights into the investigations of tissue structures

and the microenvironment. For the two dimensional case, we equate real and imaginary

69 Chapter 4

components to express equation (4.4) as a coupled system for the components Mx and

My, i.e.,ωα−1C

0 Dαt Mx(x, y, t) = Dµ2(β−1)

(∂2βMx(x,y,t)

∂|x|2β+ ∂2βMx(x,y,t)

∂|y|2β

)+ λ0(t)My(x, y, t),

ωα−1C0 D

αt My(x, y, t) = Dµ2(β−1)

(∂2βMy(x,y,t)

∂|x|2β+

∂2βMy(x,y,t)

∂|y|2β

)− λ0(t)Mx(x, y, t).

In [272], Yu et al. proposed an implicit numerical method for the two dimensional time-

Figure 4.2: The sectional view of human brain [207] and unstructured mesh partition

space fractional Bloch-Torrey equation on a finite rectangular domain. However, this

method may have low spatial accuracy (O(τα + h2 + ρh), 0 ≤ ρ ≤ 1) for problems where

the solution domain is irregular [208], for example, a human brain-like domain (see Figure

4.2(a)).

The outline of this chapter is as follows. In Section 4.2, some new definitions and prop-

erties of the fractional derivative space and fractional Sobolev space on irregular convex

domains are introduced. In Section 4.3, we derive the fully discrete finite element scheme

of the problem (4.1)-(4.3) and describe how the finite element method implemented using

an unstructured mesh can be used to solve the 2D-TSRFDE on an arbitrarily convex do-

main and present a second order temporal numerical scheme. In Section 4.4, we discuss

the stability and convergence of the method. In Section 4.5, we discuss the FEM for the

2D-TSRFDE on a non-convex domain. In Section 4.6, some numerical examples on irreg-

ular domains, including convex domains and a multiply-connected domain, are presented.

We also solve a coupled fractional Bloch-Torrey equation to verify the effectiveness of the

method. Finally, some conclusions of the work are drawn.

4.2 Preliminary knowledge

At first, we introduce some definitions and lemmas on the fractional derivative space,

which were first established by Roop and Ervin [68, 216] in the one-dimensional case. In

Chapter 4 70

the two-dimensional rectangular domains, these results are also applicable [21]. Here, we

extend them to convex domains. Referring to Figure 4.1, we define

(u, v)Ω :=

∫ b1

a1

∫ r(y)

c(y)u(x, y)v(x, y)dxdy =

∫ d1

c1

∫ m(x)

g(x)u(x, y)v(x, y)dydx

and ||u||L2(Ω) := (u, u)1/2Ω .

Definition 4.2.1 (Left fractional derivative space) For µ > 0, denote the semi-norm and

the norm respectively as

|u|JµL(Ω) :=(||c(y)D

µxu||2L2(Ω) + ||g(x)D

µyu||2L2(Ω)

)1/2, ||u||JµL(Ω) :=

(||u||2L2(Ω) + |u|2

JµL(Ω)

)1/2,

and define JµL(Ω) (JµL,0(Ω)) as the closure of C∞(Ω) (C∞0 (Ω)) with respect to || · ||JµL(Ω).

Definition 4.2.2 (Right fractional derivative space) For µ > 0, denote the semi-norm

and the norm respectively as:

|u|JµR(Ω) :=(||xDµ

r(y)u||2L2(Ω) + ||yDµ

m(x)u||2L2(Ω)

)1/2, ||u||JµR(Ω) :=

(||u||2L2(Ω) + |u|2

JµR(Ω)

)1/2,

and define JµR(Ω) (JµR,0(Ω)) as the closure of C∞(Ω) (C∞0 (Ω)) with respect to || · ||JµR(Ω).

Definition 4.2.3 (Symmetric fractional derivative space) For µ > 0, µ 6= n − 12 , n ∈ N

denote the semi-norm and the norm respectively as:

|u|JµS (Ω) :=(|(c(y)D

µxu, xD

µr(y)u)Ω|+ |(g(x)D

µyu,y D

µm(x)u)Ω|

)1/2,

||u||JµS (Ω) :=(||u||2L2(Ω) + |u|2

JµS (Ω)

)1/2,

and define JµS (Ω) (JµS,0(Ω)) as the closure of C∞(Ω) (C∞0 (Ω)) with respect to || · ||JµS (Ω).

Definition 4.2.4 (Fractional Sobolev space) For µ > 0, denote the semi-norm and the

norm respectively as:

|u|Hµ(Ω) := || |ξ|µF(u)(ξ)||L2(R2), ||u||Hµ(Ω) :=(||u||2L2(Ω) + |u|2Hµ(Ω)

)1/2,

where F(u)(ξ) is the Fourier transform of u, which is the zero extension of u outside Ω

and define Hµ(Ω) (Hµ0 (Ω)) as the closure of C∞(Ω) (C∞0 (Ω)) with respect to || · ||Hµ(Ω).

71 Chapter 4

Define the following fractional derivative and integral operators:

−∞Dµxu(x, y, t) =

1

Γ(n− µ)

∂n

∂xn

∫ x

−∞(x− s)n−µ−1u(s, y, t) ds,

xDµ+∞u(x, y, t) =

(−1)n

Γ(n− µ)

∂n

∂xn

∫ +∞

x(s− x)n−µ−1u(s, y, t) ds,

−∞Iµxu(x, y, t) =

1

Γ(µ)

∫ x

−∞(x− s)µ−1u(s, y, t) ds,

xIµ+∞u(x, y, t) =

(−1)n

Γ(µ)

∫ +∞

x(s− x)µ−1u(s, y, t) ds,

c(y)Iµxu(x, y, t) :=

1

Γ(µ)

∫ x

c(y)(x− s)µ−1u(s, y, t) ds,

xIµr(y)u(x, y, t) :=

(−1)n

Γ(µ)

∫ r(y)

x(s− x)µ−1u(s, y, t) ds.

The definitions of the operators in the y direction are similar.

Remark 4.2.1 If supp(u) ⊂ Ω, then −∞Dµxu = c(y)D

µxu, xD

µ+∞u = xD

µr(y)u, −∞I

µxu =

c(y)Iµxu and xI

µ+∞u = xI

µr(y).

Lemma 4.2.1 Let µ > 0, define the operators: (i) −∞Iµx : L2(Ω)→ L2(Ω), (ii) −∞D

µx :

JµL(Ω) → L2(Ω), (iii) xIµ+∞ : L2(Ω) → L2(Ω), (iv) xD

µ+∞ : JµR(Ω) → L2(Ω), then all the

operators are bounded operators.

Proof. Using Young’s theorem [1] ||v ∗ w||L2(Ω) ≤ ||v||L1(Ω)||w||L2(Ω) and noting that

−∞Iµxu = xµ−1

Γ(µ) ∗ u, where ∗ denotes convolution, we have

||−∞Iµxu||L2(Ω) =1

Γ(µ)||xµ−1 ∗ u||L2(Ω) ≤

1

Γ(µ)||xµ−1||L1(Ω)||u||L2(Ω)

=1

Γ(µ)

∫ b1

a1

∫ r(y)

c(y)|x|µ−1dxdy||u||L2(Ω) ≤

1

Γ(µ)

∫ b1

a1

∫ d1

c1

|x|µ−1dxdy||u||L2(Ω)

≤ (b1 − a1)(|d1|µ + |c1|µ)

Γ(µ+ 1)||u||L2(Ω) ≤ C||u||L2(Ω).

By the definition of JµL(Ω), we have

||−∞Dµxu||L2(Ω) ≤ (||u||2L2(Ω) + ||−∞Dµ

xu||2L2(Ω) + ||−∞Dµyu||2L2(Ω))

12

= (||u||2L2(Ω) + ||c(y)Dµxu||2L2(Ω) + ||g(x)D

µyu||2L2(Ω))

12 = ||u||JµL(Ω).

The proofs of (iii) and (iv) are similar.

Lemma 4.2.2 For u ∈ JµL,0(Ω) ∩ JµR,0(Ω) and 0 < s < µ, we have

||u||L2(Ω) ≤ C1||c(y)Dsxu||L2(Ω) ≤ C2||c(y)D

µxu||L2(Ω),

||u||L2(Ω) ≤ C3||g(x)Dsyu||L2(Ω) ≤ C4||g(x)D

µyu||L2(Ω),

Chapter 4 72

where C1, C2, C3 and C4 are some positive constants independent of u.

Proof. Combining Lemma 4.2.1, we have

||u||L2(Ω) = ||−∞Isx−∞Dsxu||L2(Ω) ≤ C1||−∞Ds

xu||L2(Ω) = C1||c(y)Dsxu||L2(Ω),

||c(y)Dsxu||L2(Ω) = ||−∞Ds

xu||L2(Ω) = ||−∞I(µ−s)x −∞D

µxu||L2(Ω)

≤ C2||−∞Dµxu||L2(Ω) = C2||c(y)D

µxu||L2(Ω).

The second inequality can be proved similarly.

Lemma 4.2.3 If µ > 0, then JµL(Ω), JµR(Ω) and Hµ(Ω) are equivalent with equivalent

norms and semi-norms; if µ > 0, µ 6= n − 12 , n ∈ N, then JµL,0(Ω), JµR,0(Ω), JµS,0(Ω) and

Hµ0 (Ω) are equivalent with equivalent norms and semi-norms.

Proof. The proof is similar to the 1D case in [216], therefore we omit it here.

Lemma 4.2.4 If µ ∈ (1/2, 1), u, v ∈ J2µL,0(Ω) ∩ J2µ

R,0(Ω), then(c(y)D

2µx u(x, y), v(x, y)

)Ω

=(c(y)D

µxu(x, y), xD

µr(y)v(x, y)

)Ω,(

xD2µr(y)u(x, y), v(x, y)

)Ω

=(xD

µr(y)u(x, y), c(y)D

µxv(x, y)

)Ω.

Proof. Combining the formula [216] (aIµxw, v)L2(a,b) = (w, xI

µb v)L2(a,b), we have(

c(y)D2µx u(x, y), v(x, y)

)Ω

=(D2xc(y)I

(2µ−2)x u(x, y), v(x, y)

)Ω

=(Dxc(y)I

(2µ−2)x u(x, y),−Dxv(x, y)

)Ω

=(c(y)I

(2µ−2)x Dxu(x, y),−Dxv(x, y)

)Ω

=(c(y)I

(µ−1)x Dxu(x, y),−xI(µ−1)

r(y) Dxv(x, y))

Ω=(c(y)D

µxu(x, y), xD

µr(y)v(x, y)

)Ω.

The proof of the second identity is similar.

4.3 Finite element method

4.3.1 The fully discrete finite element scheme

For convenience, in the subsequent sections, we suppose that C, C1, C2, ... are some

positive constants, which may take distinct values according to different contexts discussed

throughout this paper. Let τ = TN be the time step and tn = nτ , n = 0, 1, 2, ..., N . Using

the finite difference method we have

C0 D

γt u(x, y, tn) =

1

Γ(1− γ)

∫ tn

0(tn − s)−γ

∂u(x, y, s)

∂sds =

τ−γ

Γ(2− γ)

n∑k=0

bnku(x, y, tk) +Rnt ,

73 Chapter 4

where bnn = 1, bn0 = (n−1)1−γ−n1−γ < 0, bnk = (n−k+1)1−γ−2(n−k)1−γ+(n−k−1)1−γ <

0, k = 1, 2, ..., n− 1. Denote

∇γt u(x, y, tn) =τ−γ

Γ(2− γ)

n∑k=0

bnku(x, y, tk), n = 1, 2, ..., N, (4.5)

then [238]

||Rnt ||0 = ||C0 Dγt u(x, y, tn)−∇γt u(x, y, tn)||0 ≤ Cτ2−γ . (4.6)

Denote V = Hα0 (Ω)∩Hβ

0 (Ω). We divide the domain Ω into a number of regular triangular

regions. Let Th be this triangulation and h be the maximum diameter of the triangular

elements. We define the finite element subspace as:

Vh :=vh|vh ∈ C(Ω) ∩ V, vh|K is linear for all K ∈ Th and vh|∂Ω = 0

.

Assume that unh ∈ Vh is the approximation of u(x, y, t) at t = tn. Then, by Lemma 4.2.4,

we obtain the fully discrete finite element scheme of Eq.(4.1) is: find unh ∈ Vh for each

t = tn (n = 1, 2, ..., N) such that

(∇γt unh, vh)Ω +A(unh, vh)Ω = (fn, vh)Ω, ∀vh ∈ Vh, (4.7)

where Kx = K12 cos(απ) , Ky = K2

2 cos(βπ) and

A(u, v)Ω := Kx

(c(y)D

αxu, xD

αr(y)v

)Ω

+(xD

αr(y)u, c(y)D

αxv)

Ω

+Ky

(g(x)D

βyu, yD

βm(x)v

)Ω

+(yD

βm(x)u, g(x)D

βy v)

Ω

.

4.3.2 The implementation of FEM with an unstructured mesh

We consider the computation process for piecewise linear polynomials on the triangular

element ep, p = 1, 2, ..., Ne, where Ne is the total number of triangles. Then, within

element ep, the field function up(x, y) can be written as

up(x, y) =

3∑j0=1

uj0 ϕj0(x, y),

where the triangle vertices are numbered in a counter-clockwise order as 1, 2, 3 and the

basis function ϕj0(x, y) is defined by

ϕj0(x, y)∣∣∣(x,y)∈ep

=1

2∆ep

(aj0 x+ bj0 y + cj0), ϕj0(x, y)∣∣∣(x,y)/∈ep

= 0,

a1 = y2 − y3, a2 = y3 − y1, a3 = y1 − y2,

b1 = x3 − x2, b2 = x1 − x3, b3 = x2 − x1,

c1 = x2y3 − x3y2, c2 = x3y1 − x1y3, c3 = x1y2 − x2y1,

Chapter 4 74

where ∆ep is the area of triangle element p. It is well-known that

ϕj0(xi0 , yi0) = δi0j0 , i0, j0 = 1, 2, 3,

where δ is the Kronecker function. With these local field functions and basis functions,

we can obtain the formulation of u(x, y) for the whole triangulation:

u(x, y) =

Np∑i=1

ui li(x, y),

where li(x, y) is the new basis function whose support domain is Ωei (see Figure 4.3) and

Np is the total number of vertices on the convex domain Ω. Now, we rewrite unh in the

form of

unh =

Np∑i=1

uni li(x, y), (4.8)

where uni are the coefficients that are to be solved for. Substituting Eq.(4.8) into Eq.(4.7)

with vh = lj(x, y), j = 1, 2, . . . , Np gives

Np∑i=1

uni

[(li, lj)Ω + ω0A(li, lj)Ω

]

=−Np∑i=1

n−1∑k=1

bnkuki (li, lj)Ω − bn0 (u0, lj)Ω + ω0(fn, lj)Ω, (4.9)

where ω0 = τγΓ(2− γ). Eq.(4.9) can be expressed in matrix form as

(M + ω0B)Un = −Mn−1∑k=1

bnkUk −G0 + ω0F

n1 , (4.10)

where M is the mass matrix with elements Mij = (lj , li)Ω =∑Ne

p=1(lj , li)ep , B is the

stiffness matrix with elements Bij = A(lj , li)Ω and Un = [un1 , un2 , ..., u

nNp

]T . The vectors

G0 and Fn1 are given by

G0 = bn0 [(u0, l1)Ω, (u0, l2)Ω, ..., (u

0, lNp)Ω]T =

Ne∑p=1

bn0 [(u0, l1)ep , (u0, l2)ep , ..., (u

0, lNp)ep ]T ,

Fn1 = [(fn, l1)Ω, (fn, l2)Ω, ..., (f

n, lNp)Ω]T =

Ne∑p=1

[(fn, l1)ep , (fn, l2)ep , ..., (f

n, lNp)ep ]T .

75 Chapter 4

Due to the non-local property of the fractional derivative, matrix B is the most difficult

part to calculate. For matrix B, the (i, j) entry is given by

Bij = A(lj , li)Ω

= Kx

(c(y)D

αx lj(x, y), xD

αr(y)li(x, y)

)Ω

+(xD

αr(y)lj(x, y), c(y)D

αx li(x, y)

)Ω

+Ky

(g(x)D

βy lj(x, y), yD

βm(x)li(x, y)

)Ω

+(yD

βm(x)lj(x, y), g(x)D

βy li(x, y)

)Ω

. (4.11)

Here, we first discuss the support domains of the four fractional derivatives c(y)Dαx l(x, y),

xDαr(y)l(x, y), g(x)D

βy l(x, y), yD

βm(x)l(x, y), and denote them as ΩL

ei , ΩRei , ΩD

ei , ΩUei , respec-

tively. We adopt the notation that L, R, D, U correspond to the ‘left’, ‘right’, ‘down’ and‘up’ directions for the four fractional derivatives. They do not correspond to the actuallocation in the domain Ω. From Figure 4.3, we see node i is surrounded by the elementse1, e2, e3, e4, e5 and the boundary ∂Ωei is constituted by ∂Ωl

ei and ∂Ωrei , where ∂Ωl

ei ismade up of line segments B1B2, B2B3, B3B4 and ∂Ωr

ei is connected by the line segments

B4B5, B5B1, respectively. Then ∂Ωlei , y = yB4(x ≥ xB4), y = yB1(x ≥ xB1) and ∂Ω form

the support domain ΩLei . Similarly, we can obtain the support domains ΩR

ei , ΩDei and ΩU

ei(see Figure 4.3).

e1

e2

e3e4

e5

B1

B2

B3

B4

B5

ΩLeiΩ

Rei

∂Ω ∂ΩΩei

e1

e2

e3e4

e5

B1

B2

B3

B4

B5

ΩDei

ΩUei

∂Ω

∂Ω

Ωei

Figure 4.3: The illustration of support domains ΩLei , ΩR

ei , ΩDei , ΩU

ei

In view of the similarity of the four terms in the right hand side of Eq.(4.11), we illustrate

the computation of (c(y)Dαx lj , xD

αr(y)li)Ω as an example. By applying Gauss quadrature,

we have

(c(y)D

αx lj , xD

αr(y)li

)Ω

=

Ne∑p=1

(c(y)D

αx lj , xD

αr(y)li

)ep

=

Ne∑p=1

∫epc(y)D

αx lj xD

αr(y)li dxdy

≈Ne∑p=1

∑(xq ,yq)∈GK

ωq c(y)Dαx lj

∣∣∣(xq ,yq)

xDαr(y)li

∣∣∣(xq ,yq)

,

where GK stands for the set of all Gauss points in element ep and ωq are the weights

associated with the Gauss point P (xq, yq) (see Figure 4.4). When point P (xq, yq) is out

of the support domains ΩLei and ΩR

ei , we have c(y)Dαx lj(x, y) = 0 and xD

αr(y)li(x, y) = 0.

To evaluate c(y)Dαx lj

∣∣∣(xq ,yq)

, suppose that segment y = yq, c(yq) ≤ x ≤ xq intersects nq

Chapter 4 76

points with the triangular element of Ωej , and these points are numbered as x0q < x1

q <

x2q <, ..., x

nqq , then by the definition of the fractional derivative, we have

c(y)Dαx lj

∣∣∣(xq ,yq)

= c(yq)Dαx lj(x, yq)

∣∣∣x=xq

=

(1

Γ(1− α)

∂

∂x

∫ x

c(yq)(x− ξ)−αlj(ξ, yq)dξ

)∣∣∣x=xq

=

ni∑k=1

(1

Γ(1− α)

∂

∂x

∫ xki

xk−1i

(x− ξ)−αlj(ξ, yq)dξ)∣∣∣

x=xq. (4.12)

e1e2

e3e4

e5

ΩLejΩ

Rej

x40P0(x0, y0)

x30x20x10x00

j

Case I

e1e2

e3e4

e5

ΩLejΩ

Rej

x31x21x11x01

j

P1(x1, y1)

Case II

e1e2e3

e4

e5

ΩLejΩ

Rej

x32x22

x12x02

j

P2(x2, y2)

x42 x52

Case III

Figure 4.4: The points of intersection by y = yq with the triangle element of Ωej and ∂Ω

Then, there are three different cases that need to be discussed (see Figure 4.4). In case I,

the Gauss point P0(x0, y0) is only located in ΩLej , and we have

lj(x, y0) =

0, x00 ≤ x ≤ x1

0,

ϕj2(x, y0), x10 ≤ x ≤ x2

0,

ϕj1(x, y0), x20 ≤ x ≤ x3

0,

ϕj5(x, y0), x30 ≤ x ≤ x4

0,

0, x40 ≤ x,

where ϕjp(x, y) is the basis function of node j on element p and c(y0) = x00. Case II, the

Gauss point P1(x1, y1) is only located in the ΩRej\Ω

Lej , then we have c(y)D

αx lj

∣∣∣(x1,y1)

= 0.

Case III, the Gauss point P2(x2, y2) is located in ΩLej ∩ ΩR

ej = Ωej , then we have

lj(x, y2) =

0, x0

2 ≤ x ≤ x12,

ϕj2(x, y2), x12 ≤ x ≤ x2

2,

ϕj3(x, y2), x22 ≤ x ≤ x3

2,

ϕj4(x, y2), x32 ≤ x ≤ x4

2.

77 Chapter 4

Similarly, to evaluate xDαr(y)lj

∣∣∣(xq ,yq)

, we also need to consider three cases. In case I,

xDαr(y)lj

∣∣∣(x0,y0)

= 0. In case II,

lj(x, y1) =

0, x ≤ x0

1,

ϕj3(x, y1), x01 ≤ x ≤ x1

1,

ϕj4(x, y1), x11 ≤ x ≤ x2

1,

0, x21 ≤ x ≤ x3

1.

In case III,

lj(x, y2) =

ϕj2(x, y2), x1

2 ≤ x ≤ x22,

ϕj3(x, y2), x22 ≤ x ≤ x3

2,

ϕj4(x, y2), x32 ≤ x ≤ x4

2,

0, x42 ≤ x ≤ x5

2.

As lj(ξ, yq) is a linear function on [xk−1i , xki ], k = 1, 2, ..., ni, Eq.(4.12) can be evaluated

using integration by parts. Finally, we summarise the whole computation process in the

following algorithm (see Algorithm 3).

Algorithm 3 Compiling fractional derivative using FEM on an unstructured mesh

1: Partition the convex domain Ω with unstructured triangular elements ep and save the

element information (node number, coordinates, and element number );

2: for p = 1, 2, · · · , Ne do

3: Find the Gauss points (xq, yq) and weights ωi on each triangle element ep;

4: for j = 1, 2, · · · , Np do

5: Find the support domain Ωej ;

6: Find the points of intersection by y = yq with Ωej and calculate c(y)Dαx lj

∣∣∣(xq ,yq)

,

xDαr(y)li

∣∣∣(xq ,yq)

;

7: Find the points of intersection by x = xq with Ωej and calculate g(x)Dβy lj

∣∣∣(xq ,yq)

,

yDβm(x)li

∣∣∣(xq ,yq)

;

8: end for

9: Form stiffness matrix B;

10: end for

11: Calculate (lj , li)ep on each triangle element ep to form the mass matrix M ;

12: Calculate (u0, lk)ep and (fn, lk)ep , k = 1, 2, ..., Np and obtain G0, Fn;

13: Solve the linear system (4.10) and obtain Un.

4.3.3 Second order temporal numerical scheme

In this part, we will give a second order temporal numerical scheme. Firstly, when 0 < γ <

1, for a function W (t) for which the Caputo fractional derivative of order γ exists, we have

Chapter 4 78

the following relationship between its Caputo fractional derivative and Riemann-Liouville

fractional derivative [128]

C0 D

γtW (t) = 0D

γt [W (t)−W (0)] = 0D

γtW (t)− W (0)

Γ(1− γ)tγ. (4.13)

Then, we consider the shifted Grunwald formula [181] to approximate the Riemann-

Liouville fractional derivative. Define

Aγτ,lW (t) = τ−γ∞∑k=0

g(γ)k W (t− (k − l)τ),

where g(γ)k = (−1)k

(γk

)for k ≥ 0. To obtain the second order temporal numerical scheme,

we need the following lemma [241, 255]

Lemma 4.3.1 Let W (t) ∈ L1(R), −∞Dγ+2t W and its Fourier transform belong to L1(R),

and define the weighted and shifted Grunwald difference operator by

RL∇γtW (t) =γ − 2q

2(p− q)Aγτ,pW (t) +

2p− γ2(p− q)

Aγτ,qW (t), (4.14)

then we have

RL∇γtW (t) = −∞DγtW (t) +O(τ2),

for t ∈ R, where p and q are integers and p 6= q.

By choosing (p, q) = (0,−1) in (4.14), we obtain

RL∇γtW (tn) =2 + γ

2τγ

n∑k=0

g(γ)k W (tn−k)−

γ

2τγ

n−1∑k=0

g(γ)k W (tn−k−1)

=1

τγ

n∑k=0

ω(γ)k W (tn−k) =

1

τγ

n∑k=0

ω(γ)n−kW (tk),

where ω(γ)0 = 2+γ

2 g(γ)0 , ω

(γ)k = 2+γ

2 g(γ)k −

γ2g

(γ)k−1, k ≥ 1. Now, we transform (4.1) into the

following form

0Dγt u(x, y, t) = K1

∂2αu(x, y, t)

∂|x|2α+K2

∂2βu(x, y, t)

∂|y|2β+ f(x, y, t), (4.15)

where f(x, y, t) = f(x, y, t) + u(x,y,0)Γ(1−γ)tγ . Then, we obtain the fully discrete finite element

scheme of Eq.(4.15) is: to find unh ∈ Vh for each t = tn (n = 1, 2, ..., N) such that

(RL∇γt unh, vh)Ω +A(unh, vh)Ω = (fn, vh)Ω, ∀vh ∈ Vh. (4.16)

79 Chapter 4

Substituting Eq.(4.8) into Eq.(4.16) with vh = lj(x, y), j = 1, 2, . . . , Np gives

Np∑i=1

uni

[ω

(γ)0 (li, lj)Ω + τγA(li, lj)Ω

]= −

Np∑i=1

n−1∑k=0

ω(γ)n−ku

ki (li, lj)Ω + τγ(fn, lj)Ω. (4.17)

Writing (4.17) in matrix form, we have

(ω(γ)0 M + τγB)Un = −M

n−1∑k=0

ω(γ)n−kU

k + τγFn2 , (4.18)

where Fn2 = [(fn, l1)Ω, (fn, l2)Ω, ..., (f

n, lNp)Ω]T .

Remark 4.3.1 When the solution u(x, y, t) is not smooth enough, the modified weighted

shifted Grunwald-Letnikov formula with appropriate correction terms can be used [278].

4.4 Stability and convergence of the fully discrete scheme

Before giving the proof, we introduce some new definitions and lemmas. Here, we simplify

the notations (·, ·)Ω, || · ||L2(Ω) and || · ||Hs(Ω) as (·, ·), || · ||0, || · ||s, respectively. Let

σ = maxα, β, we define the semi-norm | · |(α,β) and norm ||| · |||(α,β) as follows

|u|(α,β) :=(K1||c(y)D

αxu||20 +K2||g(x)D

βyu||20

) 12, |||u|||(α,β) :=

(||u||20 + |u|2(α,β)

) 12.

Then we have the following lemma:

Lemma 4.4.1 Assume that u ∈ Hα0 (Ω) ∩Hβ

0 (Ω) and Ω is a convex domain, then

C1|||u|||(α,β) ≤ |u|(α,β) ≤ |||u|||(α,β) ≤ C2|u|Hσ(Ω),

where positive constants C1 < 1 and C2 are independent of u, i.e., the semi-norm |u|(α,β)

and ||u||(α,β) are equivalent.

Proof. We have that |u|(α,β) ≤ |||u|||(α,β). By Lemma 4.2.2,

||u||0 ≤ C||c(y)Dsxu||0 ≤ C|u|(α,β),

then |||u|||2(α,β) ≤ (C2 + 1)|u|2(α,β), therefore,

C1|||u|||(α,β) ≤ |u|(α,β), C1 =1√

C2 + 1.

Since |u|(α,β) ≤ C3|u|JσL(Ω), by Lemma 4.2.3, we obtain

|||u|||(α,β) ≤√C2 + 1|u|(α,β) ≤ C3

√C2 + 1|u|JσL(Ω) ≤ C2|u|Hσ(Ω).

Chapter 4 80

By the above lemma, it is straightforward to obtain, ∀ u ∈ Hα0 (Ω) ∩Hβ

0 (Ω),

A(u, v) ≤ C|||u|||(α,β)|||v|||(α,β), A(u, u) ≥ C|||u|||2(α,β).

Here we choose the interpolation operator Ih to satisfy the approximation properties of

the subspace of Hs+1(Ω) [19], then Ih : Hs+1(Ω)→ Vh satisfies

||u− Ihu||l ≤ Chµ−l||u||µ, ∀u ∈ Hµ(Ω), 0 ≤ l < µ ≤ s+ 1. (4.19)

We define the projection operator Ph : V → Vh satisfying

A(Phu, v) = A(u, v), u ∈ V, ∀v ∈ Vh.

Furthermore, we can derive the approximation property of Ph.

Lemma 4.4.2 If u ∈ H2(Ω) ∩ V , σ = max(α, β), then

|u− Phu|(α,β) ≤ Ch2−σ||u||2,

in which the constant C is independent of u and h.

Proof. Since |u− Phu|2(α,β) = A(u− Phu, u− Phu) = A(u− Phu, u− Ihu) and

A(u− Phu, u− Ihu) ≤ C|||u− Phu|||(α,β)|||u− Ihu|||(α,β) ≤ C1|u− Phu|(α,β)|u− Ihu|(α,β),

therefore, via the approximation properties (4.19) and Lemma 4.4.1, we have

|u− Phu|(α,β) ≤ C1|u− Ihu|(α,β) ≤ C2|u− Ihu|σ ≤ C3h2−σ||u||2.

As A(u, v) is continuous and coercive, we can derive the existence and uniqueness of

scheme (4.7). Now, we discuss stability and the convergence of this scheme.

Theorem 4.4.1 The fully discrete finite element scheme (4.7) is unconditionally stable.

Proof. Assume that znh (n = 1, 2, ..., N) is another solution of the fully scheme (4.7), and

let Enh = unh − znh , then

(∇γtEnh , vh) +A(Enh , vh) = 0, i.e.,

(Enh , vh) + τγΓ(2− γ)A(Enh , vh) = −(

n−1∑k=0

bnkEkh, vh).

Using the Cauchy-Schwarz inequality and taking vh = Enh and noting the positivity of

A(·, ·), we have

||Enh ||0 ≤ −n−1∑k=0

bnk ||Ekh||0.

81 Chapter 4

Utilising mathematical induction and noticing −∑n−1

k=0 bnk = 1, it is readily concluded that

||Enh ||0 ≤ ||E0h||0. Therefore, the fully discrete scheme (4.7) is unconditional stable.

Theorem 4.4.2 Suppose that u(tn), unh are the exact solution and numerical solution of

problem (4.1)-(4.3) at t = tn respectively, and u, utt,C0 D

γt u ∈ L∞(0, T ;Hs+1(Ω)). Then

for n = 1, 2, ..., N , the following error estimate holds

|||unh − u(tn)|||2(α,β) ≤ Cτ4−2γ + ||u0

h − u(t0)||2σ

+ h2s+2−2σ(||u(t0)||2s+1 + ||u(tn)||2s+1 + max

0≤t≤T||C0 D

γt u(t)||2s+1

).

Proof. As u(x, y, t) is the exact solution of problem (4.1)-(4.3), then(C0 D

γt u(tn), vh

)+A

(u(tn), vh

)= (fn, vh), ∀v ∈ Vh. (4.20)

Denote en = u(tn)− unh, and subtract (4.7) from (4.20), we have

(∇γt en, vh) +A(en, vh) = −(C0 D

γt u(tn)−∇γt u(tn), vh

).

Define en = ρn + θn, ρn = u(tn) − Phu(tn), θn = Phu(tn) − unh and let vh = ∇γt θn, we

obtain

(∇γt θn,∇γt θn) +A(θn,∇γt θn) = −(∇γt ρn,∇

γt θn)− (Rnt ,∇

γt θn). (4.21)

We have

|(∇γt ρn,∇γt θn)| ≤ 1

2||∇γt ρn||20 +

1

2||∇γt θn||20,

|(Rnt ,∇γt θn)| ≤ 1

2||Rnt ||20 +

1

2||∇γt θn||20.

Substituting them into (4.21) leads to

A

(θn,

n∑k=0

bnkθk

)≤ τγΓ(2− γ)

2||∇γt ρn||20 +

τγΓ(2− γ)

2||Rnt ||20. (4.22)

Note that

A

(θn,

n∑k=0

bnkθk

)=

1

2

[A(θn, θn) +

n−1∑k=0

bnkA(θk, θk)−n−1∑k=0

bnkA(θk − θn, θk − θn)

]. (4.23)

As A(θk− θn, θk− θn) ≥ 0, bnk < 0, k = 0, 1, ..., n−1, substituting (4.23) into (4.22), gives

A(θn, θn) ≤ −n−1∑k=0

bnkA(θk, θk) + τγΓ(2− γ)||∇γt ρn||20 + τγΓ(2− γ)||Rnt ||20. (4.24)

From (4.6) we have

||Rnt ||20 ≤ Cτ4−2γ max0≤t≤T

||utt||20, ||C0 Dγt ρ

n −∇γt ρn||0 ≤ Cτ2−γ max0≤t≤T

||ρtt||0,

Chapter 4 82

then

||∇γt ρn||20 ≤ ||C0 Dγt ρ

n||20 + Cτ4−2γ max0≤t≤T

||ρtt||20.

Using Lemma 4.2.2 and Lemma 4.4.2, we obtain

||∇γt ρn||20 ≤ Ch2s+2−2σ||C0 D

γt u

n||2s+1 + τ4−2γ

.

By Lemma 4.4.1, Lemma 4.4.2 and A(u, u) ≤ C||u||2(α,β), we have

A(θ0, θ0) ≤ C||θ0||2(α,β) ≤ C||u(t0)− u0

h||2(α,β) + ||ρ0||2(α,β)

≤ C

||u(t0)− u0

h||2σ + h2s+2−2σ||u(t0)||2s+1

.

Then, from (4.24), we obtain

A(θn, θn) ≤ Γ(2− γ)(−bn0 )C||u(t0)− u0

h||2σ + h2s+2−2σ||u(t0)||2s+1

+ Γ(2− γ)τγC

τ4−2γ + h2s+2−2σ||C0 D

γt u(tn)||2s+1

−n−1∑k=1

bnkA(θk, θk). (4.25)

Utilising mathematical induction, it is readily concluded that

A(θn, θn) ≤ Γ(1− γ)C1

||u(t0)− u0

h||2σ + h2s+2−2σ||u(t0)||2s+1

+ Γ(1− γ)nγτγC2

τ4−2γ + h2s+2−2σ max

0≤t≤T||C0 D

γt u(t)||2s+1

,

Since A(θn, θn) ≥ C||θn||2(α,β), the above mathematical induction gives

||θn||2(α,β) ≤ Cτ4−2γ + ||u(t0)− u0

h||2σ

+ h2s+2−2σ[||u(t0)||2s+1 + max

0≤t≤T||C0 D

γt u(t)||2s+1

], (4.26)

and

||en||2(α,β) ≤ C(||ρn||2(α,β) + ||θn||2(α,β)

).

Therefore, using Lemma 4.4.2, we obtain

||unh − u(tn)||2(α,β) ≤ Cτ4−2γ + ||u(t0)− u0

h||2σ

+ h2s+2−2σ(||u(t0)||2s+1 + ||u(tn)||2s+1 + max

0≤t≤T||C0 D

γt u(t)||2s+1

).

83 Chapter 4

Remark 4.4.1 It can deduced from Theorem 4.4.2 that when a triangular linear basis

function (s = 1) is used, the error satisfies

|||unh − u(tn)|||(α,β) ≤ C(τ2−γ + h2−σ).

Remark 4.4.2 It is well-known that the study of the regularities of the elliptic space-

fractional PDEs is still under development. Wang and Yang [251] analyzed the regularity

of the solution of fractional-order derivatives involving a variable coefficient in Holder s-

paces and established the well-posedness of a Petrov-Galerkin formulation. Jin et al. [118]

developed variational formulations for boundary value problems involving either Riemann-

Liouville or Caputo fractional derivatives and established the Sobolev regularity of the

variational solutions. Recently, Ervin et al. [70] investigated the regularity solution of the

steady-state fractional diffusion equation on a bounded domain. As the theoretical analysis

of regularities for the elliptic space-fractional PDEs on convex domains with both left and

right sided fractional derivatives is challenging, it is still an open research problem and

needs further exploration.

4.5 FEM for the 2D-TSRFDE on non-convex domains

In this section, we will discuss the FEM for the two-dimensional time-space Riesz frac-

tional diffusion equation on non-convex domains. When the considered solution domain

is nonconvex, the definition of the fractional derivative should change according to the

shape of the solution domain. For example, we consider the multiply-connected domain

Ω0 = [a, b] × [c, d]\[a∗, b∗] × [c∗, d∗] =⋃8i=1 Ωi shown in Figure 4.5. In this case, the

fractional derivative must be modified to take the central hole into consideration.

Ω1 Ω2 Ω3

Ω4 Ω5

Ω6 Ω7 Ω8

a a∗

b∗

bc

c∗

d∗

d

Figure 4.5: The illustration of a multiply-connected domain

Chapter 4 84

Definition 4.5.1 The definition of the fractional derivative in the x direction for the

domain Ω0 illustrated in Figure 4.5 is defined as:

aD2αx u(x, y, t) =

aD

2αx u(x, y, t), x ∈ Ω0\Ω5,

b∗D2αx u(x, y, t), x ∈ Ω5.

xD2αb u(x, y, t) =

xD

2αb u(x, y, t), x ∈ Ω0\Ω4,

xD2αa∗ u(x, y, t), x ∈ Ω4.

The definition of the fractional derivative in the y direction is defined in a similar manner.

Following Algorithm 3, we can solve the problem on a multiply-connected domain Ω0.

We now provide the theoretical analysis when the solution is nonconvex. Here we still

take the multi-connected domain Ω0 shown in Figure 4.5 as an example. The domain Ω0

can be divided into a finite union of convex domains Ωi: Ω0 =⋃8i=1 Ωi. Next, we define

the inner product and its induced norm as:

(u, v)Ω0 :=8∑i=1

(u, v)Ωi , ||u||L2(Ω0) := (u, u)1/2Ω0

=

√√√√ 8∑i=1

||u||2L2(Ωi),

where (u, v)Ωi and ||u||L2(Ωi) are the general inner product and norm defined on convex

domain Ωi. Then we define the semi-norm and the norm of the left fractional derivative

space on Ω0 as:

|u|JµL(Ω0) :=(||aDµ

xu||2L2(Ω0) + ||cDµyu||2L2(Ω0)

)1/2, ||u||JµL(Ω0) :=

(||u||2L2(Ω0) + |u|2

JµL(Ω0)

)1/2.

It is straightforward to derive

|u|JµL(Ω0) =

√√√√ 8∑i=1

|u|2JµL(Ωi)

, ||u||JµL(Ω0) =

√√√√ 8∑i=1

||u||2JµL(Ωi)

,

where |u|JµL(Ωi)and ||u||JµL(Ωi)

are defined in Definition 4.2.1. Similarly, we can define

other norm operators for the fractional derivative space on domain Ω0. In the following,

we will prove Lemmas 4.2.1 to 4.2.4 also hold on domain Ω0.

Lemma 4.5.1 Let µ > 0, define the operators: (i) −∞Iµx : L2(Ω0)→ L2(Ω0), (ii) −∞D

µx :

JµL(Ω0) → L2(Ω0), (iii) xIµ+∞ : L2(Ω0) → L2(Ω0), (iv) xD

µ+∞ : JµR(Ω0) → L2(Ω0), then

all the operators are bounded operators.

Proof. By the definition we have

||−∞Iµxu||L2(Ω0) =

√√√√ 8∑i=1

||−∞Iµxu||2L2(Ωi).

85 Chapter 4

According to Lemma 4.2.1, the operator −∞Iµx : L2(Ωi) → L2(Ωi) is bounded. Then we

have√√√√ 8∑i=1

||−∞Iµxu||2L2(Ωi)≤

√√√√ 8∑i=1

Ci||u||2L2(Ωi)≤

√√√√C0

8∑i=1

||u||2L2(Ωi)=√C0||u||L2(Ω0),

where C0 = maxCi, i = 1, 2, . . . , 8. Then we obtain

||−∞Iµxu||L2(Ω0) ≤ C||u||L2(Ω0).

By the definition of JµL(Ω0), we have

||−∞Dµxu||L2(Ω0) ≤ (||u||2L2(Ω0) + ||−∞Dµ

xu||2L2(Ω0) + ||−∞Dµyu||2L2(Ω0))

12

= (||u||2L2(Ω0) + ||c(y)Dµxu||2L2(Ω0) + ||g(x)D

µyu||2L2(Ω0))

12 = ||u||JµL(Ω0).

The proofs of (iii) and (iv) are similar.

Lemma 4.5.2 For u ∈ JµL,0(Ω0) ∩ JµR,0(Ω0) and 0 < s < µ, we have

||u||L2(Ω0) ≤ C1||c(y)Dsxu||L2(Ω0) ≤ C2||c(y)D

µxu||L2(Ω0),

||u||L2(Ω0) ≤ C3||g(x)Dsyu||L2(Ω0) ≤ C4||g(x)D

µyu||L2(Ω0),

where C1, C2, C3 and C4 are some positive constants independent of u.

Proof. Combining Lemma 4.2.2, we have

||u||L2(Ω0) =

√√√√ 8∑i=1

||u||2L2(Ωi)≤

√√√√ 8∑i=1

C1i||c(y)Dsxu||2L2(Ωi)

≤C1

√√√√ 8∑i=1

||c(y)Dsxu||2L2(Ωi)

= C1||c(y)Dsxu||L2(Ω0),

||c(y)Dsxu||L2(Ω0) =

√√√√ 8∑i=1

||c(y)Dsxu||2L2(Ωi)

≤

√√√√ 8∑i=1

C2i||c(y)Dµxu||2L2(Ωi)

≤C2||c(y)Dµxu||L2(Ω0),

where C1 = max√C1i and C2 = max

√C2i, i = 1, 2, . . . , 8. The second inequality can

be proved similarly.

Lemma 4.5.3 If µ ∈ (1/2, 1), u, v ∈ J2µL,0(Ω0) ∩ J2µ

R,0(Ω0), then(c(y)D

2µx u(x, y), v(x, y)

)Ω0

=(c(y)D

µxu(x, y), xD

µr(y)v(x, y)

)Ω0

,(xD

2µr(y)u(x, y), v(x, y)

)Ω0

=(xD

µr(y)u(x, y), c(y)D

µxv(x, y)

)Ω0

.

Chapter 4 86

Proof. Combining Lemma 4.2.4, we have

(c(y)D

2µx u(x, y), v(x, y)

)Ω0

=

√√√√ 8∑i=1

(c(y)D

2µx u(x, y), v(x, y)

)Ωi

=

√√√√ 8∑i=1

(c(y)D

µxu(x, y), xD

µr(y)v(x, y)

)Ωi

=(c(y)D

µxu(x, y), xD

µr(y)v(x, y)

)Ω0

.

The proof of the second identity is similar.

Then we can obtain the fully discrete finite element scheme of Eq.(4.1) on the multi-

connected domain Ω0: find unh ∈ Vh for each t = tn (n = 1, 2, ..., N) such that

(∇γt unh, vh)Ω0 +A(unh, vh)Ω0 = (fn, vh)Ω0 , ∀vh ∈ Vh. (4.27)

Now we will prove the stability of scheme (4.27).

Theorem 4.5.1 The fully discrete variational scheme (4.27) is unconditionally stable.

Proof. Assume that znh (n = 1, 2, ..., N) is another solution of the fully scheme (4.27),

and let Enh = unh − znh , then

(∇γtEnh , vh)Ω0 +A(Enh , vh)Ω0 = 0, i.e.,

(Enh , vh)Ω0 + τγΓ(2− γ)A(Enh , vh)Ω0 = −(

n−1∑k=0

bnkEkh, vh)Ω0 .

Then we have

8∑i=1

(Enh , vh)Ωi + τγΓ(2− γ)8∑i=1

A(Enh , vh)Ωi = −8∑i=1

(n−1∑k=0

bnkEkh, vh)Ωi .

According to Theorem 4.4.1, we have ||Enh ||L2(Ωi) ≤ ||E0h||L2(Ωi). Then we obtain√√√√ 8∑

i=1

||Enh ||2L2(Ωi)≤

√√√√ 8∑i=1

||E0h||2L2(Ωi)

,

i.e., ||Enh ||L2(Ω0) ≤ ||E0h||L2(Ω0), which means the fully discrete scheme (4.27) is uncondi-

tional stable.

Following a similar process of proof, we can obtain that Theorem 4.4.2 also hold for

multi-connected domain Ω0, which means that we can extend the analysis of stability and

convergence from convex domain Ωi to multi-connected domain Ω0. The extension of our

theory to multi-connected domain Ω0 is studied numerically in Example 4.6.4.

87 Chapter 4

We remark that the results from this initial investigation appear very promising and we

believe the theory can be generalised to a more complicated domain in a straightforward

manner.

4.6 Numerical examples

In this section, we present some numerical examples to verify the effectiveness of our

theoretical analysis presented in Section 4.4. We adopt the linear polynomials on triangles,

where h is the maximum length of the triangles and Ne is the number of triangles in Th. By

Theorem 4.4.2, it is expected that ||u(tn)−unh||0 ∼ O(h2), |||u(tn)−unh|||(α,β) ∼ O(h2−σ),

σ = max(α, β) in the spatial direction and ||u(tn) − unh||0 ∼ O(τ2−γ) in the temporal

direction. Here, we use the following formulation to calculate the convergence order:

Order =

log(||E(h1)||0/||E(h2)||0)

log(h1/h2) , in space,log(||E(τ1)||0/||E(τ2)||0)

log(τ1/τ2) , in time.

Example 4.6.1 Firstly, we consider the following 2D-TSRFDE on an elliptical domainC0 D

γt u(x, y, t) = K1

∂2αu(x,y,t)

∂|x|2α +K2∂2βu(x,y,t)

∂|y|2β+ f(x, y, t), (x, y, t) ∈ Ω× (0, T ],

u(x, y, 0) = 110(4x2 + y2 − 1)2, (x, y) ∈ Ω,

u(x, y, t) = 0, (x, y, t) ∈ ∂Ω× [0, T ],

where Ω = (x, y)| 4x2 + y2 < 1, K1 = 1, K2 = 1, T = 1,

f(x, y, t) =t2−γ

5Γ(3− γ)(4x2 + y2 − 1)2 +

t2 + 1

20 cos(απ)

16[f1(x, a0, 2α) + g1(x, b0, 2α)]

+ 8(y2 − 1)[f2(x, a0, 2α) + g2(x, b0, 2α)]

+ (y4 − 2y2 + 1)[f3(x, a0, 2α) + g3(x, b0, 2α)]

+t2 + 1

20 cos(βπ)

[f1(y, c0, 2β) + g1(y, d0, 2β)]

+ 8(x2 − 1)[f2(y, c0, 2β) + g2(y, d0, 2β)]

+ (16x4 − 8x2 + 1)[f3(y, c0, 2β) + g3(y, d0, 2β)],

a0 = −1

2

√1− y2, b0 =

1

2

√1− y2, c0 = −

√1− 4x2, d0 =

√1− 4x2,

f1(x, a, α) = aDαx (x4), f2(x, a, α) = aD

αx (x2), f3(x, a, α) = aD

αx (1),

g1(x, b, α) = xDαb (x4), g2(x, b, α) = xD

αb (x2), g3(x, b, α) = xD

αb (1).

The exact solution is u(x, y, t) = t2+110 (4x2 + y2 − 1)2.

Figure 4.6 shows the elliptical domain partitioned by different unstructured triangular

meshes. The corresponding numerical results are given in Table 4.1, which illustrates the

L(α,β) error, L2 error and corresponding convergence order of h. Table 4.2 displays the L2

error and the convergence order of τ . From these two tables, we can see that the expected

Chapter 4 88

x

-0.5 0 0.5

y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

-0.5 0 0.5

y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

-0.5 0 0.5

y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 4.6: The unstructured triangular meshes used in the calculation of the ellipticaldomain for h = 0.12558, 0.08391 and 0.04531

convergence orders O(h2−σ), O(h2), and O(τ2−γ) are attained. Table 4.3 shows the L2

error and the convergence order of τ = h for the second order temporal numerical scheme.

We can see that the numerical results are in excellent agreement with the exact solution

and we attain the second order, which demonstrates the effectiveness of the numerical

method.

Table 4.1: The L(α,β) error, L2 error and convergence order of h for different α, β at t = 1with γ = 0.7, τ = 1/1000

γ = 0.7 Ne h L(α,β) error Order L2 error Order

70 3.0312E-01 8.9507E-02 – 9.1296E-03 –α = 0.75 468 1.2558E-01 3.5831E-02 1.04 1.7483E-03 1.88β = 0.95 1142 8.3913E-02 2.1668E-02 1.25 6.9743E-04 2.28

4324 4.5308E-02 1.0303E-02 1.21 1.8504E-04 2.15

70 3.0312E-01 9.0145E-02 – 8.6868E-03 –α = 0.8 468 1.2558E-01 3.1969E-02 1.18 1.4612E-03 2.02β = 0.8 1142 8.3913E-02 1.8311E-02 1.38 5.5341E-04 2.41

4324 4.5308E-02 8.1488E-03 1.31 1.3987E-04 2.23

Table 4.2: The L2 error and convergence order of τ for γ = 0.7 at t = 1 with α = β = 0.8and h2 ≈ τ2−γ

Ne τ h L2 error Order

276 114 1.8428E-01 2.2122E-03 –

1142 146 8.3913E-02 5.5444E-04 1.16

1738 161 6.9134E-02 3.7308E-04 1.40

89 Chapter 4

Table 4.3: The L2 error and convergence order of τ = h for the second order numericalscheme with γ = 0.7, α = β = 0.8 at t = 1

Ne h L2 error Order

276 1.8428E-01 2.2182E-03 –1142 8.3913E-02 5.7249E-04 1.721738 6.9134E-02 3.9132E-04 1.964324 4.5308E-02 1.6187E-04 2.09

Example 4.6.2 Next, we consider the following 2D-TSRFDE without a source term on

a general irregular convex domain ΩC0 D

γt u(x, y, t) = ∂2αu(x,y,t)

∂|x|2α + ∂2βu(x,y,t)

∂|y|2β, (x, y, t) ∈ Ω× (0, T ],

u(x, y, 0) = 10 cos(π2xy), (x, y) ∈ Ω,

u(x, y, t) = 0, (x, y, t) ∈ ∂Ω× [0, T ].

Figure 4.7 shows the different triangular unstructured meshes used to partition the domain

Ω. Figure 4.8 shows a diffusion behaviour of u(x, y, t) at different time t = 0.2, 0.4, 0.8

that decays with increasing time. As we cannot determine the exact solution for this

problem, we use an approximate solution unh derived by choosing a very fine mesh (h =

0.0557). To observe the convergence behaviour, we choose a set of points in the domain

and derive the L2 error for different h. The corresponding numerical results are given

in Table 4.4. Again, we see that the numerical results exhibit a convergence order that

attains the expected value of O(h2), which means that the numerical method is effective

on general irregular convex domains. As other arbitrarily shaped convex domains can be

partitioned similarly, we can conclude that the method is applicable to other arbitrarily

shaped convex domains as well.

x

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 4.7: The unstructured triangular meshes used in the calculation of the generalirregular convex domain Ω for h = 0.2825, 0.1454 and 0.0557

Chapter 4 90

21.5

y

10.5

t = 0.2

021.5

1

x

0.5

-1

-0.5

0

0.5

2

2.5

1.5

-1.5

1

0

u(x,y,t)

21.5

y

10.5

t = 0.4

021.5

1

x

0.5

-1

-0.5

0

0.5

2

2.5

1.5

-1.5

1

0

u(x,y,t)

21.5

y

10.5

t = 0.8

021.5

1

x

0.5

-1

-0.5

0

0.5

2

2.5

1.5

-1.5

1

0

u(x,y,t)

Figure 4.8: The diffusion profiles of u(x, y, t) on the general irregular convex domain Ωat different t with γ = 0.9, α = β = 0.80, h = 0.1454, τ = 1/1000 at t = 1.0

Table 4.4: The L2 error and convergence order of h for α = β = 0.8, γ = 0.9, τ = 1/1000at t = 1.0 on the general irregular convex domain Ω

Ne h L2 error Order

244 2.8250E-01 5.0708E-03 –958 1.4536E-01 1.6899E-03 1.652476 8.9184E-02 5.4918E-04 2.30

Example 4.6.3 Then, we consider the following two dimensional coupled fractional

Bloch-Torrey diffusion equation with a time-varying magnetic field gradient on human

brain-like domain shown in Figure 4.2, in which we choose 41 sample points on the

boundary of the human brain (Figure 4.2) and connect the adjacent points by a line to

form an approximate boundary of the human brain (Figure 4.2(b)).

ωγ−1C0 D

γtMx(x, y, t) = Dµ2(β−1)

(∂2βMx(x,y,t)

∂|x|2β+ ∂2βMx(x,y,t)

∂|y|2β

)+λ1(t)My(x, y, t), (x, y, t) ∈ Ω× (0, T ],

ωγ−1C0 D

γtMy(x, y, t) = Dµ2(β−1)

(∂2βMy(x,y,t)

∂|x|2β+

∂2βMy(x,y,t)

∂|y|2β

)−λ1(t)Mx(x, y, t), (x, y, t) ∈ Ω× (0, T ],

Mx(x, y, 0) = 0, My(x, y, 0) = 100, x, y ∈ Ω,

Mx(x, y, t)|∂Ω = 0, My(x, y, t)|∂Ω = 0, (x, y, t) ∈ ∂Ω× [0, T ].

(4.28)

Here, we choose ω = 2, D = 1× 10−3, µ = 15, λ1(t) = t, T = 50, τ = 1/20 to observe the

behaviour of the transverse magnetization |Mxy(x, y, t)| =√Mx(x, y, t)2 +My(x, y, t)2.

The related numerical scheme for the system (4.28) is given as follows.

Assume that Xnh ∈ Vh and Y n

h ∈ Vh are the approximations of Mx(x, y, t) and My(x, y, t)

at t = tn, respectively. Then the fully discrete scheme associated with the variational

form of (4.28) is: find Xnh ∈ Vh and Y n

h ∈ Vh for each t = tn (n = 1, 2, ..., N) such that(∇γtXn

h , vh)Ω +A(Xnh , vh)Ω = λ2(tn)(Y n, vh)Ω, ∀vh ∈ Vh,

(∇γt Y nh , vh)Ω +A(Y n

h , vh)Ω = −λ2(tn)(Xn, vh)Ω, ∀vh ∈ Vh,(4.29)

91 Chapter 4

where λ2(tn) = λ1(tn)ωγ−1 . Substituting Xn

h =∑Np

i=1Xni li(x, y) and Y n

h =∑Np

i=1 Yni li(x, y)

into (4.29) with vh = lj(x, y) leads to

Np∑i=1

Xni

[(li, lj)Ω + ω0A(li, lj)Ω

]− ω0λ2(tn)

Np∑i=1

Y ni (li, lj)Ω

= −Np∑i=1

n−1∑k=1

bnkXki (lk, lj)Ω − bn0 (X0, lj)Ω,

Np∑i=1

Y ni

[(li, lj)Ω + ω0A(li, lj)Ω

]+ ω0λ2(tn)

Np∑i=1

Xni (li, lj)Ω

= −Np∑i=1

n−1∑k=1

bnkYki (lk, lj)Ω − bn0 (Y 0, lj)Ω.

(4.30)

where ω0 = τγΓ(2− γ). Eq.(4.30) can be written in matrix form as(M + ω0B)Xn − ω0λ2(tn)MY n = −M

n−1∑k=1

bnkXk −G0

X ,

ω0λ2(tn)MXn + (M + ω0B)Y n = −Mn−1∑k=1

bnkYk −G0

Y .

(4.31)

where M is the mass matrix with elements Mij = (lj , li)Ω, B is the stiffness matrix

with elements Bij = A(lj , li)Ω, Xn = [Xn1 , X

n2 , ..., X

nNp

]T , Y n = [Y n1 , Y

n2 , ..., Y

nNp

]T . The

vectors G0X and G0

Y are given by G0X = bn0 [(X0, l1)Ω, (X

0, l2)Ω, ..., (X0, lNp)Ω]T and G0

Y =

bn0 [(Y 0, l1)Ω, (Y0, l2)Ω, ..., (Y

0, lNp)Ω]T respectively. Finally, Eq.(4.31) can be recast into

the form

(M + ω0B −ω0λ2(tn)M

ω0λ2(tn)M M + ω0B

)(Xn

Y n

)=

−Mn−1∑k=1

bnkXk −G0

X

−Mn−1∑k=1

bnkYk −G0

Y

.

This linear system is then solved using general iterative methods.

Figure 4.9 shows the different triangular unstructured meshes used to partition the domainΩ.

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

0

0.2

0.4

0.6

0.8

1

1.2

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

0

0.2

0.4

0.6

0.8

1

1.2

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

0

0.2

0.4

0.6

0.8

1

1.2

Figure 4.9: The unstructured triangular meshes used in the calculation of the humanbrain-like domain for h = 0.11624, 0.06943 and 0.03278

Chapter 4 92

Figure 4.10 displays the solution behaviour for Mx(x, y, t), My(x, y, t) at the randomly

chosen point (x∗, y∗) = (0.5702, 0.8548) with different values of γ. We observe that the

effects of γ on the solution behaviour is significant and the smaller γ is, the faster it decays

from (0, 100) to (0, 0).

Mx(x∗, y∗, t)

-80 -60 -40 -20 0 20 40 60 80 100

My(x

∗,y∗,t)

-100

-80

-60

-40

-20

0

20

40

60

80

100

γ = 0.99

Mx(x∗, y∗, t)

-100 -80 -60 -40 -20 0 20 40 60 80 100

My(x

∗,y∗,t)

-100

-80

-60

-40

-20

0

20

40

60

80

100

γ = 0.90

Mx(x∗, y∗, t)

-100 -80 -60 -40 -20 0 20 40 60 80 100

My(x

∗,y∗,t)

-100

-80

-60

-40

-20

0

20

40

60

80

100

γ = 0.80

Figure 4.10: Plots of Mx(x, y, t), My(x, y, t) at point (x∗, y∗) = (0.5702, 0.8548) for differ-ent γ with β = 1.0, h = 0.06943 on the human brain-like domain

Figure 4.11 highlights a new finding of the effect of β on the solution behaviour, which

we believe is an original contribution to the literature. Although not obvious, the effects

of β on the solution behaviour is similar to that of γ, which can be reflected clearly in

Figure 4.12(b). Figure 4.12 exhibits the normalized decay of the transverse magnetization

versus t at the point (x∗, y∗) = (0.5702, 0.8548) for different γ and β. We conclude that

the smaller γ is, the sharper the decay rate is at the beginning time, but more time-

consuming in the process of decaying to 0 at latter times, which conforms with Figure

4.10. The effects of β is similar, which are aligned with Figure 4.11.

Mx(x∗, y∗, t)

-100 -80 -60 -40 -20 0 20 40 60 80 100

My(x

∗,y∗,t)

-100

-80

-60

-40

-20

0

20

40

60

80

100

β = 0.60

Mx(x∗, y∗, t)

-100 -80 -60 -40 -20 0 20 40 60 80 100

My(x

∗,y∗,t)

-100

-80

-60

-40

-20

0

20

40

60

80

100

β = 0.80

Mx(x∗, y∗, t)

-100 -80 -60 -40 -20 0 20 40 60 80 100

My(x

∗,y∗,t)

-100

-80

-60

-40

-20

0

20

40

60

80

100

β = 1.00

Figure 4.11: Plots of Mx(x, y, t), My(x, y, t) at point (x∗, y∗) = (0.5702, 0.8548) for differ-ent β with γ = 0.99, h = 0.06943 on the human brain-like domain

t

0 5 10 15 20 25 30 35 40 45 50

|Mxy/M

0|

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ=0.99γ=0.90γ=0.80

t

0 5 10 15 20 25 30 35 40 45 50

|Mxy/M

0|

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β=1.0β=0.8β=0.6

Figure 4.12: Normalized decay of the transverse magnetization versus t at point (x∗, y∗) =(0.5702, 0.8548) for different γ (with fixed β = 1.0) and β (with fixed γ = 0.99)

93 Chapter 4

Finally, we choose a very fine grid mesh (h = 0.03278) to observe the discrete L2 error of

Mxy for different h. The corresponding numerical results are given in Table 4.5 exhibiting

the convergence order of O(h2), which shows the effectiveness of the method on the

coupled fractional Bloch-Torrey equation.

Table 4.5: The L2 error and convergence order of Mxy for different h with β = 0.8,γ = 0.95, τ = 1/20 at t = 50

Ne h L2 error Order

221 1.3485E-01 4.0010E-04 –882 6.9427E-01 9.9154E-05 2.101777 5.1856E-02 5.5050E-05 2.02

Example 4.6.4 To further demonstrate the flexibility and effectiveness of our method,

we consider the following 2D-TSRFDE without a source term on a multiply-connected

domain (see Figure 4.13)C0 D

γt u(x, y, t) = ∂2αu(x,y,t)

∂|x|2α + ∂2βu(x,y,t)

∂|y|2β, (x, y, t) ∈ Ω× (0, T ],

u(x, y, 0) = 10 cos(π2xy), (x, y) ∈ Ω,

u(x, y, t) = 0, (x, y, t) ∈ ∂Ω× [0, T ].

Figure 4.14 illustrates the behaviour of u(x, y, t) at different times t = 0.2, 0.4, 0.8 ona multiply-connected domain, which decays with increasing time. We also choose a veryfine grid mesh (h = 0.0693) to observe the L2 error of u(x, y, t) for different h. Thecorresponding numerical results are given in Table 4.6 exhibiting the convergence orderof O(h2), which is in agreement with the analysis presented in Section 4.5.

x

-1 -0.5 0 0.5 1

y

-1

-0.5

0

0.5

1

Figure 4.13: The illustration of a multiply-connected domain with triangular partition

Table 4.6: The L2 error and convergence order of h for α = β = 0.8, γ = 0.9, τ = 1/1000at t = 1.0 on the multiply-connected domain

Ne h L2 error Order

418 1.7470E-01 3.4886E-03 –714 1.3974E-01 2.0598E-03 2.361710 8.7473E-02 7.1365E-04 2.26

Chapter 4 94

1

0.5

x

0

-0.5

t = 0.2

-1-1

-0.5

0

y

0.5

0.4

0.3

0.5

0.2

0.1

0

0.7

0.6

1

u(x,y,t)

1

0.5

x

0

-0.5

t = 0.4

-1-1

-0.5

0

y

0.5

0.4

0.3

0.5

0.2

0.1

0

0.7

0.6

1

u(x,y,t)

1

0.5

x

0

-0.5

t = 0.8

-1-1

-0.5

0

y

0.5

0.4

0.3

0.5

0.2

0.1

0

0.7

0.6

1

u(x,y,t)

Figure 4.14: The diffusion profiles of u(x, y, t) on a multiply-connected domain at differentt with γ = 0.9, α = β = 0.80, τ = 1/1000 at t = 1.0

4.7 Conclusions

In this chapter, we considered the Galerkin FEM to a class of two-dimensional time-

space Riesz fractional diffusion equation on irregular convex domains. We partitioned the

irregular convex domain into a sum of unstructured triangular meshes. Then utilising

FEM, we obtained the variation formulation of the problem and the associated discrete

scheme with the accuracy of O(τ2−γ + h2). Furthermore, we reduced the computation of

inner products from the whole domain Ω to a single triangular element and evaluated it

approximately by the Gauss quadrature technique. Moreover, we derived a second order

temporal numerical scheme for the problem. Finally, numerical examples on irregular

convex domains were conducted, which verified the effectiveness and reliability of the

method. Furthermore, with our numerical method, we are able to exhibit the effects of

the time and space fractional indices for the coupled two-dimensional fractional Bloch-

Torrey equation. We concluded that the numerical method can be extended to other

arbitrarily shaped convex domains and even some non-convex domains. In future work,

we shall investigate the FEM to other fractional problems on irregular convex domains,

such as the two-dimensional FDE with variable coefficients.

Chapter 5

An unstructured mesh control volume method for two-dimensional

space fractional diffusion equations with variable coefficients on

convex domains

5.1 Introduction

In the past two decades, fractional differential equations have been applied in many fields

of science [60, 80, 128, 174, 220, 230, 274], in which space fractional diffusion equations

are used to model the anomalous transport of solute in groundwater hydrology [148, 181].

For space fractional diffusion equations with constant coefficients, analytical solutions can

be obtained by utilising the Fourier transform methods. However, many practical prob-

lems involve variable coefficients [9, 38], in which the diffusion velocity can vary over the

solution domain. The work involving space fractional diffusion equations with variable co-

efficients is numerous. Meerschaert et al. [181, 183] considered the finite difference method

for the one-dimensional one-sided and two-sided space fractional diffusion equations with

variable coefficients, respectively. Zhang et al. [284] explored the homogeneous space-

fractional advection-dispersion equation with space-dependent coefficients. Ding et al.

[63] presented the weighted finite difference methods for a class of space fractional partial

differential equations with variable coefficients. Moroney and Yang [194, 195] proposed

some fast preconditioners for the numerical solution of a class of two-sided nonlinear space-

fractional diffusion equations with variable coefficients. Chen and Deng [46] discussed the

alternating direction implicit method to solve a two-dimensional two-sided space fractional

convection-diffusion equation on a finite domain. Wang and Zhang [253] developed a high-

accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem

of a one-sided variable-coefficient conservative fractional diffusion equation. Feng et al.

[77] proposed the finite volume method for a two-sided space-fractional diffusion equation

with variable coefficients. Chen et al. [49] considered an inverse problem for identify-

ing the fractional derivative indices in a two-dimensional space-fractional nonlocal model

with variable diffusivity coefficients. Jia and Wang [110] presented a fast finite volume

method for conservative space-fractional diffusion equations with variable coefficients. In

[81], Feng et al. presented a new second order finite difference scheme for a two-sided

space-fractional diffusion equation with variable coefficients.

In fact, many mathematical models and problems from science and engineering must be

computed on irregular domains and therefore seeking effective numerical methods to solve

these problems on such domains is important. Although existing numerical methods for

fractional diffusion equations are numerous [20, 79, 88, 116, 138, 149, 165, 168, 238, 278,

297, 298], most of them are limited to regular domains and uniform meshes. Research

95

Chapter 5 96

involving unstructured meshes and irregular domains is sparse. Yang et al. [261] proposed

the finite volume scheme for a two-dimensional space-fractional reaction-diffusion equation

based on the fractional Laplacian operator −(−∇2)α2 , which was computed using unstruc-

tured triangular meshes on a unit disk. Burrage et al. [28] developed some techniques for

solving fractional-in-space reaction diffusion equations using the finite element method on

both structured and unstructured grids. Qiu et al. [210] developed the nodal discontinu-

ous Galerkin method for fractional diffusion equations on a two-dimensional domain with

triangular meshes. Liu et al. [157] presented the semi-alternating direction method for

a two-dimensional fractional FitzHugh-Nagumo monodomain model on an approximate

irregular domain. Qin et al. [208] also used the implicit alternating direction method to

solve a two-dimensional fractional Bloch-Torrey equation using an approximate irregular

domain. Karaa et al. [120] proposed a finite volume element method implemented on

an unstructured mesh for approximating the anomalous subdiffusion equations with a

temporal fractional derivative. Yang et al. [262] established the unstructured mesh finite

element method for the nonlinear Riesz space fractional diffusion equations on irregular

convex domains. Fan et al. [72] extended the unstructured mesh finite element method

developed by Yang et al. [262] to the time-space fractional wave equation. Feng et al.

[82] investigated the unstructured mesh finite element method for a two-dimensional time-

space Riesz fractional diffusion equation on irregular arbitrarily shaped convex domains

and a multiply-connected domain. Le et al. [134] studied the finite element approxima-

tion for a time-fractional diffusion problem on a domain with a re-entrant corner. To the

best of our knowledge, the control volume finite element method (see Carr et al. [35] for

an illustration of the method applied to wood drying) has not been generalised to allow

the solution of space fractional diffusion equations with variable coefficients.

In this paper, we will consider the unstructured mesh control volume method for the

following two-dimensional space fractional diffusion equation with variable coefficients

(2D-SFDE-VC) [49] on an arbitrarily shaped convex domain:

∂u(x, y, t)

∂t=∂

∂x

[K1(x, y, t)

∂αu(x, y, t)

∂xα−K2(x, y, t)

∂αu(x, y, t)

∂(−x)α

]+∂

∂y

[K3(x, y, t)

∂βu(x, y, t)

∂yβ−K4(x, y, t)

∂βu(x, y, t)

∂(−y)β

]+f(x, y, t), (x, y, t) ∈ Ω× (0, T ], (5.1)

subject to the initial condition

u(x, y, 0) = φ(x, y), (x, y) ∈ Ω, (5.2)

and boundary condition

u(x, y, t) = 0, (x, y, t) ∈ ∂Ω× [0, T ], (5.3)

97 Chapter 5

where 0 < α, β < 1, Ki(x, y, t) ≥ 0, i = 1, 2, 3, 4, f(x, y, t) and φ(x, y) are assumed to be

two known smooth functions. When the solution domain is rectangular Ω = (a, b)×(c, d),

we define the Riemman-Liouville fractional derivative as [203]:

∂αu(x, y, t)

∂xα= aD

αxu(x, y, t) =

1

Γ(1− α)

∂

∂x

∫ x

a(x− s)−αu(s, y, t) ds,

∂αu(x, y, t)

∂(−x)α= xD

αb u(x, y, t) =

−1

Γ(1− α)

∂

∂x

∫ b

x(s− x)−αu(s, y, t) ds,

∂βu(x, y, t)

∂yβ= cD

βyu(x, y, t) =

1

Γ(1− β)

∂

∂y

∫ y

c(y − s)−βu(x, s, t) ds,

∂βu(x, y, t)

∂(−y)β= yD

βdu(x, y, t) =

−1

Γ(1− β)

∂

∂y

∫ d

y(s− y)−βu(x, s, t) ds.

When the boundary of the solution domain is nonconstant or curved, for example a

a(y) b(y)

c(x)

d(x)

Figure 5.1: An illustration of a solution domain with curved boundary

convex domain shown in Figure 5.1 with left boundary a(y), right boundary b(y), lower

boundary c(x) and upper boundary d(x), we define the Riemman-Liouville fractional

derivative as [82]:

∂αu(x, y, t)

∂xα= a(y)D

αxu(x, y, t) =

1

Γ(1− α)

∂

∂x

∫ x

a(y)(x− s)−αu(s, y, t) ds,

∂αu(x, y, t)

∂(−x)α= xD

αb(y)u(x, y, t) =

−1

Γ(1− α)

∂

∂x

∫ b(y)

x(s− x)−αu(s, y, t) ds,

∂βu(x, y, t)

∂yβ= c(x)D

βyu(x, y, t) =

1

Γ(1− β)

∂

∂y

∫ y

c(x)(y − s)−βu(x, s, t) ds,

∂βu(x, y, t)

∂(−y)β= yD

βd(x)u(x, y, t) =

−1

Γ(1− β)

∂

∂y

∫ d(x)

y(s− y)−βu(x, s, t) ds.

Chapter 5 98

Remark 5.1.1 When Ki(x, y, t) i = 1, 2, 3, 4 take the special form

K1(x, y, t) = K2(x, y, t) = − Kx

2 cos π(1+α)2

,

K3(x, y, t) = K4(x, y, t) = − Ky

2 cos π(1+β)2

,

Eq.(5.1) can be written as the following Riesz space fractional diffusion equation [157, 262]

∂u(x, y, t)

∂t= Kx

∂1+αu(x, y, t)

∂|x|1+α+Ky

∂1+βu(x, y, t)

∂|y|1+β+ f(x, y, t), (5.4)

where

∂1+αu(x, y, t)

∂|x|1+α= − 1

2 cos π(1+α)2

[∂1+αu(x, y, t)

∂x1+α+∂1+αu(x, y, t)

∂(−x)1+α

],

∂1+βu(x, y, t)

∂|y|1+β= − 1

2 cos π(1+β)2

[∂1+βu(x, y, t)

∂y1+β+∂1+βu(x, y, t)

∂(−y)1+β

].

One important application of Eq.(5.4) is in the study of cardiac arrhythmias. In two

dimensions, the fractional FitzHugh-Nagumo monodomain model can be rewritten as a

two-dimensional Riesz space fractional reaction-diffusion model, which can be used to

describe the propagation of the electrical potential in heterogeneous cardiac tissue [23,

157]. This electrophysiological model of the heart can describe how electrical currents flow

through the heart controlling its contraction and can be used to ascertain the effects of

certain drugs designed to treat heart problems.

The major contribution of this paper is as follows.

• Different from [120] and [261], we consider the control volume method for the two-

dimensional space fractional diffusion equation with variable coefficients, in which

the space fractional operator is either the Riemman-Liouville fractional derivative

or Riesz space fractional derivative. To the best of our knowledge, this is a new

contribution to the literature.

• We propose a novel technique utilizing the control volume method implemented with

an unstructured triangular mesh to deal with the space fractional derivative on an

irregular convex domain, which we believe provides a very flexible solution strategy

because our considered solution domain can be arbitrarily convex. Compared to

the finite difference method in [157, 208], our method requires fewer grid nodes to

generate the meshes in the solution domain partition.

• For the methods considered in this paper, we construct the control volumes using

triangular meshes and transform the problem (5.1) from the solution domain to a

single control volume. Then we integrate problem (5.1) over an arbitrary control

volume and change the control volume integral to a line integral over the control

volume faces, which is approximated by the midpoint approximation. Moreover,

we utilise the linear basis function to approximate the fractional derivatives at the

99 Chapter 5

midpoints of the control volume faces, in which some numerical techniques are used

to handle the non-locality of the fractional derivative of the basis function.

• We explore the property of the stiffness matrix generated by the integral of space

fractional derivative. We find that the stiffness matrix is sparse and not regular.

Specially, the more small the maximum edge of the triangulation is, the more sparse

of the stiffness matrix becomes. Therefore, we choose a suitable sparse storage for-

mat for the stiffness matrix and utilise the bi-conjugate gradient stabilized method

(Bi-CGSTAB) iterative method to solve the linear system, which is more efficient

than using the Gaussian elimination method.

• We present several examples to verify our method, in which we make a comparison

of our method with the finite element method proposed in [262] for solving the Riesz

space fractional diffusion equation (5.4) on a circular domain. In [262], the authors

develop an algorithm to form the stiffness matrix and derive the bilinear operator

as

A(u, v) =Kx

2 cos π(1+α)2

(a(y)D

(1+α)2

x u, xD(1+α)

2

b(y) v)

+(xD

(1+α)2

b(y) u, a(y)D(1+α)

2x v

)+

Ky

2 cos π(1+β)2

(c(x)D

(1+β)2

y u, yD(1+β)

2

d(x) v)

+(yD

(1+β)2

d(x) u, c(x)D(1+β)

2y v

).

The bilinear form involves eight fractional derivative terms and the approximation

of two-fold multiple integrals, which are approximated by Gauss quadrature. While

for the control volume method, we use the following form to generate the stiffness

matrix form,

Kx

2 cos π(1+α)2

∮Γi

[∂αu(x, y, t)

∂xα− ∂αu(x, y, t)

∂(−x)α

]dy

− Ky

2 cos π(1+β)2

∮Γi

[∂βu(x, y, t)

∂yβ− ∂βu(x, y, t)

∂(−y)β

]dx,

in which we only need to calculate four fractional derivative terms and the approx-

imation of line integrals. The numerical results demonstrate that our method can

reduce CPU time significantly while retaining the same accuracy and approximation

property as the finite element method. The numerical results also illustrate that

our method is effective and reliable and can be applied to problems on arbitrarily

convex domains.

The outline of this chapter is as follows. In Section 5.2, the unstructured mesh control

volume method for the problem (5.1) is proposed and the full implementation details

are provided. Then the property of the stiffness matrix is explored and a fast iterative

solver is developed for the linear system. In Section 5.3, several numerical examples

are presented to verify the effectiveness of the method and comparisons are made with

existing methods to highlight its computational performance. Finally, some conclusions

of the work are drawn.

Chapter 5 100

5.2 Control volume finite element method

In this section, we will generalise the control volume method to solve Eq.(5.1), placing par-

ticular emphasis on the way the Riemman-Liouville fractional derivatives are discretised

in space. Firstly, we divide the solution domain Ω into a number of regular triangular re-

gions. Let Th denote this triangulation and h be the maximum diameter of the triangular

elements. Then we introduce the control volumes, which are constructed as follows. Let

Mh be a set of vertice,

Mh = Pi : Pi is a vertex of the element K ∈ Th and Pi ∈ Ω,

and M0h be the set of interior nodes in Th. We denote P0 as the interior node of the

triangulation Th and Pi (i = 1, 2, · · · ,m) as its adjacent nodes (see Figure 5.2 withm = 6).

P0

P1

P2

P3

P4

P5 P6

S1

S2

S3

S4

S5

S6

Q1

Q2

Q3

Q4

Q5

Q6K∗

P0

Figure 5.2: An illustration of a control volume

Let Si (i = 1, 2, · · · ,m) be the midpoints of the line segments P0Pi andQi (i = 1, 2, · · · ,m)

the barycenters of the triangle ∆P0PiPi+1 with Pm+1 = P1. The control volume K∗P0is

constructed by joining successively S1, Q1, · · · , Sm, Qm, S1 (see Figure 5.2). We call

the line segments SiQi and QiSi+1 (i = 1, 2, · · · ,m and Sm+1 = S1) control volume faces.

Consequently, each of the triangular elements is divided into three sub-domains by these

control surfaces. These quadrilateral shapes are called sub-control volumes and are illus-

trated in Figure 5.2 (for example, the quadrilateral S1Q1S2P0). Thus, a control volume

consists of the sum of all neighbouring sub-control volumes that surround the given node

P0. The control volume is polygonal in shape and can be assembled in a straightforward

and efficient manner at the element level. The flow across each control surface must be

determined by an integral. Therefore, the finite volume method discretization process is

initiated by utilising the integrated form of Eq.(5.1).

101 Chapter 5

Integrating (5.1) over an arbitrary control volume Vi (i = 1, 2, · · · , Np), yields∫Vi

∂u(x, y, t)

∂tdVi =

∫Vi

∂

∂x

[K1(x, y, t)

∂αu(x, y, t)

∂xα−K2(x, y, t)

∂αu(x, y, t)

∂(−x)α

]dVi

+

∫Vi

∂

∂y

[K3(x, y, t)

∂βu(x, y, t)

∂yβ−K4(x, y, t)

∂βu(x, y, t)

∂(−y)β

]dVi

+

∫Vi

f(x, y, t) dVi. (5.5)

Utilising a lumped mass approach for the time derivative and source term and applying

Green’s theorem to the other two integrals terms, gives

∆Vi∂u(x, y, t)

∂t

∣∣∣∣(xi,yi)

=

∮Γi

[K1(x, y, t)

∂αu(x, y, t)

∂xα−K2(x, y, t)

∂αu(x, y, t)

∂(−x)α

]dy

−∮

Γi

[K3(x, y, t)

∂βu(x, y, t)

∂yβ−K4(x, y, t)

∂βu(x, y, t)

∂(−y)β

]dx

+∆Vif(xi, yi, t), (5.6)

where Γi is the boundary of control volume Vi. We assume the finite volume integrationis an anticlockwise traversal and the outward unit normal surface vector to the controlsurface is shown in Figure 5.3 with ∆x = xb − xa and ∆y = yb − ya.

∆y

∆x

(xa, ya)

(xb, yb)

ni

Figure 5.3: A control volume face and the outward normal unit vector

Denote ∆Vi and ∆Vij the area of the control volume and the sub-control volume sur-

rounding the point (xi, yi), then we have

∆Vi =

mi∑j=1

∆Vij ,

where mi is the total number of sub-control volumes that make up the control volume

associated with the node i. The integral term on the right-hand side of Eq.(5.1) is a

line integral, which can be approximated by the midpoint approximation for each control

Chapter 5 102

surface. Hence, the first integral term in Eq.(5.6) can be rewritten as∮Γi

[K1(x, y, t)

∂αu(x, y, t)

∂xα−K2(x, y, t)

∂αu(x, y, t)

∂(−x)α

]dy

=

mi∑j=1

2∑r=1

[K1(x, y, t)

∂αu(x, y, t)

∂xα−K2(x, y, t)

∂αu(x, y, t)

∂(−x)α

]∣∣∣∣(xr,yr)

∆yij,r, (5.7)

where (xr, yr) is the mid-point of the control face (CF) (see Figure 5.4).

CF2

CF1

(xr, yr)

Figure 5.4: The illustration of control faces with mid-points

Similarly, for the second integral term in Eq.(5.6), we have∮Γi

[K3(x, y, t)

∂βu(x, y, t)

∂yβ−K4(x, y, t)

∂βu(x, y, t)

∂(−y)β

]dx

=

mi∑j=1

2∑r=1

[K3(x, y, t)

∂βu(x, y, t)

∂yβ−K4(x, y, t)

∂βu(x, y, t)

∂(−y)β

]∣∣∣∣(xr,yr)

∆xij,r. (5.8)

Substituting Eqs.(5.7) and (5.8) into (5.6), we obtain

∆Vi∂u(x, y, t)

∂t

∣∣∣∣(xi,yi)

=

mi∑j=1

2∑r=1

[K1(x, y, t)

∂αu(x, y, t)

∂xα−K2(x, y, t)

∂αu(x, y, t)

∂(−x)α

]∣∣∣∣(xr,yr)

∆yij,r

−mi∑j=1

2∑r=1

[K3(x, y, t)

∂βu(x, y, t)

∂yβ−K4(x, y, t)

∂βu(x, y, t)

∂(−y)β

]∣∣∣∣(xr,yr)

∆xij,r

+∆Vif(xi, yi, t). (5.9)

To discretise the time derivative in Eq.(5.9) at t = tn, we use the backward Euler difference

scheme

∂u(x, y, tn)

∂t=u(x, y, tn)− u(x, y, tn−1)

τ+O(τ). (5.10)

In the following, we discuss the spatial discretisation of u(x, y, tn). We consider the

computation process for piecewise linear polynomials on the triangular element ep, p =

103 Chapter 5

1, 2, ..., Ne, where Ne is the total number of triangles. Then, within element ep, the field

function up(x, y) can be written as

up(x, y) =

3∑j=1

uj ϕj(x, y) +O(h2),

where the triangle vertices are numbered in a counter-clockwise order as 1, 2, 3 and the

basis function ϕj(x, y) is defined as

ϕj(x, y)∣∣∣(x,y)∈ep

=1

2∆ep

(aj x+ bj y + cj), ϕj(x, y)∣∣∣(x,y)/∈ep

= 0,

a1 = y2 − y3, a2 = y3 − y1, a3 = y1 − y2,

b1 = x3 − x2, b2 = x1 − x3, b3 = x2 − x1,

c1 = x2y3 − x3y2, c2 = x3y1 − x1y3, c3 = x1y2 − x2y1,

where ∆ep is the area of triangle element p. It is well-known that

ϕj(xi, yi) = δij , i, j = 1, 2, 3,

where δ is the Kronecker function. With these local field functions and basis functions,

we can obtain a global approximation of u(x, y) for the whole triangulation:

u(x, y) =

Np∑k=1

uk lk(x, y) +O(h2),

where lk(x, y) is the new basis function whose support domain is Ωek (see Figure 5.5 the

green polygonal domain) and Np is the total number of vertices on the convex domain

Ω. Now, we denote uh(x, y, tn) as the approximation solution of u(x, y, tn) and write

uh(x, y, tn) in the form

uh(x, y, tn) =

Np∑k=1

unk lk(x, y), (5.11)

where unk are the coefficients that are to be solved for. Substituting Eqs.(5.10) and (5.11)

into Eq.(5.9), we discretise equation (5.9) at t = tn as follows:

∆Vi

Np∑k=1

unk − un−1k

τlk(xi, yi)

=

Np∑k=1

mi∑j=1

2∑r=1

unk

[K1(x, y, t)

∂αlk(x, y)

∂xα−K2(x, y, t)

∂αlk(x, y)

∂(−x)α

]∣∣∣∣(xr,yr)

∆yij,r

−Np∑k=1

mi∑j=1

2∑r=1

unk

[K3(x, y, t)

∂βlk(x, y)

∂yβ−K4(x, y, t)

∂βlk(x, y)

∂(−y)β

]∣∣∣∣(xr,yr)

∆xij,r

+∆Vif(xi, yi, tn). (5.12)

Chapter 5 104

Using the fact that

lk(xi, yi) =

1, i = k,

0, i 6= k,

we obtain

∆Viuni − u

n−1i

τ

=

Np∑k=1

mi∑j=1

2∑r=1

unk

[K1(x, y, t)

∂αlk(x, y)

∂xα−K2(x, y, t)

∂αlk(x, y)

∂(−x)α

]∣∣∣∣(xr,yr)

∆yij,r

−Np∑k=1

mi∑j=1

2∑r=1

unk

[K3(x, y, t)

∂βlk(x, y)

∂yβ−K4(x, y, t)

∂βlk(x, y)

∂(−y)β

]∣∣∣∣(xr,yr)

∆xij,r

+∆Vif(xi, yi, tn). (5.13)

Eq.(5.13) can be written in the following matrix form

AUn −Un−1

τ= MUn + AFn, (5.14)

where A =diag [∆V1,∆V2, . . . ,∆VNp ], Un = [un1 , un2 , . . . , u

nNp

]T , Fn = [f(x1, y1, tn), f(x2,

y2, tn), . . . , f(xNp , yNp , tn)]T . Rearranging we obtain

(A− τM)Un = AUn−1 + τAFn. (5.15)

To form matrix M, we need to calculate the fractional derivative of the basis function

lk(x, y). In the following, we focus on the calculation of ∂αlk(x,y)∂xα , ∂αlk(x,y)

∂(−x)α , ∂β lk(x,y)∂yβ

and

∂β lk(x,y)∂(−y)β

at (xr, yr). To evaluate ∂αlk(x,y)∂xα

∣∣(xr,yr)

and ∂αlk(x,y)∂(−x)α

∣∣(xr,yr)

, suppose that line

y = yr intersects nq points with the support domain Ωek of lk(x, y) (see Figure 5.5 withnq = 5).

k

e1

e2e3

e4

e5 e6

e7

(xr, yr)

x1 x2 x3 x4 x5a b

Figure 5.5: The illustration of line y = yr intersecting nq points with the support domainΩek of lk(x, y), where (xr, yr) locates out of Ωek

105 Chapter 5

Then we have

∂αlk(x, y)

∂xα

∣∣∣∣(xr,yr)

=∂αlk(x, yr)

∂xα

∣∣∣∣x=xr

,

∂αlk(x, y)

∂(−x)α

∣∣∣∣(xr,yr)

=∂αlk(x, yr)

∂(−x)α

∣∣∣∣x=xr

.

Using the important observation that

lk(x, yr) =

0, a ≤ x ≤ x1,

ϕk4(x, yr), x1 ≤ x ≤ x2,

ϕk3(x, yr), x2 ≤ x ≤ x3,

ϕk2(x, yr), x3 ≤ x ≤ x4,

ϕk1(x, yr), x4 ≤ x ≤ x5,

0, x5 ≤ x ≤ b,

where ϕkp(x, y) is the basis function of node k on the triangular element ep, we obtain

∂αlk(x, yr)

∂xα

∣∣∣∣x=xr

=

(1

Γ(1− α)

∂

∂x

∫ x

a(x− ξ)−αlk(ξ, yr)dξ

)∣∣∣∣x=xr

=

[1

Γ(1− α)

∂

∂x

(∫ x1

a+

∫ x2

x1

+

∫ x3

x2

+

∫ x4

x3

+

∫ x5

x4

+

∫ x

x5

)(x− ξ)−αlk(ξ, yr)dξ

]∣∣∣∣x=xr

=

[1

Γ(1− α)

∂

∂x

(∫ x2

x1

+

∫ x3

x2

+

∫ x4

x3

+

∫ x5

x4

)(x− ξ)−αlk(ξ, yr)dξ

]∣∣∣∣x=xr

. (5.16)

As lk(x, yr) is a linear function on each sub integral interval, Eq.(5.16) can be evaluat-

ed using integration by parts over each sub integral interval. For the right fractional

derivative of lk(x, yr) at (xr, yr), we obtain

∂αlk(x, yr)

∂(−x)α

∣∣∣∣x=xr

=

(−1

Γ(1− α)

∂

∂x

∫ b

x(ξ − x)−αlk(ξ, yr)dξ

)∣∣∣∣x=xr

= 0. (5.17)

Now we consider the case that point (xr, yr) is in the support domain Ωek of lk(x, y).

Suppose that line y = yr intersects nq points with the support domain Ωek (see Figure

5.6 with nq = 4). In this case, we have

lk(x, yr) =

0, a ≤ x ≤ x1,

ϕk5(x, yr), x1 ≤ x ≤ x2,

ϕk6(x, yr), x2 ≤ x ≤ x3,

ϕk7(x, yr), x3 ≤ x ≤ x4,

0, x4 ≤ x ≤ b.

Chapter 5 106

k

e1

e2e3

e4

e5 e6

e7

(xr, yr)

x1 x2 x3 x4a b

Figure 5.6: The illustration of line y = yr intersecting nq points with the support domainΩek of lk(x, y), where (xr, yr) locates in Ωek

Then

∂αlk(x, yr)

∂xα

∣∣∣∣x=xr

=

(1

Γ(1− α)

∂

∂x

∫ x

a(x− ξ)−αlk(ξ, yr)dξ

)∣∣∣∣x=xr

=

[1

Γ(1− α)

∂

∂x

(∫ x1

a+

∫ x2

x1

+

∫ x

x2

)(x− ξ)−αlk(ξ, yr)dξ

]∣∣∣∣x=xr

=

[1

Γ(1− α)

∂

∂x

(∫ x2

x1

+

∫ x

x2

)(x− ξ)−αlk(ξ, yr)dξ

]∣∣∣∣x=xr

, (5.18)

and

∂αlk(x, yr)

∂(−x)α

∣∣∣∣x=xr

=

(−1

Γ(1− α)

∂

∂x

∫ b

x(ξ − x)−αlk(ξ, yr)dξ

)∣∣∣∣x=xr

=

[−1

Γ(1− α)

∂

∂x

(∫ x3

x+

∫ x4

x3

+

∫ b

x4

)(ξ − x)−αlk(ξ, yr)dξ

]∣∣∣∣x=xr

=

[−1

Γ(1− α)

∂

∂x

(∫ x3

x+

∫ x4

x3

)(ξ − x)−αlk(ξ, yr)dξ

]∣∣∣∣x=xr

. (5.19)

If line y = yr intersects zero points with the support domain Ωek , then we have

∂αlk(x, yr)

∂xα

∣∣∣∣x=xr

= 0,∂αlk(x, yr)

∂(−x)α

∣∣∣∣x=xr

= 0. (5.20)

The calculation of ∂β lk(x,y)∂yβ

and ∂β lk(x,y)∂(−y)β

at (xr, yr) can be derived in a similar manner for

the y direction. Finally, we summarise the whole computation process in the following

algorithm (see Algorithm 4).

Remark 5.2.1 When the boundary of the solution domain is nonconstant or curved, all

of the above calculation is applicable as well.

107 Chapter 5

Algorithm 4 Unstructured mesh CVM for solving 2D-SFDE-VC

1: Partition the convex domain Ω with unstructured triangular elements ep and save theelement information (node number, coordinates, and element number );

2: for p = 1, 2, · · · , Ne do3: Find the barycenters of each triangular element ep, form the control faces, sub-

control volumes and save the sub-control volume information (the midpoint coor-dinates of each side of the triangular elements ep, the midpoint coordinates (xr, yr)of each control faces, etc.);

4: Calculate the areas of the sub-control volumes and control volumes, form matrixA;

5: for k = 1, 2, · · · , Np do6: Find the support domain Ωek ;7: Find the points of intersection by y = yr with Ωek and calculate

∂αlk(x,y)∂xα

∣∣(xr,yr)

,∂αlk(x,y)∂(−x)α

∣∣(xr,yr)

;

8: Find the points of intersection by x = xr with Ωer and calculate ∂β lk(x,y)∂yβ

∣∣(xr,yr)

,

∂β lk(x,y)∂(−y)β

∣∣(xr,yr)

;

9: end for10: Form the matrix M;11: Form the vector Fn;12: end for13: Solve the linear system (5.15) and obtain Un.

Here, we discuss the structure of matrix M. Firstly, the matrix M generated by scheme

(5.13) is sparse and not regular (see Figure 5.7). Then we explore the sparsity of matrix

M for different h. Table 5.1 shows the size and density (nonzero entries percentage) of

matrix M for different h where we can observe that as h decreases the density of matrix

M reduces significantly. We can infer that when h is small enough, the matrix M is

extremely sparse and this facilitates the use of a sparse matrix storage format to reduce

the memory usage of our computational method. Furthermore, we employ an efficient

sparse iterative solver Bi-CGSTAB [246] to solve the linear system (5.15) (see Algorithm

5), which is more efficient than using the Gaussian elimination method. The CPU time

comparison of the two methods is studied numerically in Example 5.3.1.

Table 5.1: The size and density of matrix M for different h on a square domain [0, 1]×[0, 1]

h Size Density

5.2693E-01 4×4 100%3.1123E-01 15×15 86.667%1.6759E-01 64×64 57.715%8.6682E-02 258×258 34.002%4.3719E-02 1115×1115 17.705%2.3063E-02 5255×5255 8.517%

Chapter 5 108

0 10 20 30 40 50 600

10

20

30

40

50

60

Figure 5.7: Sparsity pattern of matrix M for h = 1.6759× 10−1. The size of M is 64×64.Blue points indicate the nonzero entries

Algorithm 5 The Bi-CGSTAB algorithm

1: Define A0 = A− τM, use a sparse matrix storage format to store A0;2: In each time level tn, x0 = Un−1, b = AUn−1 + τAFn;3: Compute r0 = b−A0x0, r0 is an arbitrary vector, such that (r0, r0) 6= 0. We choose

r0 = r0;4: Let ρ0 = α0 = ω0 = 1, v0 = p0 = 0;5: for i = 1, 2, 3, · · · , do6: ρi = (r0, ri−1);7: β0 = (ρi/ρi−1)(αi−1/ωi−1);8: pi = ri−1 + β0(pi−1 − ωi−1vi−1);9: vi = A0pi, αi = ρi/(r0,vi);

10: s = ri−1 − αivi, t0 = A0s;11: ωi = (t0, s)/(t0, t0);12: xi = xi−1 + αipi + ωis;13: if xi is accurate enough then quit;14: ri = s− ωit0;15: end for16: Un = xi.

5.3 Discussion of Numerical Results

In this section, we provide some numerical examples to verify the effectiveness of our

method presented in Section 5.2. We adopt linear polynomials on triangles and define h

as the maximum length of the triangle edges. Ne is taken as the number of triangles in

Th. Here, the numerical computations were carried out using MATLAB R2014b on a Dell

desktop with configuration: Intel(R) Core(TM) i7-4790, 3.60 GHz and 16.0 GB RAM.

We use the following formula to calculate the convergence order:

Order =log(E(h1)/E(h2))

log(h1/h2).

109 Chapter 5

Example 5.3.1 Firstly, we consider the following 2D-SFDE-VC on a rectangular domain

∂u(x, y, t)

∂t=∂

∂x

[K1(x, y, t)

∂αu(x, y, t)

∂xα−K2(x, y, t)

∂αu(x, y, t)

∂(−x)α

]+∂

∂y

[K3(x, y, t)

∂βu(x, y, t)

∂yβ−K4(x, y, t)

∂βu(x, y, t)

∂(−y)β

]+f(x, y, t), (x, y, t) ∈ Ω× (0, T ],

subject to

u(x, y, 0) = x2(1− x)2y2(1− y)2, (x, y) ∈ Ω,

u(x, y, t) = 0, (x, y, t) ∈ ∂Ω× [0, T ],

where Ω = (0, 1)× (0, 1), T = 1,

f(x, y, t) = 2tx2(1− x)2y2(1− y)2 −[∂K1(x, y, t)

∂x· p(x, α) +K1(x, y, t) · p(x, 1 + α)

− ∂K2(x, y, t)

∂x· p(1− x, α) +K2(x, y, t) · p(1− x, 1 + α)

]y2(1− y)2(t2 + 1)

−[∂K3(x, y, t)

∂y· p(y, β) +K3(x, y, t) · p(y, 1 + β)− ∂K4(x, y, t)

∂y· p(1− y, β)

+K4(x, y, t) · p(1− y, 1 + β)]x2(1− x)2(t2 + 1),

p(z, r) =Γ(3)

Γ(3− r)z2−r − 2Γ(4)

Γ(4− r)z3−r +

Γ(5)

Γ(5− r)z4−r.

This is a two-dimensional anomalous diffusion model, which can describe anomalous trans-port in heterogeneous porous media and can be used to explain the region-scale anomalousdispersion with heavy tails [49].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.8: The rectangular domain partitioned by unstructured meshes with controlvolumes for h ≈ 3.1123× 10−1, 1.6759× 10−1, 8.6682× 10−2, 4.3719× 10−2, respectively

Chapter 5 110

The exact solution of this problem is given by u(x, y, t) = (t2 + 1)x2(1 − x)2y2(1 − y)2.Figure 5.8 shows the rectangular domain partitioned by unstructured triangular meshesand control volumes for different h. Here, we consider three different coefficient cases[81]: linear coefficients K1(x, y, t) = 2 − x, K2(x, y, t) = 2 + x, K3(x, y, t) = 2 − y,K4(x, y, t) = 2 + y, quadratic coefficients K1(x, y, t) = 2 − x2, K2(x, y, t) = 2 + x2,K3(x, y, t) = 2− y2, K4(x, y, t) = 2 + y2 and exponential coefficients K1(x, y, t) = 3− ex,K2(x, y, t) = 3 + ex, K3(x, y, t) = 3 − ey, K4(x, y, t) = 3 + ey. The numerical results aregiven in Tables 5.2 to 5.4. Table 5.2 illustrates the L2 error, L∞ error and correspondingconvergence order of h for the linear coefficient case for different α, β with τ = 10−3 att = 1. Tables 5.3 and 5.4 show the L2 error, L∞ error and corresponding convergenceorder of h for the quadratic coefficient case and exponential coefficient case, respectively.From these tables we can see that the convergence order of both the L2 error and L∞error is 2−maxα, β order [77] and the numerical results are in excellent agreement withthe exact solution, which demonstrates the effectiveness of the numerical method.

Table 5.2: The L2 error, L∞ error, convergence order and CPU time of h with τ = 10−3

for the linear coefficient case at t = 1

h L2 error Order L∞ error Order Time

3.1123E-01 3.5684E-04 – 1.4774E-03 – 4.90sα = 0.3 1.6759E-01 1.0880E-04 1.92 4.3735E-04 1.97 19.50sβ = 0.5 8.6682E-02 2.2391E-05 2.40 1.3895E-04 1.74 2.30min

4.3719E-02 6.9379E-06 1.71 3.7632E-05 1.91 28.42min

3.1123E-01 3.7935E-04 – 1.4827E-03 – 4.91sα = 0.4 1.6759E-01 1.2435E-04 1.80 4.2971E-04 2.00 19.98sβ = 0.8 8.6682E-02 2.5152E-05 2.42 1.3725E-04 1.73 2.36min

4.3719E-02 7.2675E-06 1.81 3.5722E-05 1.97 28.56min

3.1123E-01 3.9259E-04 – 1.3844E-03 – 4.91sα = 0.7 1.6759E-01 1.4100E-04 1.65 4.1957E-04 1.93 19.87sβ = 0.9 8.6682E-02 2.8670E-05 2.42 1.4117E-04 1.65 2.37min

4.3719E-02 7.5385E-06 1.95 3.3666E-05 2.09 28.47min

Table 5.3: The L2 error, L∞ error, convergence order and CPU time of h with τ = 10−3

for the quadratic coefficient case at t = 1

h L2 error Order L∞ error Order Time

3.1123E-01 3.1608E-04 – 1.3430E-03 – 4.97sα = 0.3 1.6759E-01 1.0064E-04 1.85 4.0906E-04 1.92 20.48sβ = 0.5 8.6682E-02 2.0661E-05 2.40 1.3852E-04 1.64 2.45min

4.3719E-02 6.2709E-06 1.74 3.7584E-05 1.91 28.69min

3.1123E-01 3.6299E-04 – 1.4108E-03 – 4.88sα = 0.4 1.6759E-01 1.2145E-04 1.77 4.1614E-04 1.97 20.51sβ = 0.8 8.6682E-02 2.4646E-05 2.42 1.3823E-04 1.67 2.46min

4.3719E-02 6.7517E-06 1.89 3.3858E-05 2.06 28.78min

3.1123E-01 3.8524E-04 – 1.3424E-03 – 4.97sα = 0.7 1.6759E-01 1.3952E-04 1.64 4.0669E-04 1.93 20.56sβ = 0.9 8.6682E-02 2.8522E-05 2.41 1.4126E-04 1.60 2.44min

4.3719E-02 7.1520E-06 2.02 3.1880E-05 2.17 28.68min

111 Chapter 5

We can also observe that with h decreasing, the CPU time grows considerably, which we

believe is mainly due to the non-locality of the fractional derivative of the basis function

and the computational cost to generate the matrix M. In addition, we give a comparison

between the Bi-CGSTAB and Gaussian elimination. In the Bi-CGSTAB solver, we set

10−10 as the stopping criterion and the maximum iteration number is 102. Table 5.5

displays the consumed CPU time of these two algorithms at t = 1 with τ = 10−3, α = 0.3,

β = 0.5, K1(x, y, t) = 2 − x, K2(x, y, t) = 2 + x, K3(x, y, t) = 2 − y, K4(x, y, t) = 2 + y

for different h. Compared to the Gaussian elimination, Bi-CGSTAB has significantly

reduced 90% of the computational time for h = 4.3719E − 02. Another advantage of

Bi-CGSTAB to be mentioned is that the average iteration number does not appear to

increase significantly as h decreases. Here, the average iteration number is approximately

10 regardless of the model dimensions. We conclude that the Bi-CGSTAB solver is more

efficient than Gaussian elimination for solving this problem.

Table 5.4: The L2 error, L∞ error, convergence order and CPU time of h with τ = 10−3

for the exponential coefficient case at t = 1

h L2 error Order L∞ error Order Time

3.1123E-01 5.1809E-04 – 1.9033E-03 – 4.97sα = 0.3 1.6759E-01 1.6296E-04 1.87 5.3973E-04 2.04 20.62sβ = 0.5 8.6682E-02 3.8817E-05 2.18 1.6032E-04 1.84 2.45min

4.3719E-02 1.1574E-05 1.77 4.8226E-05 1.76 28.46min

3.1123E-01 4.5022E-04 – 1.6750E-03 – 4.93sα = 0.4 1.6759E-01 1.4896E-04 1.79 1.0117E-04 2.01 20.52sβ = 0.8 8.6682E-02 3.4126E-05 2.24 4.8309E-04 1.84 2.45min

4.3719E-02 1.1238E-05 1.62 4.3016E-05 1.76 28.66min

3.1123E-01 4.2412E-04 – 1.4994E-03 – 4.93sα = 0.7 1.6759E-01 1.5286E-04 1.65 4.6520E-04 1.89 20.50sβ = 0.9 8.6682E-02 3.3401E-05 2.31 1.4533E-04 1.76 2.45min

4.3719E-02 1.0565E-05 1.68 4.0322E-05 1.87 28.56min

Table 5.5: Comparison of the consumed CPU time of Gaussian elimination versus Bi-CGSTAB method

Ne h Gauss elimination Bi-CGSTAB

44 3.1123E-01 4.90s 4.90s158 1.6759E-01 22.57s 19.50s578 8.6682E-02 5.39min 2.30min2356 4.3719E-02 5.48h 28.42min

Example 5.3.2 Next, we consider the following two-dimensional Riesz space fractional

diffusion equation on a circular domain, which can be used to describe the propagation

Chapter 5 112

of the electrical potential in heterogeneous cardiac tissue [23, 157, 262].∂u(x,y,t)

∂t = Kx∂1+αu(x,y,t)

∂|x|1+α +Ky∂1+βu(x,y,t)

∂|y|1+β+ f(x, y, t), (x, y, t) ∈ Ω× (0, T ],

u(x, y, 0) = (x2 + y2 − 1)2, (x, y) ∈ Ω,

u(x, y, t) = 0, (x, y, t) ∈ ∂Ω× [0, T ],

(5.21)

where Ω = (x, y)|x2 + y2 < 1, Kx = 1, Ky = 1, T = 1,

f(x, y, t) = −e−t(x2 + y2 − 1)2 +e−t

2 cos((1 + α)/2π)

[(f1(x, a0, α) + g1(x, b0, α)

)+ (2y2 − 2)

(f2(x, a0, α) + g2(x, b0, α)

)+ (y2 − 1)2

(f3(x, a0, α) + g3(x, b0, α)

)]+

e−t

2 cos((1 + β)/2π)

[(f1(y, c0, β) + g1(y, d0, β)

)+ (2x2 − 2)

(f2(y, c0, β) + g2(y, d0, β)

)+ (x2 − 1)2

(f3(y, c0, β) + g3(y, d0, β)

)],

a0 = −√

1− y2, b0 =√

1− y2, c0 = −√

1− x2, d0 =√

1− x2,

f1(x, a, α) = aD1+αx (x4), f2(x, a, α) = aD

1+αx (x2), f3(x, a, α) = aD

1+αx (1),

g1(x, b, α) = xD1+αb (x4), g2(x, b, α) = xD

1+αb (x2), g3(x, b, α) = xD

1+αb (1).

The exact solution is given by u(x, y, t) = e−t(x2 + y2 − 1)2.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 5.9: The unstructured meshes with control volumes for h ≈ 2.8917×10−1, 1.6444×10−1, 8.6550× 10−2, 4.5873× 10−2, respectively

113 Chapter 5

Figure 5.9 shows the circular domain partitioned by unstructured triangular meshes and

control volumes for different h. In [262], Yang et al. applied the Galerkin finite element

method for solving the two-dimensional Riesz space fractional diffusion equation with a

nonlinear source term on convex domains. They developed an algorithm to form the

stiffness matrix on triangular meshes, which can deal with space fractional derivatives

on any convex domain. Here, we will make a comparison between our method (CVM)

and Yang’s method (FEM) for solving the two-dimensional Riesz space fractional diffusion

equation (5.21) on a circular domain using the same triangular meshes. Firstly, we present

a comparison of the density of the two stiffness matrices generated by FEM and CVM

for different h in Table 5.6. We can see that with h decreasing the density of the two

stiffness matrices reduces significantly. Compared to the stiffness matrix generated by

FEM, the stiffness matrix generated by CVM is slightly more sparse. Next, we present a

comparison of the error and convergence. Table 5.7 displays the L2 error, L∞ error and

corresponding convergence order of h for different α, β with τ = 10−3 at t = 1 by applying

FEM. Table 5.8 highlights the error and convergence order by using FVM. We can see

that the accuracy of our method is similar to FEM, both of which are second order. Then,

we present a comparison of CPU time for the two methods in Table 5.9 both using the

Bi-CGSTAB solver. We choose α = β = 0.8 and τ = 10−3 at t = 1 to observe the running

time for different h. We observe that compared to the running time of FEM, CVM can

reduce the running time significantly, which illustrates that CVM is more effective for

solving the two-dimensional Riesz space fractional diffusion equation on convex domains.

This is mainly due to the bilinear form in [262] that involves eight fractional derivative

terms and the approximation of two-fold multiple integrals, which are approximated by

Gauss quadrature, while for CVM we only need to calculate four fractional derivative

terms and the approximation of line integrals. In addition, we give a comparison of the

exact solution u(x, y, t) and numerical solution uh(x, y, t) in Figure 5.10 and the error plot

of u(x, y, t)−uh(x, y, t) in Figure 5.11 for h = 4.5873×10−2, α = β = 0.8 with τ = 10−3 at

t = 1 by applying CVM. We can see that the numerical solution is in excellent agreement

with the exact solution, which demonstrates the effectiveness of our numerical method

again.

Table 5.6: The comparison of the density of stiffness matrix generated by FEM and CVMfor different h

Ne h Size FEM CVM

174 2.8917E-01 74× 74 65.413 % 55.332%570 1.6444E-01 260× 260 41.814 % 33.521%2310 8.6550E-02 1104× 1104 22.233 % 17.469%8744 4.5873E-02 4271× 4271 11.712 % 9.107%

Chapter 5 114

Table 5.7: The L2 error, L∞ error and convergence order of h for FEM with τ = 10−3 att = 1

FEM h L2 error Order L∞ error Order

2.8917E-01 6.7022E-03 – 5.8841E-03 –α = 0.80 1.6444E-01 2.0787E-03 2.07 2.8557E-03 1.28β = 0.80 8.6550E-02 5.2077E-04 2.16 8.1791E-04 1.95

4.5873E-02 1.3554E-04 2.12 2.3520E-04 1.96

2.8917E-01 6.9018E-03 – 5.5925E-03 –α = 0.70 1.6444E-01 2.1713E-03 2.05 2.7718E-03 1.24β = 0.90 8.6550E-02 5.4452E-04 2.16 7.9048E-04 1.95

4.5873E-02 1.4147E-04 2.12 2.2242E-04 2.00

Table 5.8: The L2 error, L∞ error and convergence order of h for CVM with τ = 10−3 att = 1

CVM h L2 error Order L∞ error Order

2.8917E-01 1.4782E-02 – 2.1786E-02 –α = 0.80 1.6444E-01 4.5014E-03 2.11 7.5230E-03 1.88β = 0.80 8.6550E-02 1.2275E-03 2.02 1.8279E-03 2.20

4.5873E-02 3.4069E-04 2.02 5.4557E-04 1.90

2.8917E-01 1.4950E-02 – 2.1864E-02 –α = 0.70 1.6444E-01 4.5530E-03 2.11 7.6462E-03 1.86β = 0.90 8.6550E-02 1.2566E-03 2.01 1.8659E-03 2.20

4.5873E-02 3.4898E-04 2.02 5.4606E-04 1.94

Table 5.9: The comparison of running time between FEM and CVM for different h withα = β = 0.80, τ = 10−3 at t = 1

Ne h FEM CVM

174 2.8917E-01 3.49 min 35.01 s570 1.6444E-01 12.90 min 2.63min2310 8.6550E-02 1.38 h 28.41min8744 4.5873E-02 17.89h 6.59h

1

0.5

x

0

Exact solution

-0.5

-1-1

-0.5

0

y

0.5

0.4

0

0.1

0.2

0.3

1

u(x,y,t=

1)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1

0.5

x

0

Numerical solution

-0.5

-1-1

-0.5

0

y

0.5

0.4

0

0.1

0.2

0.3

1

uh(x,y,t=

1)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 5.10: The comparison of the exact solution u(x, y, t) and numerical solutionuh(x, y, t) for h = 4.5873× 10−2, α = β = 0.8 with τ = 10−3 at t = 1

115 Chapter 5

1

0.5

x

0

-0.5

-1-1

-0.5

0

y

0.5

-6

-4

-2

0

2

1

×10-4

u−

uh

×10-4

-5

-4

-3

-2

-1

0

1

Figure 5.11: The error plot of u(x, y, t) − uh(x, y, t) for h = 4.5873 × 10−2, α = β = 0.8with τ = 10−3 at t = 1

Example 5.3.3 Finally, we consider the following 2D-SFDE-VC without a source term

on different convex domains

∂u(x, y, t)

∂t=∂

∂x

[K1(x, y, t)

∂αu(x, y, t)

∂xα−K2(x, y, t)

∂αu(x, y, t)

∂(−x)α

]+∂

∂y

[K3(x, y, t)

∂βu(x, y, t)

∂yβ−K4(x, y, t)

∂βu(x, y, t)

∂(−y)β

], (x, y, t) ∈ Ω× (0, T ],

subject to

u(x, y, 0) = 100, (x, y) ∈ Ω,

u(x, y, t) = 0, (x, y, t) ∈ ∂Ω× [0, T ].

where K1(x, y, t) = 2 − x, K2(x, y, t) = 2 + x, K3(x, y, t) = 2 − y, K4(x, y, t) = 2 + y,

T = 0.5.

Here, we choose α = β = 0.8 and τ = 10−3 to observe the diffusion behavior of u(x, y, t).Figure 5.12 shows the different diffusion profiles of u(x, y, t) at t = 0.5 on different convexdomains. We can see that the diffusive behaviour of u(x, y, t) is different on differentconvex domains, in which the diffusive velocity on domain 1 is the fastest and the diffusivevelocity on domain 4 is the slowest. We also can observe that our method is effective andis applicable for all these convex domains.

Chapter 5 116

0.5

x

0

-0.5-0.5

0

y

1

2

3

4

00.5

×10-10

uh(x,y,t)

×10-10

0

0.5

1

1.5

2

2.5

3

x

-0.5 0 0.5

y

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5 ×10-10

0

0.5

1

1.5

2

2.5

3

10.5

x

0-0.5

-1-1

-0.5

0

y

0.5

0.02

0.01

0.005

0

0.015

1

uh(x,y,t)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

x

-1 -0.5 0 0.5 1

y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

21.5

x

10.5

00

0.5

1

y

1.5

0.01

0.025

0.005

0

0.03

0.02

0.015

2

uh(x,y,t)

0

0.005

0.01

0.015

0.02

0.025

x

0 0.5 1 1.5 2

y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.005

0.01

0.015

0.02

0.025

32.5

y

21.5

10.5

03

2

x

1

0

0.5

0

2

1.5

1

2.5

uh(x,y,t)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

0 0.5 1 1.5 2 2.5 3

y

0

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

21.5

x

10.5

00

1

2

y

0.8

1

1.2

0

0.2

0.4

0.6

3

×10-3

uh(x,y,t)

×10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

0 0.5 1 1.5 2

y

0

0.5

1

1.5

2

2.5 ×10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5

x

0

-0.5-1

-0.5

0

y

0.5

4

0

1

2

3

1

×10-7

uh(x,y,t)

×10-7

0

0.5

1

1.5

2

2.5

3

x

-1 -0.5 0 0.5 1

y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 ×10-7

0

0.5

1

1.5

2

2.5

3

Figure 5.12: The diffusion profiles of u(x, y, t) at t = 0.5 on different convex domains withα = β = 0.8 and τ = 10−3

117 Chapter 5

5.4 Conclusions

In this chapter, we considered the unstructured mesh control volume method for the

two-dimensional space fractional diffusion equation with variable coefficients on convex

domains. We partitioned the irregular convex domain using triangular meshes. Then

we constructed the control volumes and solved the space fractional diffusion equation

by utilising the finite volume method. Finally, numerical examples on irregular convex

domains were studied, which verified the effectiveness and reliability of the method. We

concluded that the numerical method can be extended to other arbitrarily shaped convex

domains. Furthermore, according to the property of the stiffness matrix generated by

the finite volume method, we chose a suitable sparse matrix format for the stiffness

and utilised the Bi-CGSTAB iterative method to solve the linear system, which is more

efficient than using Gauss elimination method. In addition, we made a comparison of

our method with the finite element method proposed in [262], which demonstrated that

our method can reduce CPU time significantly while retaining the same accuracy and

approximation property as the finite element method. In future work, we shall investigate

the unstructured mesh control volume method applied to other fractional problems on

irregular convex domains, such as the two-dimensional multi-term time-space fractional

diffusion equation with variable coefficients or three-dimensional space fractional diffusion

equations with variable coefficients.

Chapter 6

Numerical methods and analysis for simulating the flow of a

generalised Oldroyd-B fluid between two infinite parallel rigid plates

6.1 Introduction

During the past few decades, fluids have been widely applied in engineering and industry,

such as oil and gas well drilling, extrusion of molten plastic, metallic plate cooling, the

flow of polymer solutions, etc., which exhibit a linear relationship between stress and the

rate of strain. Generally, the constitutive equation, a relation between the stress and the

local properties of the fluid, is used to specify the rheological properties of the material,

in which the simplest is the Newtonian constitutive equation. Based on this, the classical

Navier Stokes’ theory is successfully applied to describe the mechanical behavior of many

fluids. However, some fluids produced industrially do not obey the Newtonian postulate

that the stress tensor is directly proportional to the rate of deformation tensor, such

as for molten plastics, slurries, emulsions, pulps, petroleum drilling, manufacturing of

food and other similar activities, which are called non-Newtonian fluids. Quite different

from the characteristics of Newtonian fluids, the governing equations of non-Newtonian

fluids gives rise to highly nonlinear differential equations, which are usually classified as

fluids of integral, differential and rate types [65]. The integral models are utilised to

describe the response of fluids that have considerable memory, such as polymer melting,

whereas differential and rate type models are applied to describe the response of fluids

that have slight memory, such as dilute polymeric solutions. One particular subclass

of non-Newtonian fluids is the generalised Oldroyd-B fluids, which have been found to

approximate the response of many dilute polymeric liquids.

In [242], Tong et al. dealt with unsteady transient rotational flows and unsteady unidi-

rectional transient flows of an Oldroyd-B fluid in an annular pipe. In [122], Khan et al.

developed the exact analytical solutions for the magnetohydrodynamic (MHD) flows of

an Oldroyd-B fluid. In [84], Fetecau et al. considered the velocity field and the adequate

shear stress corresponding to the decay of a potential vortex in a generalised Oldroyd-B

fluid by means of Hankel and Laplace transforms. In [206], Qi et al. presented the analyt-

ical solutions corresponding to two types of unsteady unidirectional flows of a generalised

Oldroyd-B fluid between two parallel plates. In [163], Zheng et al. presented a research

for the magnetohydrodynamic (MHD) flow of an incompressible generalised Oldroyd-B

fluid due to an infinite accelerating plate. Zheng et al. [294] also studied the 3D flow of

a generalised Oldroyd-B fluid due to a constant pressure gradient between two side walls

perpendicular to a plate. Recently, Zhao et al. [286] investigated natural convection heat

transfer of a generalised Oldroyd-B fluid in a porous medium with a modified fractional

118

119 Chapter 6

Darcy’s law. Jiang et al. [114] presented an analytical solution of the unsteady elec-

troosmotic flow of an Oldroyd-B fluid in a circular microchannel under the Debye-Huckel

approximation.

One important model is the following model, which describes the flow problem of an

incompressible Oldroyd-B fluid bounded by two rigid plates:

(1 + λDαt )∂u

∂t= ν(1 + θDβ

t )∂2u

∂y2, (6.1)

where λ is the relaxation time, θ is the retardation time, µ is the dynamic viscosity, ρ is

the constant density of the fluid and ν = µρ . We will give the detailed derivation of the

problem (6.1) in Section 6.2. The system is still initially and starts to move with some

acceleration at t = 0+. In [204], Qi et al. considered the constant velocity case and Khan

et al. [124] discussed the constant acceleration and variable acceleration cases. A two-

dimensional case is studied in [223]. Ming et al. [187] give the general solution expressed

by multivariate Mittag-Leffler function using a separation variables method. All of this

research is limited to the study of the analytical solution, no numerical solution techniques

are addressed.

In this paper, we will consider the following multi-term time fractional diffusion equation:

a1Dγt u+ a2

∂u

∂t= a3

∂2u

∂x2+ a4D

βt

∂2u

∂x2+ f(x, t), (x, t) ∈ Ω, (6.2)

subject to the initial conditions

u(x, 0) = φ1(x), ut(x, 0) = φ2(x), 0 ≤ x ≤ L, (6.3)

and the boundary conditions:

u(0, t) = 0, u(L, t) = 0, 0 ≤ t ≤ T, (6.4)

where a1 > 0, a2 ≥ 0, a3 > 0, a4 ≥ 0, 1 < γ < 2, 0 < β < 1 and Ω = (0, L)× (0, T ]. The

Caputo time fractional derivatives Dγt u(x, t) and Dβ

t u(x, t) are defined as [203]

Dγt u(x, t) =

1

Γ(2− γ)

∫ t

0(t− s)1−γ ∂

2u(x, s)

∂s2ds, 1 < γ < 2,

Dβt u(x, t) =

1

Γ(1− β)

∫ t

0(t− s)−β ∂u(x, s)

∂sds, 0 < β < 1.

Remark 6.1.1 Eq.(6.2) contains different types of fractional diffusion equations. It in-

corporates the generalised Oldroyd-B fluid model (6.1), the time fractional diffusion-wave

equation (a2 = a4 = 0) [238], and the generalised Maxwell fluid model (a3 = 0) [293].

Chapter 6 120

Although there is some literature involving the exact solution of the generalised Oldroyd-

B fluid model, they are typically given in series form with generalised G or R function,

which are difficult to express explicitly. Therefore, numerical solution of this model is

a promising tool to provide for the insight on the behaviour of the solution. In [11],

Bazhlekova et al. proposed a finite difference method to solve the viscoelastic flow with

generalised fractional Oldroyd-B constitutive model. They chose the Riemann-Liouville

time fractional derivative and utilised the Grunwald-Letnikov formula to approximate

it, which has low accuracy and the overall scheme lacked theoretical analysis. To date,

numerical methods to solve fractional diffusion equations are mainly based on finite dif-

ference methods [78, 81, 148, 149, 157], finite element methods [72, 79, 290], finite volume

methods [77, 140, 155], spectral method [297, 298] and parameter estimation methods

[49, 71, 150, 151, 268].

The main contributions of this paper are as follows. Although [11] presented some nu-

merical results using the Grunwald-Letnikov formula, they did not provide the theoretical

analysis associated with the method. For the time fractional derivative in the L.H.S. of

Eq.(6.2), we use the so-called L1 or L2 scheme for approximation. For the coupled oper-

ator (time fractional operator on the spatial derivative) in the R.H.S. of Eq.(6.2), various

techniques can be applied [299]. However, Eq.(6.2) involves these two terms simultane-

ously. Therefore, the derivation of the numerical solution becomes difficult and it is more

challenging to establish the theoretical analysis. In this paper, firstly, we use the L1

scheme to approximate the coupled operator and propose a new scheme to discretise the

time fractional derivative, which is similar to the L2 scheme. Then we give an important

and useful lemma, which can also be used in other time fractional diffusion problems.

Furthermore, we derive the implicit scheme of the problem (6.2) and establish the sta-

bility and convergence analysis. In particular, we prove our method is unconditionally

stable and convergent under discrete L2 and L∞ norms. Moreover, we give a high time

order scheme. Although some analytical solutions of problem (6.1) can be derived via

transform techniques, this approach will not be possible when the initial and boundary

conditions become complex. However, our numerical method will not be influenced by

this, which is one advantage of our method.

The outline of this chapter is as follows. In Section 6.2 we give the detailed derivation of

the flow problem. In Section 6.3, some preliminary knowledge is given, in which a new

numerical scheme to discretise the time fractional derivative is proposed. In Section 6.4,

we develop the finite difference method for Eq.(6.2) and derive the implicit scheme. We

also discuss the solvability of the numerical scheme. We proceed with the proof of the

stability and convergence of the scheme by the energy method in Section 6.5. In Section

6.6, we propose a high time order scheme. In Section 6.7, we present two numerical

examples to demonstrate the effectiveness of our method and some conclusions are drawn

finally.

121 Chapter 6

6.2 Formulation of the flow problem

Here, we consider the flow of an incompressible Olyroyd-B fluid bounded by two infinite

parallel rigid plates (y = 0 and y = L) with distance L > 0 (see Figure 6.1), where

the positive y-axis is taken perpendicular to the plates and the x-axis is parallel to the

direction of flow. Define the velocity field

Figure 6.1: The illustration of the flow problem

V := ui + vj + wk,

where i, j,k is the standard basis of R3. Suppose that the main flow only takes place

along the x-axis, then v = w = 0, i.e.,

V = u(y, t)i. (6.5)

The fundamental equations of an incompressible fluid are given by

div V = 0, (6.6)

ρdV

dt= −∇p+ div S + Fb, (6.7)

where div is the divergence operator, ρ is the density of the fluid, ∇ is the gradient

operator, p is the pressure, S is the extra-stress tensor, Fb = (Fbx, Fby, Fbz) is the body

force. By (6.5), the continuous equation (6.6) is satisfied automatically. The constitutive

equation for a generalised Oldroyd-B fluid is defined as [103]:(1 + λ

Dα

Dtα

)S = µ

(1 + θ

Dβ

Dtβ

)A, (0 < α, β < 1), (6.8)

where λ is the relaxation time, µ is the dynamic viscosity coefficient of the fluid, θ is the

retardation time, and A = L + LT (L = ∇V) denotes the first Rivlin-Ericksen tensor.

The operators Dα

Dtα and Dβ

Dtβare material derivatives and can be expressed as [103]

DαS

Dtα= Dα

t S + (V · ∇)S− LS− SLT , (6.9)

DβA

Dtβ= Dβ

t A + (V · ∇)A− LA−ALT , (6.10)

Chapter 6 122

where Dαt and Dβ

t are the time fractional derivative operators of order α and β, respec-

tively. For unidirectional flow, we consider the extra-stress tensor of the form

S = S(y, t) :=

Sxx Sxy Sxz

Syx Syy Syz

Szx Szy Szz

.Taking into account the initial condition S(y, 0) = 0 and substituting (6.9) and (6.10)

into (6.8), we obtain the following equations:

Syy = Sxz = Szx = Syz = Szy = Szz = 0, Sxy = Syx, (6.11)

(1 + λDαt )Sxy = µ(1 + θDβ

t )∂u

∂y, (6.12)

(1 + λDαt )Sxx = 2λSxy

∂u

∂y− 2µθ

(∂u∂y

)2. (6.13)

Next, substituting Eqs.(6.11)-(6.13) into Eq.(6.7) and neglecting the body forces and

pressure gradient, gives

(1 + λDαt )∂u

∂t= ν(1 + θDβ

t )∂2u

∂y2,

where ν = µρ is the kinematic viscosity coefficient of the fluid. Here we are interested in

the following fluid problem between two infinite plates. Initially, the whole system is at

rest and the upper plate is fixed. Then at time t = 0+, the lower plate starts to move

with some acceleration ϕ(t) in the x-direction. Due to the influence of shear, the fluid

is gradually set in motion. Thus we can write the corresponding initial and boundary

conditions as:

u(y, 0) = 0,∂u

∂t(y, 0) = 0, 0 ≤ y ≤ L,

u(0, t) = ϕ(t), u(L, t) = 0, t > 0.

Now we transform the non-homogeneous boundary condition into the homogeneous bound-

ary condition by setting

u(y, t) = W (y, t) + (1− y

L)ϕ(t).

This leads us to the requirement to solve the following problem for W (y, t):

(1 + λDαt )∂W

∂t= ν(1 + θDβ

t )∂2W

∂y2+ f(y, t),

W (y, 0) = φ1(y),∂W

∂t(y, 0) = φ2(y),

W (0, t) = 0, W (L, t) = 0,

123 Chapter 6

where

f(y, t) = −(1− y

L)(1 + λDα

t )∂ϕ

∂t,

φ1(y) = −(1− y

L)ϕ(0), φ2(y) = −(1− y

L)ϕ′(0).

Note: When u(L, t) 6= 0, a similar transform technique can be used. Without ambiguity,

we replace the notation y with x in Eq.(6.2) throughout the rest of the paper.

6.3 Preliminary knowledge

For convenience, in the subsequent sections, we suppose that C,C1, C2, . . . are posi-

tive constants, which may take distinct values according to different contexts discussed

throughout this paper.

Firstly, in the interval [0, L], we take the mesh points xi = ih, i = 0, 1, · · · ,M , and

tn = nτ , n = 0, 1, · · · , N , where h = L/M , τ = T/N are the uniform spatial step size and

temporal step size, respectively. Denote Ωτ ≡ tn|0 ≤ n ≤ N and Ωh ≡ xi|0 ≤ i ≤M.Define uni as the numerical solution of u(xi, tn). We introduce the following notations:

∇tuni =uni − u

n−1i

τ, u

n− 12

i =uni + un−1

i

2,

∇xuni =uni − uni−1

h, δ2

xuni =

uni−1 − 2uni + uni+1

h2.

Let

Vh = v | v is a grid function on Ωh and v0 = vM = 0.

For any χ, v ∈ Vh, we define the following discrete inner products and induced norms:

(χ, v) = hM−1∑i=1

χivi, 〈∇xχ,∇xv〉 = hM∑i=1

∇xχi · ∇xvi,

||v||0 =√

(v, v), |v|1 =√〈∇xv,∇xv〉.

It is straightforward to check that

(δ2xvk, vn) = −〈∇xvk,∇xvn〉, (6.14)

(δ2xvk,∇tvn) = −1

τ〈∇xvk,∇xvn −∇xvn−1〉. (6.15)

Denote the maximum norm

||v||∞ = max0≤i≤M

|vi|,

then we have the following lemma [239].

Chapter 6 124

Lemma 6.3.1 Let v ∈ Vh, then it holds that

||v||∞ ≤√L

2|v|1, ||v||0 ≤

L√6|v|1.

To discretise the time fractional derivative Dγt u(x, t) (1 < γ < 2) at (xi, tn), we have

Dγt u(xi, tn)

=1

Γ(2− γ)

∫ tn

0(tn − s)1−γ ∂

2u(xi, s)

∂s2ds

=1

Γ(2− γ)

n∑k=1

∫ tk

tk−1

(tn − s)1−γ ∂2u(xi, s)

∂s2ds

=1

Γ(2− γ)

n∑k=1

∫ tk

tk−1

(tn − s)1−γ · u(xi, tk)− 2u(xi, tk−1) + u(xi, tk−2)

τ2ds+ rn

=τ1−γ

Γ(3− γ)

[a

(γ)0 ∇tu(xi, tn)−

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu(xi, tk)− a

(γ)n−1

∂u(xi, 0)

∂t

]+ rn,

where u(xi, t−1) = u(xi, 0)− τ ∂u(xi,0)∂t and a

(γ)k = (k+ 1)2−γ −k2−γ , k = 0, 1, 2, . . . For the

truncation error rn, we have [138]

|rn| ≤ CT 2−γ

Γ(3− γ)max

0≤t≤T

∣∣∣∣∂3u(xi, t)

∂t3

∣∣∣∣ · τ +O(τ2).

Remark 6.3.1 According to the problem we considered here, utt(x, 0) = 0. When utt(x, 0) 6=0, the error rn satisfies |rn| ≤ Cτ2−γ.

Then we can obtain the discrete scheme for the time fractional derivative Dγt u(x, t) at

mesh points (xi, tn)

Dγt u(xi, tn) =

τ1−γ

Γ(3− γ)

[a

(γ)0 ∇tu(xi, tn)−

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu(xi, tk)

− a(γ)n−1

∂u(xi, 0)

∂t

]+R1, (6.16)

where |R1| ≤ Cτ .

Lemma 6.3.2 For 1 < γ < 2, define a(γ)k = (k + 1)2−γ − k2−γ, k = 0, 1, 2, . . . , n and

S = S1, S2, S3, . . . and Q, then it holds that

τ1−γ

Γ(3− γ)

N∑n=1

[a

(γ)0 Sn −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)Sk − a

(γ)n−1Q

]Sn

≥ T 1−γ

2Γ(2− γ)

N∑n=1

S2n −

T 2−γ

2τΓ(3− γ)Q2, N = 1, 2, 3, . . .

125 Chapter 6

Proof. See [238].

To discretise the time fractional derivative Dβt u(x, t) (0 < β < 1) we have the following

formula [238]

Dβt u(xi, tn) =

τ1−β

Γ(2− β)

n−1∑k=0

a(β)k

u(xi, tn−k)− u(xi, tn−k−1)

τ+R2

=τ1−β

Γ(2− β)

n∑k=1

a(β)n−k

u(xi, tk)− u(xi, tk−1)

τ+R2,

where a(β)k = (k + 1)1−β − k1−β, k = 0, 1, 2, . . . , n and |R2| ≤ C max

0<t≤T|∂

2u(x,t)∂t2

|τ2−β. It is

straightforward to derive the following properties of a(β)k .

Lemma 6.3.3 For 0 < β < 1, define a(β)k = (k + 1)1−β − k1−β, k = 0, 1, 2, . . . then

1. a(β)k > 0, a

(β)0 = 1, a

(β)k > a

(β)k+1, lim

k→∞a

(β)k = 0,

2.n−1∑k=0

(a(β)k − a

(β)k+1) + a

(β)n = 1,

3. a(β)k+1 − 2a

(β)k + a

(β)k−1 ≥ 0,

4. (1− β)(k + 1)−β ≤ a(β)k ≤ (1− β)k−β.

Since

∂2u(xi, tn)

∂x2= δ2

xu(xi, tn)− h2

12

∂4u(ξi, tn)

∂x4,

where xi−1 ≤ ξi ≤ xi, then we have

Dβt

∂2u(xi, tn)

∂x2=

τ−β

Γ(2− β)

n−1∑k=0

a(β)k

[∂2u(xi, tn−k)

∂x2− ∂2u(xi, tn−k−1)

∂x2

]+R3,

=τ−β

Γ(2− β)

n−1∑k=0

a(β)k [δ2

xu(xi, tn−k)− δ2xu(xi, tn−k−1)] +R4 +R3,

where |R3| ≤ C max(x,t)∈Ω

|∂4u(x,t)∂x2∂t2

|τ2−β and

R4 = − h2τ−β

12Γ(2− β)

n−1∑k=0

a(β)k

[∂4u(ξi, tn−k)

∂x4− ∂4u(ξi, tn−k−1)

∂x4

]= − h2τ1−β

12Γ(2− β)

n−1∑k=0

a(β)k

∂5u(ξi, ςn−k)

∂x4∂t,

Chapter 6 126

namely,

|R4| ≤h2τ1−β

12Γ(2− β)max

(x,t)∈Ω

∣∣∣∣∂5u(ξi, ςn−k)

∂x4∂t

∣∣∣∣ n−1∑k=0

a(β)k

=h2τ1−β

12Γ(2− β)max

(x,t)∈Ω

∣∣∣∣∂5u(ξi, ςn−k)

∂x4∂t

∣∣∣∣ · n1−β

≤ h2T 1−β

12Γ(2− β)max

(x,t)∈Ω

∣∣∣∣∂5u(ξi, ςn−k)

∂x4∂t

∣∣∣∣,where tn−k−1 < ςn−k < tn−k. Then we have

Dβt

∂2u(xi, tn)

∂x2

=τ−β

Γ(2− β)

n−1∑k=0

a(β)k [δ2

xu(xi, tn−k)− δ2xu(xi, tn−k−1)] +R5

=τ1−β

Γ(2− β)

n∑k=1

a(β)n−k

δ2xu(xi, tk)− δ2

xu(xi, tk−1)

τ+R5

=τ−β

Γ(2− β)

[a

(β)0 δ2

xu(xi, tn)−n−1∑k=1

(a(β)n−k−1 − a

(β)n−k)δ

2xu(xi, tk)

−a(β)n−1δ

2xu(xi, t0)

]+R5, (6.17)

where |R5| ≤ C(τ2−β + h2).

Lemma 6.3.4 [172] Let d0, d1, . . . , dn, . . . be a sequence of real numbers with the prop-

erties

dn ≥ 0, dn − dn−1 ≤ 0, dn+1 − 2dn + dn−1 ≥ 0.

Then for any positive integer M , and for each vector [V1, V2, . . . , VM ] with M real entries,

M∑n=1

( n−1∑p=0

dp Vn−p

)Vn ≥ 0.

Now we give a very useful and important lemma.

Lemma 6.3.5 For 0 < β < 1, define a(β)k = (k + 1)1−β − k1−β, k = 0, 1, 2, . . . , n, then

for any positive integer N and real vector P = [v1, v2, . . . , vN−1, vN ] ∈ RN , we have

N∑n=1

n∑k=1

a(β)n−k v

kvn ≥ 0.

127 Chapter 6

Proof. It is easy to check that

N∑n=1

n∑k=1

a(β)n−k v

kvn =N∑n=1

( n−1∑k=0

a(β)k vn−k

)vn.

Then using Lemma 6.3.3 and 6.3.4, we have

N∑n=1

( n−1∑k=0

a(β)k vn−k

)vn ≥ 0,

which completes the proof.

6.4 Derivation and solvability of the numerical scheme

In this section, we will derive the implicit finite difference scheme of Eq.(6.2). Suppose

that u(x, t) ∈ C4,3x,t (Ω). Firstly, from Eq.(6.2), we have

a1Dγt u(xi, tn) + a2

∂u(xi, tn)

∂t= a3

∂2u(xi, tn)

∂x2+ a4D

βt

∂2u(xi, tn)

∂x2+ f(xi, tn). (6.18)

From Eqs.(6.16)-(6.17), we have

a1µ1

[a

(γ)0 ∇tu(xi, tn)−

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu(xi, tk)− a

(γ)n−1φ2(xi)

]+a2∇tu(xi, tn) = a3δ

2xu(xi, tn) + a4µ2

[a

(β)0 δ2

xu(xi, tn)

−n−1∑k=1

(a(β)n−k−1 − a

(β)n−k)δ

2xu(xi, tk)− a

(β)n−1δ

2xu(xi, t0)

]+ f(xi, tn) +Rni , (6.19)

where µ1 = τ1−γ

Γ(3−γ) , µ2 = τ−β

Γ(2−β) and |Rni | ≤ C(τ + h2), in which C is independent of τ

and h. Then, omitting the error term, we obtain the following implicit finite difference

scheme for Eq.(6.2) at point (xi, tn)

a1µ1 + a2

τuni − (a3 + a4µ2)δ2

xuni

=a1µ1 + a2

τun−1i + a1µ1

[ n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki + a

(γ)n−1φ2(xi)

]−a4µ2

[ n−1∑k=1

(a(β)n−k−1 − a

(β)n−k)δ

2xu

ki + a

(β)n−1δ

2xu

0i

]+ fni , (6.20)

with initial and boundary conditions

u0i = φ1(xi), 0 ≤ i ≤M, un0 = unM = 0, n ≥ 1.

Now, we discuss the solvability of the finite difference scheme (6.20).

Theorem 6.4.1 The finite difference scheme (6.20) is uniquely solvable.

Chapter 6 128

Proof. At each time level, the coefficient matrix B is linear tridigonal of the form

B =

b1 + 2b2 −b2 0 · · · 0 0

−b2 b1 + 2b2 −b2 · · · 0 0

0 −b2 b1 + 2b2 · · · 0 0...

......

. . ....

...

0 0 0 · · · b1 + 2b2 −b20 0 0 · · · −b2 b1 + 2b2

,

where b1 = a1µ1+a2τ > 0 and b2 = a3+a4µ2

h2> 0. Note that B is a strictly diagonally

dominant matrix, and therefore nonsingular This implies that the numerical scheme (6.20)

is uniquely solvable.

6.5 Stability and convergence

6.5.1 Stability

We will analyze the stability of the scheme by using the energy method.

Theorem 6.5.1 The finite difference scheme (6.20) is unconditionally stable.

Proof. Eq.(6.20) can be recast into

a1µ1

[a

(γ)0 ∇tu

ni −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki − a

(γ)n−1φ2(xi)

]+ a2∇tuni

=a3δ2xu

ni + a4µ2τ

n∑k=1

a(β)n−k∇t(δ

2xu

ki ) + fni . (6.21)

Multiplying Eq.(6.21) by hτ∇tuni and summing i from 1 to M − 1 and summing n from

1 to l, 1 ≤ l ≤ N , we obtain

a1µ1τl∑

n=1

M−1∑i=1

h[a

(γ)0 ∇tu

ni −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki − a

(γ)n−1φ2(xi)

]∇tuni

+a2τl∑

n=1

M−1∑i=1

h(∇tuni )2 = a3τl∑

n=1

M−1∑i=1

hδ2xu

ni ∇tuni

+a4µ2τ2

l∑n=1

M−1∑i=1

h

n∑k=1

a(β)n−k∇t(δ

2xu

ki )∇tuni + τ

l∑n=1

M−1∑i=1

hfni ∇tuni . (6.22)

129 Chapter 6

Using Lemma 6.3.2, we have

a1µ1τl∑

n=1

M−1∑i=1

h[a

(γ)0 ∇tu

ni −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki − a

(γ)n−1φ2(xi)

]∇tuni

≥ a1τT1−γ

2Γ(2− γ)

l∑n=1

M−1∑i=1

h(∇tuni )2 − a1T2−γ

2Γ(3− γ)

M−1∑i=1

hφ2(xi)2

=a1τT

1−γ

2Γ(2− γ)

l∑n=1

||∇tun||20 −a1T

2−γ

2Γ(3− γ)||φ2||20.

Then the L.H.S. of (6.22) is bounded by

a1µ1τ

l∑n=1

M−1∑i=1

h[a

(γ)0 ∇tu

ni −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki − a

(γ)n−1φ2(xi)

]∇tuni

+a2τ

l∑n=1

M−1∑i=1

h(∇tuni )2 ≥ τ(

a1T1−γ

2Γ(2− γ)+ a2

) l∑n=1

||∇tun||20 −a1T

2−γ

2Γ(3− γ)||φ2||20. (6.23)

Utilising (6.15), we have

a3τl∑

n=1

M−1∑i=1

hδ2xu

ni ∇tuni = a3τ

l∑n=1

(δ2xu

n,∇tun)

=− a3

l∑n=1

〈∇xun,∇xun −∇xun−1〉

≤ − a3

2

l∑n=1

(|un|21 − |un−1|21) =a3

2(|u0|21 − |ul|21), (6.24)

where we use the inequality a(a − b) ≥ 12(a2 − b2). Combining (6.15) and Lemma 6.3.5,

we have

a4µ2τ2

l∑n=1

M−1∑i=1

h

n∑k=1

a(β)n−k∇t(δ

2xu

ki )∇tuni = a4µ2τ

2l∑

n=1

n∑k=1

a(β)n−k

(∇t(δ2

xuk),∇tun

)=− a4µ2τ

2l∑

n=1

n∑k=1

a(β)n−k

⟨∇t(∇xuk),∇t(∇xun)

⟩≤ 0. (6.25)

Using the inequality ab ≤ εa2 + b2

4ε , we have

τl∑

n=1

M−1∑i=1

hfni ∇tuni

≤τ(

a1T1−γ

2Γ(2− γ)+ a2

) l∑n=1

M−1∑i=1

h(∇tuni )2 + τ1

4

(a1T 1−γ

2Γ(2−γ) + a2

) l∑n=1

M−1∑i=1

h(fni )2

=τ

(a1T

1−γ

2Γ(2− γ)+ a2

) l∑n=1

||∇tun||20 +τΓ(2− γ)

2(a1T 1−γ + 2a2Γ(2− γ))

l∑n=1

||fn||20. (6.26)

Chapter 6 130

Substituting (6.23)-(6.26) into (6.22), we have

τ

(a1T

1−γ

2Γ(2− γ)+ a2

) l∑n=1

||∇tun||20 −a1T

2−γ

2Γ(3− γ)||φ2||20

≤a3

2(|u0|21 − |ul|21) + τ

(a1T

1−γ

2Γ(2− γ)+ a2

) l∑n=1

||∇tun||20

+τΓ(2− γ)

2(a1T 1−γ + 2a2Γ(2− γ))

l∑n=1

||fn||20,

then we have

a3

2|ul|21 ≤

a3

2|u0|21 +

a1T2−γ

2Γ(3− γ)||φ2||20 +

τΓ(2− γ)

2(a1T 1−γ + 2a2Γ(2− γ))

l∑n=1

||fn||20,

i.e.,

|ul|21 ≤ |u0|21 +a1T

2−γ

a3Γ(3− γ)||φ2||20 +

τΓ(2− γ)

a3[a1T 1−γ + 2a2Γ(2− γ)]

l∑n=1

||fn||20. (6.27)

From Lemma 6.3.1, we have

||ul||20 ≤L2

6|ul|21 ≤

L2

6

|u0|21 +

a1T2−γ

a3Γ(3− γ)||φ2||20 +

τΓ(2− γ)

a3[a1T 1−γ + 2a2Γ(2− γ)]

l∑n=1

||fn||20,

||ul||2∞ ≤L

4|ul|21 ≤

L

4

|u0|21 +

a1T2−γ

a3Γ(3− γ)||φ2||20 +

τΓ(2− γ)

a3[a1T 1−γ + 2a2Γ(2− γ)]

l∑n=1

||fn||20,

which means that the scheme (6.20) is unconditionally stable.

6.5.2 Convergence

Define Un = [un1 , un2 , . . . , u

nM−1]T and un = [u(x1, tn), u(x2, tn), . . . , u(xM−1, tn)]T as the

numerical solution and exact solution vectors, respectively. Then we obtain the following

theorem of convergence.

Theorem 6.5.2 Suppose that the solution of problem (6.2)-(6.4) satisfies u(x, t) ∈ C4,3x,t (Ω),

then there exists a positive constant C independent of h and τ such that

||un − Un||0 ≤ CL

√Γ(2− γ)TL

6a3[a1T 1−γ + 2a2Γ(2− γ)](τ + h2),

||un − Un||∞ ≤CL

2

√Γ(2− γ)T

a3[a1T 1−γ + 2a2Γ(2− γ)](τ + h2).

131 Chapter 6

Proof. Denote eni = u(xi, tn) − uni , en = [en1 , en2 , . . . , e

nM−1]. Subtracting (6.19) from

(6.18), we have

a1µ1

[a

(γ)0 ∇te

ni −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇te

ki

]+ a2∇teni

=a3δ2xeni + a4µ2

[a

(β)0 δ2

xeni −

n−1∑k=1

(a(β)n−k−1 − a

(β)n−k)δ

2xeki − a

(β)n−1δ

2xe

0i

]+Rni ,

with

e0i = 0, en0 = enM = 0.

Then from (6.27), we have

|en|21 ≤Γ(2− γ)τh

a3[a1T 1−γ + 2a2Γ(2− γ)]

n∑k=1

M−1∑i=1

(Rni )2

≤ Γ(2− γ)τh

a3[a1T 1−γ + 2a2Γ(2− γ)]

n∑k=1

M−1∑i=1

C2(τ + h2)2

≤ Γ(2− γ)C2nτ(M − 1)h

a3[a1T 1−γ + 2a2Γ(2− γ)](τ + h2)2

≤ Γ(2− γ)C2TL

a3[a1T 1−γ + 2a2Γ(2− γ)](τ + h2)2,

i.e.

|en|1 ≤ C

√Γ(2− γ)TL

a3[a1T 1−γ + 2a2Γ(2− γ)](τ + h2).

From Lemma 6.3.1, we have

||en||0 ≤L√6|en|1 ≤ CL

√Γ(2− γ)TL

6a3[a1T 1−γ + 2a2Γ(2− γ)](τ + h2),

||en||∞ ≤√L

2|en|1 ≤

CL

2

√Γ(2− γ)T

a3[a1T 1−γ + 2a2Γ(2− γ)](τ + h2).

6.6 Improve the time order of the scheme

To discretise the time fractional derivative Dγt u(x, t) (1 < γ < 2) at (xi, tn− 1

2), we have

the following formula [238]

Dγt u(xi, tn− 1

2) =

τ1−γ

Γ(3− γ)

[a

(γ)0 ∇tu(xi, tn)−

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu(xi, tk)

−a(γ)n−1

∂u(xi, 0)

∂t

]+R6, (6.28)

Chapter 6 132

where |R6| ≤ Cτ3−γ . Here we write Eq.(6.17) as:

Dβt

∂2u(xi, tn)

∂x2=

τ−β

Γ(2− β)[bn,βn δ2

xu(xi, tn) +n−1∑k=1

bn,βk δ2xu(xi, tk) + bn,β0 δ2

xu(xi, t0)] +R5,

where bn,β0 = (n− 1)1−β − n1−β, bn,βn = 1 and bn,βk = (n− k+ 1)1−β − 2(n− k)1−β + (n−k − 1)1−β, k = 1, 2, . . . , n− 1. Furthermore, when n ≥ 2, we have

Dβt

∂2u(xi, tn)

∂x2+Dβ

t

∂2u(xi, tn−1)

∂x2

=τ−β

Γ(2− β)[bn,βn δ2

xu(xi, tn) +n−1∑k=0

(bn,βk + bn−1,βk )δ2

xu(xi, tk)] +R5 (6.29)

and when n = 1, we have

Dβt

∂2u(xi, t1)

∂x2+Dβ

t

∂2u(xi, t0)

∂x2=

τ−β

Γ(2− β)[δ2xu(xi, t1)− δ2

xu(xi, t0)] +R5. (6.30)

To derive the finite difference scheme we also need the following lemma.

Lemma 6.6.1 If u(x, t) ∈ C0,3x,t (Ω), then we have [43]

u(xi, tn− 12) =

u(xi, tn) + u(xi, tn−1)

2+O(τ2), (6.31)

∂

∂tu(xi, tn− 1

2) =

u(xi, tn)− u(xi, tn−1)

τ+O(τ2). (6.32)

Now we derive a high time order numerical scheme. Suppose that u(x, t) ∈ C4,3x,t (Ω), from

Eq.(6.2), we have

a1

2[Dγ

t u(xi, tn) +Dγt u(xi, tn−1)] +

a2

2[∂u(xi, tn)

∂t+∂u(xi, tn−1)

∂t]

=a3

2[∂2u(xi, tn)

∂x2+∂2u(xi, tn−1)

∂x2] +

a4

2[Dβ

t

∂2u(xi, tn)

∂x2+Dβ

t

∂2u(xi, tn−1)

∂x2]

+1

2[f(xi, tn) + f(xi, tn−1)]. (6.33)

From Eqs.(6.28)-(6.32), we have

2a1µ1

[a

(γ)0 ∇tu(xi, tn)−

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu(xi, tk)− a

(γ)n−1φ2(xi)

]+2a2∇tu(xi, tn) = a3[δ2

xu(xi, tn) + δ2xu(xi, tn−1)] + a4µ2

[δ2xu(xi, tn)

+n−1∑k=0

(bn,βk + bn−1,βk )δ2

xu(xi, tk)]

+ f(xi, tn) + f(xi, tn−1) +R7, (6.34)

133 Chapter 6

where |R7| ≤ C(τmin3−γ,2−β + h2). Then, omitting the error term, we obtain the

numerical scheme of Eq.(6.34)

2a1µ1 + a2

τuni − (a3 + a4µ2)δ2

xuni

=2a1µ1 + a2

τun−1i + a3δ

2xu

n−1i

+2a1µ1

[ n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki + a

(γ)n−1φ2(xi)

]+a4µ2

n−1∑k=0

(bn,βk + bn−1,βk )δ2

xuki + fni + fn−1

i , (6.35)

with initial and boundary conditions

u0i = φ1(xi), 0 ≤ i ≤M, un0 = unM = 0, n ≥ 1.

When n = 1, we have

2a1µ1 + a2

τu1i − (a3 + a4µ2)δ2

xu1i

=2a1µ1 + a2

τu0i + (a3 − a4µ2)δ2

xu0i + 2a1µ1φ2(xi) + f1

i + f0i .

Remark 6.6.1 To improve the order further, we can use the second order scheme in [79,

299], modified weighted shifted Grunwald-Letnikov (WSGL) formula [156] or Richardson

extrapolation method [78].

6.7 Numerical examples

Example 6.7.1 At first, we consider the following multi-term time fractional diffusion

equationa1D

γt u+ a2

∂u∂t = a3

∂2u∂x2

+ a4Dβt∂2u∂x2

+ f(x, t), (x, t) ∈ (0, 1)× (0, T ],

u(x, 0) = 0, ut(x, 0) = 0, 0 ≤ x ≤ 1,

u(0, t) = t3, u(1, t) = et3, 0 ≤ t ≤ T,

where 0 < β < 1, 1 < γ < 2,

f(x, t) = ex[a1Γ(4)t3−γ

Γ(4− γ)+ 3a2t

2 − a3t3 − a4Γ(4)t3−β

Γ(4− β)

],

and the exact solution is u(x, t) = t3ex.

Without loss of generality, we take a1 = a2 = a3 = a4 = 1 and T = 1. Firstly, we use theimplicit finite difference scheme (6.20) (Scheme I) to solve the equation and the numericalresults are given in Tables 6.1 and 6.2. Table 6.1 shows the L2 error and L∞ error and

Chapter 6 134

convergence order of τ for different β with fixed γ = 1.1 and h = 1/1000 at t = 1. Table6.2 displays the L2 error and L∞ error and convergence order of τ for different γ with fixedβ = 0.3 and h = 1/1000 at t = 1. We can see that the numerical results are in perfectagreement with the exact solution and the convergence order reaches the expected firstorder. Then we consider the high time order scheme (6.35) (Scheme II) and the relatednumerical results are shown in Tables 6.3 and 6.4. Table 6.3 shows the L2 error andL∞ error and convergence order of τ for different β with fixed γ = 1.1 and h = 1/1000at t = 1. Table 6.4 displays the L2 error and L∞ error and convergence order of τ fordifferent γ with fixed β = 0.3 and h = 1/1000 at t = 1. We can find that the numericalresults are in good agreement with the exact solution and the convergence order reachesthe expected order 2− β and 3− γ, respectively, which demonstrates the effectiveness ofour numerical method and confirms the theoretical analysis.

Table 6.1: The temporal error and convergence order of Scheme I for different β withγ = 1.1 and h = 1/1000

β = 0.6, γ = 1.1 L2 error Order L∞ error Order

1/20 1.6370E-02 – 2.2385E-02 –1/40 8.4017E-03 0.96 1.1489E-02 0.961/80 4.2780E-03 0.97 5.8498E-03 0.971/160 2.1669E-03 0.98 2.9630E-03 0.981/320 1.0937E-03 0.99 1.4955E-03 0.99

β = 0.8, γ = 1.1 L2 error Order L∞ error Order

1/20 1.3324E-02 – 1.8219E-02 –1/40 6.8976E-03 0.95 9.4311E-03 0.951/80 3.5483E-03 0.96 4.8516E-03 0.961/160 1.8167E-03 0.97 2.4839E-03 0.971/320 9.2666E-04 0.97 1.2670E-03 0.97

Table 6.2: The temporal error and convergence order of Scheme I for different γ withβ = 0.3 and h = 1/1000

β = 0.3, γ = 1.6 L2 error Order L∞ error Order

1/20 2.3769E-02 – 3.2559E-02 –1/40 1.1783E-02 1.01 1.6140E-02 1.011/80 5.8367E-03 1.01 7.9949E-03 1.011/160 2.8934E-03 1.01 3.9631E-03 1.011/320 1.4361E-03 1.01 1.9670E-03 1.01

β = 0.3, γ = 1.8 L2 error Order L∞ error Order

1/20 2.7877E-02 – 3.8225E-02 –1/40 1.3692E-02 1.03 1.8773E-02 1.031/80 6.7208E-03 1.03 9.2139E-03 1.031/160 3.3024E-03 1.03 4.5272E-03 1.031/320 1.6252E-03 1.02 2.2278E-03 1.02

135 Chapter 6

Table 6.3: The temporal error and convergence order of Scheme II for different β withγ = 1.1 and h = 1/1000

β = 0.6, γ = 1.1 L2 error Order L∞ error Order

1/20 1.0645E-03 – 1.4546E-03 –1/40 4.5169E-04 1.24 6.1737E-04 1.241/80 1.8353E-04 1.30 2.5088E-04 1.301/160 7.2710E-05 1.34 9.9401E-05 1.341/320 2.8360E-05 1.36 3.8773E-05 1.36

β = 0.8, γ = 1.1 L2 error Order L∞ error Order

1/20 2.9664E-03 – 4.0548E-03 –1/40 1.3637E-03 1.12 1.8642E-03 1.121/80 6.1238E-04 1.16 8.3719E-04 1.151/160 2.7141E-04 1.17 3.7105E-04 1.171/320 1.1939E-04 1.18 1.6322E-04 1.18

Table 6.4: The temporal error and convergence order of Scheme II for different γ withβ = 0.3 and h = 1/1000

β = 0.3, γ = 1.6 L2 error Order L∞ error Order

1/10 5.1584E-03 – 7.0821E-03 –1/20 1.9363E-03 1.41 2.6580E-03 1.411/40 7.3473E-04 1.40 1.0085E-03 1.401/80 2.8031E-04 1.39 3.8472E-04 1.391/160 1.0719E-04 1.39 1.4710E-04 1.39

β = 0.3, γ = 1.8 L2 error Order L∞ error Order

1/10 1.1935E-02 – 1.6390E-02 –1/20 5.1435E-03 1.21 7.0605E-03 1.211/40 2.2358E-03 1.20 3.0683E-03 1.201/80 9.7587E-04 1.20 1.3391E-03 1.201/160 4.2650E-04 1.19 5.8520E-04 1.19

Example 6.7.2 Next, we consider the following accelerated flows for a generalised

Oldroyd-B fluid between two infinite parallel rigid plates.(1 + λαDα

t )∂u∂t = (1 + θβDβt )∂

2u∂x2

, (x, t) ∈ (0, 100)× (0, T ]

u(x, 0) = 0, ut(x, 0) = 0, 0 ≤ x ≤ 100,

u(100, t) = 0, 0 ≤ t ≤ T,

where 0 < α, β < 1. Here, we discuss two different acceleration flows: constant accelera-

tion flow u(0, t) = t and variable acceleration flow u(0, t) = t2.

In the calculation, we choose h = 1/1000, τ = 1/100. The variations of velocity u(x, t)with x for distinct values of λ, θ, α and β at a fixed time (t = 2) are illustrated in Figures6.2-6.5. We can see from these figures, the smaller the λ (or α), the more slowly thevelocity decays for the two different acceleration flows. However, an opposite trend for

Chapter 6 136

the variation of θ (or β) can be seen. Therefore, we can conclude that the relaxationtime λ and retardation time θ and the fractional order α, β have strong effects on thevelocity u(x, t). Figure 6.6 demonstrates the influence of time on the velocity and we cannote that the flow velocity increases with time for the two different acceleration flows.Qualitatively, the observations for variable acceleration flow and constant accelerationflow are very similar, however, are not in quantity of different magnitude. Comparedto the constant acceleration flow, we can observe that the velocity profiles in variableacceleration flow are much larger.

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2λ=4λ=6λ=8λ=10

(a) Constant acceleration flow

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4λ=4λ=6λ=8λ=10

(b) Variable acceleration flow

Figure 6.2: Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different λ with α = 0.3, β = 0.8, θ = 2 at t = 2

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2θ=2θ=4θ=6θ=8

(a) Constant acceleration flow

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4θ=2θ=4θ=6θ=8

(b) Variable acceleration flow

Figure 6.3: Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different θ with α = 0.3, β = 0.8, λ = 10 at t = 2

137 Chapter 6

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2α=0.1α=0.3α=0.5α=0.7

(a) Constant acceleration flow

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4α=0.1α=0.3α=0.5α=0.7

(b) Variable acceleration flow

Figure 6.4: Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different α with β = 0.8, λ = 5, θ = 3 at t = 2

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2β=0.2β=0.4β=0.6β=0.8

(a) Constant acceleration flow

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4β=0.2β=0.4β=0.6β=0.8

(b) Variable acceleration flow

Figure 6.5: Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different β with α = 0.3, λ = 5, θ = 3 at t = 2

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2t=0.5t=1.0t=1.5t=2.0

(a) Constant acceleration flow

x0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4t=0.5t=1.0t=1.5t=2.0

(b) Variable acceleration flow

Figure 6.6: Numerical solution profiles of velocity u(x, t) of two different accelerationflows for different t with α = 0.3, β = 0.8, λ = 5, θ = 3

Chapter 6 138

6.8 Conclusions

In this chapter, we proposed the finite difference method to solve the multi-term time

fractional diffusion equation of a generalized Oldroyd-B fluid with accuracy of O(τ + h2)

and O(τmin3−γ,2−β+h2), respectively. We also established the stability and convergence

analysis for these schemes. Two numerical examples were exhibited to verify the effec-

tiveness and reliability of our method. We can conclude that our numerical method is

robust and can be extended to other multi-term time fractional diffusion equations, such

as the generalized Oldroyd-B fluid in a rotating system and the generalized Maxwell fluid

model. In future work, we shall investigate the finite difference method to the problems

in a two-dimensional case.

Chapter 7

Novel numerical analysis of multi-term time fractional viscoelastic

non-Newtonian fluid models for simulating unsteady MHD Couette

flow of a generalised Oldroyd-B fluid

7.1 Introduction

Generally, a constitutive equation is used to specify the rheological properties of a ma-

terial, which is a relation between the stress and the local properties of the fluid. Some

common fluids, such as water, oil, air, ethanol, and benzene, exhibit a linear relationship

between the stress tensor and the rate of deformation tensor, which are called Newtonian

fluids. The Newtonian constitutive equation is the simplest linear viscoelastic model. For

small deformations, low stress, low rate, and linear materials, linear viscoelasticity are

usually applicable. However, some fluids produced industrially do not obey the Newtoni-

an postulate, such as molten plastics, slurries, emulsions, pulps, and these are termed as

non-Newtonian fluids. This means that the rapport between the stress tensor and the rate

of deformation tensor is non-linear. In reality, about 90% of fluids are nonlinear with large

deformations, therefore nonlinear viscoelastic mathematical models are needed. Research

related to non-Newtonian fluid mechanics is of great importance to the industry. Since a

rheometer cannot provide the necessary information of important rheological properties,

the constitutive equations are the best available tools for understanding the complex be-

haviour of a material. Due to the nonlinear relationship between stress and deformation

and there being no standard form universally valid for each non-Newtonian fluid, the

constitutive equation of non-Newtonian fluids is much more complex than its Newtonian

counterpart. The constitutive equations involving fractional calculus have proved to be

a valuable tool for handling viscoelastic properties [7, 90] and some results are obtained

that are in good agreement with experimental data [102, 178].

One particular subclass of non-Newtonian fluids is the generalised Oldroyd-B fluid, which

has been found to approximate the response of many dilute polymeric liquids. Consider

the flow of an incompressible Olyroyd-B fluid bounded by two infinite parallel rigid plates.

Initially, the whole system is at rest and the upper plate is fixed. Then at time t = 0+,

the lower plate starts to move with some acceleration. Due to the shear effects, the fluid

over the plate is gradually disturbed. The fundamental equations of an incompressible

fluid are

div V = 0, ρdV

dt= div T,

where div is the divergence operator, ρ is the density of the fluid, T is the Cauchy stress

tensor and ddt is the general time derivative. The constitutive equation for a generalised

139

Chapter 7 140

Oldroyd-B fluid is defined as [103]:

T = −pI + S,

(1 + λ

Dα

Dtα

)S = µ

(1 + θ

Dβ

Dtβ

)A,

where p is the pressure, I is the identity tensor, S is the extra-stress tensor, λ is the

relaxation time, µ is the dynamic viscosity coefficient of the fluid, θ is the retardation

time, and A = L + LT (L = ∇V) denotes the first Rivlin-Ericksen tensor. The operatorsDα

Dtα and Dβ

Dtβare material derivatives and can be expressed as

DαS

Dtα= Dα

t S + (V · ∇)S− LS− SLT ,

DβA

Dtβ= Dβ

t A + (V · ∇)A− LA−ALT ,

where Dαt and Dβ

t are the time fractional derivative operators of order α and β, respec-

tively. Assume that the velocity field and stress tensor have the form

V = u(y, t)i, S = S(y, t).

Taking into account the initial condition S(y, 0) = 0 and in the absence of the pressure

gradient, one can obtain the following equation with fractional derivative of the velocity

of the main flow [124, 204]

(1 + λDαt )∂u(y, t)

∂t= ν(1 + θDβ

t )∂2u(y, t)

∂y2, (7.1)

where ν = µρ . When a magnetic field is imposed on the above flow under the assumption of

low magnetic Reynolds number, the following velocity equation can be derived [163, 296]

(1 + λDαt )∂u(y, t)

∂t= ν(1 + θDβ

t )∂2u(y, t)

∂y2−K(1 + λDα

t )u(y, t), (7.2)

where K =σB2

0ρ , B0 is the magnetic intensity and σ is the electrical conductivity. When

the fluid medium is porous, the following magnetohydrodynamic (MHD) flow of a gener-

alised Oldroyd-B fluid with an effect of Hall current can be obtained [123]

(1 + λDαt )∂u(y, t)

∂t= ν(1 + θDβ

t )∂2u(y, t)

∂y2− νϕ1

k(1 + θDβ

t )u(y, t)

− σB20

ρ(1− iφ)(1 + λDα

t )u(y, t), (7.3)

where k is the permeability of the porous medium, ϕ1 is the porosity of the medium, i is

the imaginary unit, and φ is the Hall parameter. As Eqs.(7.1)-(7.3) contain similar terms,

they can be expressed in a generalised form.

141 Chapter 7

In this paper, we will consider the following novel multi-term time fractional non-Newtonian

diffusion equation:

s∑j=1

bj Dγjt u(x, t) + a1

∂u(x, t)

∂t+

q∑l=1

clDαlt u(x, t) + a2u(x, t)

= a3∂2u(x, t)

∂x2+ a4D

βt

∂2u(x, t)

∂x2+ f(x, t), (x, t) ∈ Ω× (0, T ], (7.4)

subject to the initial conditions

u(x, 0) = φ1(x), ut(x, 0) = φ2(x), 0 ≤ x ≤ L, (7.5)

and the boundary conditions

u(0, t) = 0, u(L, t) = 0, 0 ≤ t ≤ T, (7.6)

where ai > 0, i = 1, 2, 3, 4, bj > 0, j = 1, 2, . . . , s, cl ≥ 0, l = 1, 2, . . . , q, 1 < γ1 < γ2 <

. . . < γs < 2, 0 < α1 < α2 < . . . < αq < 1 and Ω = (0, L). The Caputo time fractional

derivative Dβt u(x, t) (0 < β < 1) and Dγ

t u(x, t) (1 < γ < 2) are given by [156, 203]

Dβt u(x, t) =

1

Γ(1− β)

∫ t

0(t− s)−β ∂u(x, s)

∂sds, 0 < β < 1,

Dγt u(x, t) =

1

Γ(2− γ)

∫ t

0(t− s)1−γ ∂

2u(x, s)

∂s2ds, 1 < γ < 2.

The general multi-term time fractional diffusion equation only contains the multi-term

time fractional derivative terms without the special term Dβt∂2u∂x2

. Its solution has been

investigated both theoretically and numerically. Some authors used the method of separa-

tion of variables to obtain analytical solutions of the multi-term time fractional diffusion-

wave equations and the multi-term time fractional diffusion equation [55, 111, 171]. Nu-

merical solutions for the multi-term time fractional diffusion-wave equation can be found

in [56, 264] and for multi-term time fractional diffusion equation in [117, 209, 298], respec-

tively. There is also some research on the numerical solution of the multi-term time frac-

tional diffusion equation, of which the indices belong to (0, 2) or greater than 2 [87, 154].

Different to the general multi-term time fractional diffusion equation, the new model

(7.4) not only has a multi-term time derivative, of which the fractional order indices are

from 0 to 2 but also possesses a special time fractional operator on the spatial derivative,

which is challenging to approximate. Although there is some literature [163, 187, 296]

involving the exact solution of the generalised Oldroyd-B fluid, the solution is typically

given in series form with special functions, such as the Fox H-function or the multivariate

Mittag-Leffler function, and both of these functions are difficult to express explicitly.

Therefore, a numerical solution of (7.4) is a promising tool to provide insight into the

behaviour of the model. In [11], Bazhlekova and Bazhlekov presented a finite difference

method to solve the viscoelastic flow of a generalised Oldroyd-B fluid (7.1). They utilised

the Grunwald-Letnikov formula to approximate the Riemann-Liouville time fractional

Chapter 7 142

derivative, which has first order accuracy, however, no theoretical analysis was given.

Recently, Feng et al. [80] proposed a finite difference method for the generalised fractional

Oldroyd-B fluid (7.1) between two rigid plates and gave the stability and convergence

analysis, which has low order accuracy as well. To the best of the authors’ knowledge,

there is no literature reported on the numerical solution of Eqs.(7.2) and (7.3). Therefore,

the numerical solution of Eq.(7.4) has also not appeared. For the two kinds of time

fractional derivatives in the L.H.S. of Eq.(7.4), the so-called L1 or L2 schemes can be

used for approximation. For the coupled operator (time fractional operator on the spatial

derivative) in the R.H.S. of Eq.(7.4), few techniques can be applied. As Eq.(7.4) involves

these terms simultaneously, the derivation of the numerical solution becomes difficult and

it is more challenging to establish the theoretical analysis. The main contributions of this

paper are as follows:

• We propose two new different finite difference schemes to approximate the coupled

operator Dβt∂2u∂x2

, in which the mixed L scheme is used to discretise the equation

at mesh point (xi, tn− 12) directly. We also establish the L2 scheme for the term

Dγt u(x, t) with first order accuracy. In addition, we give an important and useful

lemma, which can be extended to other multi-term time fractional diffusion prob-

lems;

• We derive two different finite difference schemes for problem (7.4) with accuracy

O(τ + h2) and O(τmin3−γs,2−αq ,2−β +h2), respectively and establish the stabili-

ty and convergence analysis. We prove our method is unconditionally stable and

convergent under the discrete H1 norm;

• Our numerical methods are robust and flexible, which can be used to deal with

the problem (7.4) under different initial and boundary conditions, for which an

analytical solution may be not feasible;

• Our numerical solution is more general and can be used to solve other time fractional

diffusion problems, such as the generalised Oldroyd-B fluid model with or without

a magnetic field effect, the generalised Maxwell fluid model, the generalised second-

grade fluid model and the generalised Burgers’ fluid model.

The outline of this chapter is as follows. In Section 7.2, some preliminary knowledge

is given, in which two numerical schemes to discretise the time fractional derivative are

proposed. In Section 7.3, we develop the finite difference schemes for Eq.(7.4). We proceed

with the proof of the stability and convergence of the scheme using the energy method

and discuss the solvability of the numerical scheme in Section 7.4. In Section 7.5, we

present two numerical examples to demonstrate the effectiveness of our method and some

conclusions are summarised.

7.2 Preliminary knowledge

For convenience, in the subsequent sections, we suppose that C,C1, C2, . . . are positive

constants, whose values will be implicitly determined by the surrounding context. Firstly,

143 Chapter 7

in the interval [0, L], we take the mesh points xi = ih, i = 0, 1, · · · ,M , and tn = nτ ,

n = 0, 1, · · · , N , where h = L/M , τ = T/N are the uniform spatial step size and temporal

step size, respectively. Denote Ωτ ≡ tn| 0 ≤ n ≤ N and Ωh ≡ xi| 0 ≤ i ≤M. Define

the grid function uni = u(xi, tn) and fni = f(xi, tn). We introduce the following notations:

∇tuni =uni − u

n−1i

τ, u

n− 12

i =uni + un−1

i

2,

∇xuni =uni − uni−1

h, δ2

xuni =

uni−1 − 2uni + uni+1

h2.

Denote

Vh = v | v is a grid function on Ωh and v0 = vM = 0.

For any χ, v ∈ Vh, we define the following discrete inner products and induced norms:

(χ, v) = h

M−1∑i=1

χivi, 〈∇xχ,∇xv〉 = h

M∑i=1

∇xχi · ∇xvi,

||v||0 =√

(v, v), ||v||∞ = max0≤i≤M

|vi|,

|v|1 =√〈∇xv,∇xv〉, ||v||1 =

√a2||v||20 + a3|v|21.

It is straightforward to check that

(δ2xvk, vn) = −〈∇xvk,∇xvn〉, (7.7)

(δ2xvk,∇tvn) = −1

τ〈∇xvk,∇xvn −∇xvn−1〉 = −〈∇xvk,∇t(∇xvn)〉. (7.8)

To discretise the time fractional derivative Dγt u(x, t) (1 < γ < 2) at (xi, tn), we have

Dγt u(xi, tn) =

1

Γ(2− γ)

∫ tn

0(tn − s)1−γ ∂

2u(xi, s)

∂s2ds

=1

Γ(2− γ)

n∑k=1

∫ tk

tk−1

(tn − s)1−γ ∂2u(xi, s)

∂s2ds

=1

Γ(2− γ)

n∑k=1

∫ tk

tk−1

(tn − s)1−γ ·uki − 2uk−1

i + uk−2i

τ2ds+ rn

=τ1−γ

Γ(3− γ)

[a

(γ)0 ∇tu

ni −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki − a

(γ)n−1

∂u(xi, 0)

∂t

]+ rn,

where u−1i = u0

i −τ∂u(xi,0)

∂t and a(γ)k = (k+1)2−γ−k2−γ , k = 0, 1, 2, . . . For the truncation

error rn, we have [138]

|rn| ≤ CT 2−γ

Γ(3− γ)max

0≤t≤T

∣∣∣∣∂3u(xi, t)

∂t3

∣∣∣∣ · τ +O(τ2).

Chapter 7 144

Then we can obtain the discrete scheme for the time fractional derivative Dγt u(x, t) at

mesh points (xi, tn)

Dγt u(xi, tn)

=τ1−γ

Γ(3− γ)

[a

(γ)0 ∇tu

ni −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki − a

(γ)n−1

∂u(xi, 0)

∂t

]+R1, (7.9)

where |R1| ≤ Cτ .

To discretise the time fractional derivative Dγt u(x, t) (1 < γ < 2) at (xi, tn− 1

2), we have

the following so-called L2 formula [238]

Dγt u(xi, tn− 1

2)

=τ1−γ

Γ(3− γ)

[a

(γ)0 ∇tu

ni −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

ki − a

(γ)n−1

∂u(xi, 0)

∂t

]+R2, (7.10)

where |R2| ≤ Cτ3−γ .

Lemma 7.2.1 For 1 < γ < 2, define a(γ)k = (k + 1)2−γ − k2−γ, k = 0, 1, 2, . . . , n and

vector S = [S1, S2, S3, . . . , SN ] and constant P , then it holds that

τ1−γ

Γ(3− γ)

N∑n=1

[a

(γ)0 Sn −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)Sk − a

(γ)n−1P

]Sn

≥ T 1−γ

2Γ(2− γ)

N∑n=1

S2n −

T 2−γ

2τΓ(3− γ)P 2, N = 1, 2, 3, . . .

Proof. See [238].

To discretise the time fractional derivative Dβt u(x, t) (0 < β < 1) at (xi, tn), we have the

following so-called L1 formula [238]

Dβt u(xi, tn) =

τ−β

Γ(2− β)

[d

(β)0 uni −

n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)u

ki − d

(β)n−1u

0i

]+R3

=τ1−β

Γ(2− β)

n∑k=1

d(β)n−k∇tu

ki +R3, (7.11)

where d(β)k = (k+ 1)1−β − k1−β, k = 0, 1, 2, . . . , n and |R3| ≤ Cτ2−β. It is straightforward

to derive the following lemma on the properties of d(β)k [149].

Lemma 7.2.2 For 0 < β < 1, define d(β)k = (k + 1)1−β − k1−β, k = 0, 1, 2, . . . then

1. d(β)k > 0, d

(β)0 = 1, d

(β)k > d

(β)k+1, lim

k→∞d

(β)k = 0,

2.n−1∑k=0

(d(β)k − d

(β)k+1) + d

(β)n = 1,

145 Chapter 7

3. d(β)k+1 − 2d

(β)k + d

(β)k−1 ≥ 0, k ≥ 1,

4. (1− β)(k + 1)−β ≤ a(β)k ≤ (1− β)k−β.

Since

∂2u(xi, tn)

∂x2= δ2

xuni −

h2

12

∂4u(ξi, tn)

∂x4,

then we have

Dβt

∂2u(xi, tn)

∂x2=

τ−β

Γ(2− β)

[d

(β)0 δ2

xuni −

n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)δ

2xu

ki − d

(β)n−1δ

2xu

0i

]+R4

=τ1−β

Γ(2− β)

n∑k=1

d(β)n−k∇t(δ

2xu

ki ) +R4, (7.12)

where |R4| ≤ C(τ2−β + h2).

Lemma 7.2.3 For 0 < β < 1, it holds that

τ1−β

Γ(2− β)

n∑k=1

d(β)n−k

(∇t(δ2

xuk),∇tun

)= − τ1−β

Γ(2− β)

n∑k=1

d(β)n−k

⟨∇t(∇xuk),∇t(∇xun)

⟩.

Proof. Combining (7.8) and (7.12), we obtain

τ1−β

Γ(2− β)

n∑k=1

d(β)n−k

(∇t(δ2

xuk),∇tun

)=

τ−β

Γ(2− β)

[d

(β)0

(δ2xu

n,∇tun)−n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)

(δ2xu

k,∇tun)− d(β)

n−1

(δ2xu

0,∇tun)]

=− τ−β

Γ(2− β)

[d

(β)0

⟨∇xun,∇t(∇xun)

⟩−n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)

⟨∇xuk,∇t(∇xun)

⟩− d(β)

n−1

⟨∇xu0,∇t(∇xun)

⟩]=− τ−β

Γ(2− β)

⟨d

(β)0 ∇xu

n −n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)∇xu

k − d(β)n−1∇xu

0,∇t(∇xun)⟩

=− τ1−β

Γ(2− β)

n∑k=1

d(β)n−k

⟨∇t(∇xuk),∇t(∇xun)

⟩.

Now we consider the discretization of Dβt u(x, t) at grid points (xi, tn− 1

2). From (7.12), we

have

Dβt u(xi, tn− 1

2) ≈1

2

[Dβt u(xi, tn) +Dβ

t u(xi, tn−1)]

=τ1−β

2Γ(2− β)

[ n∑k=1

d(β)n−k∇tu

ki +

n−1∑k=1

d(β)n−1−k∇tu

ki

]+R3. (7.13)

Chapter 7 146

Similarly, we have

Dβt

∂2u(xi, tn− 12)

∂x2≈1

2

[Dβt

∂2u(xi, tn)

∂x2+Dβ

t

∂2u(xi, tn−1)

∂x2

]=

τ1−β

2Γ(2− β)

[ n∑k=1

d(β)n−k∇t(δ

2xu

ki ) +

n−1∑k=1

d(β)n−1−k∇t(δ

2xu

ki )]

+R4. (7.14)

Lemma 7.2.4 [172] Let g0, g1, . . . , gn, . . . be a sequence of real numbers with the prop-

erties

gn ≥ 0, gn − gn−1 ≤ 0, gn+1 − 2gn + gn−1 ≥ 0.

Then for any positive integer M , and for each vector [V1, V2, . . . , VM ] with M real entries,

M∑n=1

( n−1∑p=0

gp Vn−p

)Vn ≥ 0.

Now we will prove a very important and useful lemma.

Lemma 7.2.5 For 0 < β < 1, define d(β)k = (k + 1)1−β − k1−β, k = 0, 1, 2, . . . , n, then

for any positive integer N and vector Q = [v1, v2, . . . , vN−1, vN ] ∈ RN , we have

N∑n=1

n∑k=1

d(β)n−k v

kvn ≥ 0, (7.15)

N∑n=1

n∑k=1

d(β)n−k v

kvn +

N∑n=1

n−1∑k=1

d(β)n−1−k v

kvn ≥ 0. (7.16)

Proof. It is easy to check that

N∑n=1

n∑k=1

d(β)n−k v

kvn =

N∑n=1

( n−1∑k=0

d(β)k vn−k

)vn.

Then using Lemmas 7.2.2 and 7.2.4, we have

N∑n=1

( n−1∑k=0

d(β)k vn−k

)vn ≥ 0,

i.e.,

N∑n=1

n∑k=1

d(β)n−k v

kvn ≥ 0.

147 Chapter 7

The sum in (7.16) can be rewritten in the following form

N∑n=1

n∑k=1

d(β)n−k v

kvn +N∑n=1

n−1∑k=1

d(β)n−1−k v

kvn = QAQT ,

where

A =

d(β)0 0 · · · 0 0

d(β)0 + d

(β)1 d

(β)0 · · · 0 0

d(β)1 + d

(β)2 d

(β)0 + d

(β)1 · · · 0 0

......

. . ....

...

d(β)N−3 + d

(β)N−2 d

(β)N−4 + d

(β)N−3 · · · d

(β)0 0

d(β)N−2 + d

(β)N−1 d

(β)N−3 + d

(β)N−2 · · · d

(β)0 + d

(β)1 d

(β)0

.

Note that the proof of the inequality in (7.16) is equivalent to proving the matrix A is

positive definite. Therefore we only need to prove HN = A+AT

2 is positive definite [211].

HN is a real symmetric Toeplitz matrix and has the form

HN =1

2

2d(β)0 d

(β)0 + d

(β)1 d

(β)1 + d

(β)2 · · · d

(β)N−2 + d

(β)N−1

d(β)0 + d

(β)1 2d

(β)0 d

(β)0 + d

(β)1 · · · d

(β)N−3 + d

(β)N−2

d(β)1 + d

(β)2 d

(β)0 + d

(β)1 2d

(β)0 · · · d

(β)N−4 + d

(β)N−3

......

.... . .

...

d(β)N−3 + d

(β)N−2 d

(β)N−4 + d

(β)N−3 d

(β)N−5 + d

(β)N−4 · · · d

(β)0 + d

(β)1

d(β)N−2 + d

(β)N−1 d

(β)N−3 + d

(β)N−2 d

(β)N−4 + d

(β)N−3 · · · 2d

(β)0

.

In the following, we will prove det(HN ) > 0. It is straightforward to verify that det(H1) =

d(β)0 > 0, det(H2) = (d

(β)0 )2 − (d

(β)0 +d

(β)1 )2

4 > 0. For a finite integer N , we can explicitly

calculate the value of det(HN ) > 0. When N is sufficiently large, according to [18]

(Propositions 10.2 and 10.4), we have

det(HN )

det(HN+1)> 0.

Then we can conclude that det(HN+1) > 0. To illustrate this, we plot det(HN )det(HN+1) with

different β and N (see Figure 7.1). We can see that det(HN )det(HN+1) > 0, particularly, when

almost N > 250, det(HN )det(HN+1) ≈ Cβ > 0. As matrix Hi, i = 1, 2, . . . , N are the principal

minors of matrix HN+1 and det(Hk) > 0, k = 1, 2, . . . , N + 1, then the real symmetric

Toeplitz matrix HN+1 is positive definite. The proof is completed.

To derive the finite difference scheme we also need the following lemma.

Chapter 7 148

N0 50 100 150 200 250

det(H

N)

det(H

N+1)

0

2

4

6

8

10

12 β=0.1β=0.2β=0.3β=0.4β=0.5β=0.6β=0.7β=0.8β=0.9

Figure 7.1: The ratio of det(HN )det(HN+1) for different β and N

Lemma 7.2.6 If u(x, t) ∈ C0,3x,t (Ω), then we have [43]

u(xi, tn− 12) =

u(xi, tn) + u(xi, tn−1)

2+O(τ2), (7.17)

∂

∂tu(xi, tn− 1

2) =

u(xi, tn)− u(xi, tn−1)

τ+O(τ2). (7.18)

For the discretization of the time fractional derivative Dαt u(x, t) (0 < α < 1), it is the

same as the discretization of Dβt u(x, t).

7.3 Derivation of the numerical schemes

In this section, we will give two different finite difference schemes for Eq.(7.4).

7.3.1 Scheme I: first order implicit scheme

Assume that u(x, t) ∈ C4,3x,t (Ω), from Eq.(7.4), we have

s∑j=1

bj Dγjt u(xi, tn) + a1

∂u(xi, tn)

∂t+

q∑l=1

clDαlt u(xi, tn) + a2u(xi, tn)

=a3∂2u(xi, tn)

∂x2+ a4D

βt

∂2u(xi, tn)

∂x2+ f(xi, tn). (7.19)

149 Chapter 7

Using Eqs.(7.9), (7.11) and (7.12), we obtain

s∑j=1

bjµ1,j

[a

(γj)0 ∇tuni −

n−1∑k=1

(a(γj)n−k−1 − a

(γj)n−k)∇tu

ki − a

(γj)n−1φ2(xi)

]+a1∇tuni +

q∑l=1

clµ2,l

n∑k=1

d(αl)n−k∇tu

ki + a2u

ni

=a3δ2xu

ni + a4µ3

n∑k=1

d(β)n−k∇t(δ

2xu

ki ) + fni +Rn1,i, (7.20)

where µ1,j = τ1−γj

Γ(3−γj) , µ2,l = τ1−αlΓ(2−αl) , µ3 = τ1−β

Γ(2−β) and |Rn1,i| ≤ C(τ + h2), in which C is

independent of τ and h. Then, omitting the error term and denoting Uni as the numerical

approximation to uni , the implicit finite difference scheme for Eq.(7.4) at point (xi, tn) is

given by

s∑j=1

bjµ1,j

[a

(γj)0 ∇tUni −

n−1∑k=1

(a(γj)n−k−1 − a

(γj)n−k)∇tU

ki − a

(γj)n−1φ2(xi)

]+a1∇tUni +

q∑l=1

clµ2,l

n∑k=1

d(αl)n−k∇tU

ki + a2U

ni

=a3δ2xU

ni + a4µ3

n∑k=1

d(β)n−k∇t(δ

2xU

ki ) + fni , (7.21)

with initial and boundary conditions

U0i = φ1(xi), 0 ≤ i ≤M, Un0 = UnM = 0, 1 ≤ n ≤ N.

7.3.2 Scheme II: mixed L scheme

Assume that u(x, t) ∈ C4,3x,t (Ω), from Eq.(7.4), we have

s∑j=1

bj Dγjt u(xi, tn− 1

2) + a1

∂u(xi, tn− 12)

∂t+

q∑l=1

clDαlt u(xi, tn− 1

2) + a2u(xi, tn− 1

2)

=a3

∂2u(xi, tn− 12)

∂x2+ a4D

βt

∂2u(xi, tn− 12)

∂x2+ f(xi, tn− 1

2). (7.22)

Applying Eqs.(7.10), (7.13) and (7.14) and Lemma 7.2.6, we have

s∑j=1

bjµ1,j

[a

(γj)0 ∇tuni −

n−1∑k=1

(a(γj)n−k−1 − a

(γj)n−k)∇tu

ki − a

(γj)n−1φ2(xi)

]+ a1∇tuni

+

q∑l=1

clµ2,l

2

[ n∑k=1

d(αl)n−k∇tu

ki +

n−1∑k=1

d(αl)n−1−k∇tu

ki

]+ a2u

n− 12

i (7.23)

=a3δ2xu

n− 12

i +a4µ3

2

[ n∑k=1

d(β)n−k∇t(δ

2xu

ki ) +

n−1∑k=1

d(β)n−1−k∇t(δ

2xu

ki )]

+ fn− 1

2i +Rn2,i,

Chapter 7 150

where |Rn2,i| ≤ C(τmin3−γs,2−αq ,2−β + h2). Then, omitting the error term, we obtain the

mixed L finite difference scheme for Eq.(7.4) at point (xi, tn− 12)

s∑j=1

bjµ1,j

[a

(γj)0 ∇tUni −

n−1∑k=1

(a(γj)n−k−1 − a

(γj)n−k)∇tU

ki − a

(γj)n−1φ2(xi)

]+ a1∇tUni

+

q∑l=1

clµ2,l

2

[ n∑k=1

d(αl)n−k∇tU

ki +

n−1∑k=1

d(αl)n−1−k∇tU

ki

]+ a2U

n− 12

i (7.24)

=a3δ2xU

n− 12

i +a4µ3

2

[ n∑k=1

d(β)n−k∇t(δ

2xU

ki ) +

n−1∑k=1

d(β)n−1−k∇t(δ

2xU

ki )]

+ fn− 1

2i .

Remark 7.3.1 Compared to the scheme I, scheme II has high order accuracy. However,

more terms are added in the scheme II.

7.4 Theoretical analysis

7.4.1 Solvability

Firstly, we discuss the solvability of the finite difference scheme (7.21).

Theorem 7.4.1 The finite difference scheme (7.21) is uniquely solvable.

Proof. At each time level, the coefficient matrix B is linear tridiagonal

B =

d1 + 2d2 −d2 0 · · · 0 0

−d2 d1 + 2d2 −d2 · · · 0 0

0 −d2 d1 + 2d2 · · · 0 0...

......

. . ....

...

0 0 0 · · · d1 + 2d2 −d2

0 0 0 · · · −d2 d1 + 2d2

,

where d1 =s∑j=1

bjµ1,jτ + a1

τ +q∑l=1

clµ2,lτ + a2 > 0 and d2 = a3

h2+ a4µ3

τh2> 0. Then B is a

strictly diagonally dominant matrix. Therefore B is nonsingular, which means that the

numerical scheme (7.21) is uniquely solvable.

The solvability of the finite difference scheme (7.24) is similar.

7.4.2 Stability

Here, we will analyze the stability of the schemes (7.21) and (7.24) using the energy

method.

151 Chapter 7

Theorem 7.4.2 The implicit finite difference scheme (7.21) is unconditionally stable and

it holds that

||U l||21 ≤ ||U0||21 +

s∑j=1

bjT2−γj

Γ(3− γj)||φ2||20 +

T

2ε0max

1≤n≤l||fn||20,

where ε0 =s∑j=1

bjT1−γj

2Γ(2−γj) + a1, 1 ≤ l ≤ N and U l = [U l1, Ul2, . . . , U

lM−1]T is the solution

vector of (7.21).

Proof. Multiplying Eq.(7.21) by hτ∇tUni and summing i from 1 to M − 1 and n from 1

to l, 1 ≤ l ≤ N , we obtain

τs∑j=1

bjµ1,j

l∑n=1

M−1∑i=1

h[a

(γj)0 ∇tUni −

n−1∑k=1

(a(γj)n−k−1 − a

(γj)n−k)∇tU

ki

−a(γj)n−1φ2(xi)

]∇tUni + a1τ

l∑n=1

M−1∑i=1

h(∇tUni )2

+τ

q∑l=1

clµ2,l

l∑n=1

M−1∑i=1

hn∑k=1

d(αl)n−k∇tU

ki ∇tUni + a2τ

l∑n=1

M−1∑i=1

hUni ∇tUni

=a3τl∑

n=1

M−1∑i=1

hδ2xU

ni ∇tUni + a4µ3τ

l∑n=1

M−1∑i=1

hn∑k=1

d(β)n−k∇t(δ

2xU

ki )∇tUni

+τl∑

n=1

M−1∑i=1

hfni ∇tUni . (7.25)

Using Lemma 7.2.1, we have

τs∑j=1

bjµ1,j

l∑n=1

M−1∑i=1

h[a

(γj)0 ∇tUni −

n−1∑k=1

(a(γj)n−k−1 − a

(γj)n−k)∇tU

ki − a

(γj)n−1φ2(xi)

]∇tUni

≥s∑j=1

bj

(τT 1−γj

2Γ(2− γj)

l∑n=1

M−1∑i=1

h(∇tUni )2 − T 2−γj

2Γ(3− γj)

M−1∑i=1

hφ2(xi)2

)

=

s∑j=1

bjτT 1−γj

2Γ(2− γj)

l∑n=1

||∇tUn||20 −s∑j=1

bjT 2−γj

2Γ(3− γj)||φ2||20. (7.26)

For the second term, we have

a1τl∑

n=1

M−1∑i=1

h(∇tUni )2 = a1τ

l∑n=1

||∇tUn||20. (7.27)

Chapter 7 152

Using (7.15), we obtain

τ

q∑l=1

clµ2,l

l∑n=1

M−1∑i=1

hn∑k=1

d(αl)n−k∇tU

ki ∇tUni

=τ

q∑l=1

clµ2,l

l∑n=1

n∑k=1

d(αl)n−k(∇tU

k,∇tUn) ≥ 0. (7.28)

Utilising the inequality a(a− b) ≥ 12(a2 − b2), we have

a2τl∑

n=1

M−1∑i=1

hUni ∇tUni = a2

l∑n=1

(Un, Un − Un−1)

≥a2

2

l∑n=1

(||Un||20 − ||Un−1||20) =a2

2(||U l||20 − ||U0||20). (7.29)

Applying (7.8) and the inequality a(a− b) ≥ 12(a2 − b2) again, we obtain

a3τl∑

n=1

M−1∑i=1

hδ2xU

ni ∇tUni = a3τ

l∑n=1

(δ2xU

n,∇tUn) = −a3

l∑n=1

〈∇xUn,∇xUn −∇xUn−1〉

≤ − a3

2

l∑n=1

(|Un|21 − |Un−1|21) =a3

2(|U0|21 − |U l|21). (7.30)

Combining (7.8), (7.16) and Lemma 7.2.3, we have

a4µ3τl∑

n=1

M−1∑i=1

hn∑k=1

d(β)n−k∇t(δ

2xU

ki )∇tUni = a4µ3τ

l∑n=1

n∑k=1

d(β)n−k

(∇t(δ2

xUk),∇tUn

)=− a4µ3τ

l∑n=1

n∑k=1

d(β)n−k

⟨∇t(∇xUk),∇t(∇xUn)

⟩≤ 0. (7.31)

Using the important inequality ab ≤ εa2 + b2

4ε(ε > 0), we have

τl∑

n=1

M−1∑i=1

hfni ∇tUni ≤ τε0

l∑n=1

M−1∑i=1

h(∇tUni )2 +τ

4ε0

l∑n=1

M−1∑i=1

h(fni )2

=τε0

l∑n=1

||∇tUn||20 +τ

4ε0

l∑n=1

||fn||20 ≤ τε0

l∑n=1

||∇tUn||20 +T

4ε0max

1≤n≤l||fn||20, (7.32)

where ε0 =s∑j=1

bjT1−γj

2Γ(2−γj) + a1. Substituting (7.26)-(7.32) into (7.25), we have

τε0

l∑n=1

||∇tUn||20 −s∑j=1

bjT2−γj

2Γ(3− γj)||φ2||20 +

a2

2(||U l||20 − ||U0||20)

≤a3

2(|U0|21 − |U l|21) + τε0

l∑n=1

||∇tUn||20 +T

4ε0max

1≤n≤l||fn||20,

153 Chapter 7

then rearranging gives

a2||U l||20 + a3|U l|21

≤a2||U0||20 + a3|U0|21 +s∑j=1

bjT2−γj

Γ(3− γj)||φ2||20 +

T

2ε0max

1≤n≤l||fn||20, (7.33)

namely,

||U l||21 ≤ ||U0||21 +s∑j=1

bjT2−γj

Γ(3− γj)||φ2||20 +

T

2ε0max

1≤n≤l||fn||20,

which proves that the scheme (7.22) is unconditionally stable.

Theorem 7.4.3 The implicit finite difference scheme (7.24) is unconditionally stable and

it holds that

||U l||21 ≤ ||U0||21 +s∑j=1

bjT2−γj

Γ(3− γj)||φ2||20 +

T

2ε0max

1≤n≤l||fn−

12 ||20,

where ε0 =s∑j=1

bjT1−γj

2Γ(2−γj) + a1, 1 ≤ l ≤ N and U l = [U l1, Ul2, . . . , U

lM−1]T is the solution

vector of (7.24).

Proof. Multiplying Eq.(7.24) by hτ∇tUni and summing i from 1 to M − 1 and n from

1 to l, 1 ≤ l ≤ N , we obtain

τs∑j=1

bjµ1,j

l∑n=1

M−1∑i=1

h[a

(γj)0 ∇tUni −

n−1∑k=1

(a(γj)n−k−1 − a

(γj)n−k)∇tU

ki

−a(γj)n−1φ2(xi)

]∇tUni + a1τ

l∑n=1

M−1∑i=1

h(∇tUni )2

+τ

q∑l=1

clµ2,l

2

l∑n=1

M−1∑i=1

h[ n∑k=1

d(αl)n−k∇tU

ki +

n−1∑k=1

d(αl)n−1−k∇tU

ki

]∇tUni

+a2τ

l∑n=1

M−1∑i=1

hUn− 1

2i ∇tUni = a3τ

l∑n=1

M−1∑i=1

hδ2xU

n− 12

i ∇tUni

+a4µ3τ

2

l∑n=1

M−1∑i=1

h[ n∑k=1

d(β)n−k∇t(δ

2xU

ki ) +

n−1∑k=1

d(β)n−1−k∇t(δ

2xU

ki )]∇tUni

+τl∑

n=1

M−1∑i=1

hfn− 1

2i ∇tUni . (7.34)

Chapter 7 154

Using (7.16), we obtain

τ

q∑l=1

clµ2,l

2

l∑n=1

M−1∑i=1

h[ n∑k=1

d(αl)n−k∇tU

ki +

n−1∑k=1

d(αl)n−1−k∇tU

ki

]∇tUni

=τ

q∑l=1

clµ2,l

2

[ l∑n=1

n∑k=1

d(αl)n−k

(∇tUk,∇tUn

)+

l∑n=1

n−1∑k=1

d(αl)n−1−k

(∇tUk,∇tUn

)]≥0. (7.35)

For the fourth term, we have

a2τ

l∑n=1

M−1∑i=1

hUn− 1

2i ∇tUni =

a2

2

l∑n=1

(Un + Un−1, Un − Un−1)

=a2

2

l∑n=1

(||Un||20 − ||Un−1||20) =a2

2(||U l||20 − ||U0||20). (7.36)

For the fifth term, we obtain

a3τl∑

n=1

M−1∑i=1

hδ2xU

n− 12

i ∇tUni = a3τl∑

n=1

(δ2xU

n− 12 ,∇tUn)

=− a3

2

l∑n=1

(|Un|21 − |Un−1|21) =a3

2(|U0|21 − |U l|21). (7.37)

Combining (7.8), (7.16) and Lemma 7.2.3, we have

a4µ3τ

2

l∑n=1

M−1∑i=1

h[ n∑k=1

d(β)n−k∇t(δ

2xU

ki ) +

n−1∑k=1

d(β)n−1−k∇t(δ

2xU

ki )]∇tUni

=a4µ3τ

2

l∑n=1

[ n∑k=1

d(β)n−k

(∇t(δ2

xUk),∇t(Un)

)+n−1∑k=1

d(β)n−1−k

(∇t(δ2

xUk),∇t(Un)

)]=− a4µ3τ

2

[ l∑n=1

n∑k=1

d(β)n−k

⟨∇t(∇xUk),∇t(∇xUn)

⟩+

l∑n=1

n−1∑k=1

d(β)n−1−k

⟨∇t(∇xUk),∇t(∇xUn)

⟩]≤ 0. (7.38)

Using the important inequality ab ≤ εa2 + b2

4ε(ε > 0), we have

τ

l∑n=1

M−1∑i=1

hfn− 1

2i ∇tUni ≤τε0

l∑n=1

||∇tUn||20 +τ

4ε0

l∑n=1

||fn−12 ||20

≤τε0

l∑n=1

||∇tUn||20 +T

4ε0max

1≤n≤l||fn−

12 ||20, (7.39)

155 Chapter 7

where ε0 =s∑j=1

bjT1−γj

2Γ(2−γj) + a1. Substituting (7.26), (7.27) and (7.35)-(7.39) into (7.34), we

have

τε0

l∑n=1

||∇tUn||20 −s∑j=1

bjT2−γj

2Γ(3− γj)||φ2||20 +

a2

2(||U l||20 − ||U0||20)

≤a3

2(|U0|21 − |U l|21) + τε0

l∑n=1

||∇tUn||20 +T

4ε0max

1≤n≤l||fn−

12 ||20,

then we have

a2||U l||20 + a3|U l|21

≤a2||U0||20 + a3|U0|21 +

s∑j=1

bjT2−γj

Γ(3− γj)||φ2||20 +

T

2ε0max

1≤n≤l||fn−

12 ||20,

namely,

||U l||21 ≤ ||U0||21 +s∑j=1

bjT2−γj

Γ(3− γj)||φ2||20 +

T

2ε0max

1≤n≤l||fn−

12 ||20,

which proves that the scheme (7.24) is unconditionally stable.

7.4.3 Convergence

Now we discuss the convergence of the schemes (7.21) and (7.24).

Theorem 7.4.4 Suppose that the solution of problem (7.4)-(7.6) satisfies u(x, t) ∈ C4,3x,t (Ω).

Define un = [un1 , un2 , . . . , u

nM−1]T as the exact solution vector, Un = [Un1 , U

n2 , . . . , U

nM−1]T

as the numerical solution vector of (7.21), and Un = [Un1 , Un2 , . . . , U

nM−1]T as the numeri-

cal solution vector of (7.24), respectively. Then there exists two positive constants C1 and

C2 independent of h and τ such that

||un − Un||1 ≤ C1

√TL

2ε0(τ + h2),

||un − Un||1 ≤ C2

√TL

2ε0(τmin3−γs,2−αq ,2−β + h2),

where ε0 =s∑j=1

bjT1−γj

2Γ(2−γj) + a1.

Chapter 7 156

Proof. Denote eni = uni − Uni , en = [en1 , en2 , . . . , e

nM−1]T . Subtracting (7.21) from (7.20),

we have

s∑j=1

bjµ1,j

[a

(γj)0 ∇teni −

n−1∑k=1

(a(γj)n−k−1 − a

(γj)n−k)∇te

ki

]+ a1∇teni

+

q∑l=1

clµ2,l

n∑k=1

d(αl)n−k∇te

ki + a2e

ni = a3δ

2xeni + a4µ3

n∑k=1

d(β)n−k∇t(δ

2xeki ) +Rn1,i,

with e0i = 0, en0 = enM = 0. Then from (7.32) and (7.33), we have

a2||en||20 + a3|en|21 ≤τh

2ε0

n∑k=1

M−1∑i=1

(Rn1,i)2

≤ τh2ε0

n∑k=1

M−1∑i=1

C21 (τ + h2)2 ≤ C2

1TL

2ε0(τ + h2)2,

namely,

||un − Un||21 ≤C2

1TL

2ε0(τ + h2)2.

Similarly, we can obtain

||un − Un||21 ≤C2

2TL

2ε0

(τmin3−γs,2−αq ,2−β + h2

)2.

7.5 Numerical examples

Example 7.5.1 We consider the following multi-term time fractional viscoelastic non-

Newtonian fluid model.

Dγt u(x, t) + ∂u(x,t)

∂t +Dαt u(x, t) + u(x, t)

= ∂2u(x,t)∂x2

+Dβt∂2u(x,t)∂x2

+ f(x, t), (x, t) ∈ (0, 1)× (0, 1],

u(x, 0) = sinπx, ut(x, 0) = 0, 0 ≤ x ≤ 1,

u(0, t) = 0, u(1, t) = 0, 0 ≤ t ≤ 1,

where 0 < α, β < 1, 1 < γ < 2,

f(x, t) = sinπx[Γ(4)t3−γ

Γ(4− γ)+ 3t2 +

Γ(4)t3−α

Γ(4− α)+ (1 + π2)(t3 + 1) +

π2Γ(4)t3−β

Γ(4− β)

]and the exact solution is u(x, t) = (t3 + 1) sinπx.

Firstly, we use the implicit finite difference scheme (7.21) (Scheme I) to solve the equation

and the numerical results are given in Table 7.1. The table lists the L2 error and L∞

error and the convergence order of τ for different α, β, γ with h = 1/1000 at t = 1. We

157 Chapter 7

can see that the numerical results are in perfect agreement with the exact solution and

the convergence order reaches the expected first order.

Table 7.1: The temporal error and convergence of Scheme I for different α, β and γ withh = 1/1000

α = 0.7, β = 0.6, γ = 1.5 ||E(h, τ)||0 Order ||E(h, τ)||∞ Order

1/40 7.0478E-03 – 9.9671E-03 –1/80 3.2211E-03 1.13 4.5553E-03 1.131/160 1.4899E-03 1.11 2.1071E-03 1.111/320 6.9792E-04 1.09 9.8700E-04 1.091/640 3.3092E-04 1.08 4.6799E-04 1.08

α = 0.7, β = 0.8, γ = 1.6 ||E(h, τ)||0 Order ||E(h, τ)||∞ Order

1/40 1.0895E-02 – 1.5408E-02 –1/80 5.0166E-03 1.12 7.0946E-03 1.121/160 2.3164E-03 1.11 3.2759E-03 1.111/320 1.0742E-03 1.11 1.5191E-03 1.111/640 5.0073E-04 1.10 7.0814E-04 1.10

α = 0.5, β = 0.3, γ = 1.6 ||E(h, τ)||0 Order ||E(h, τ)||∞ Order

1/40 5.5522E-03 – 7.8520E-03 –1/80 2.6575E-03 1.06 3.7583E-03 1.061/160 1.2862E-03 1.05 1.8190E-03 1.051/320 6.2820E-04 1.03 8.8841E-04 1.031/640 3.0906E-04 1.02 4.3707E-04 1.02

Table 7.2: The temporal error and convergence of Scheme II for different α, β and γ withh = 1/1000

α = 0.7, β = 0.6, γ = 1.5 ||E(h, τ)||0 Order ||E(h, τ)||∞ Order

1/40 2.8002E-03 – 3.9601E-03 –1/80 1.0828E-03 1.37 1.5313E-03 1.371/160 4.1712E-04 1.38 5.8989E-04 1.381/320 1.6057E-04 1.38 2.2708E-04 1.381/640 6.2009E-05 1.37 8.7694E-05 1.37

α = 0.7, β = 0.8, γ = 1.6 ||E(h, τ)||0 Order ||E(h, τ)||∞ Order

1/40 6.7356E-03 – 9.5256E-03 –1/80 2.9253E-03 1.20 4.1370E-03 1.201/160 1.2677E-03 1.21 1.7928E-03 1.211/320 5.4905E-04 1.21 7.7648E-04 1.211/640 2.3795E-04 1.21 3.3652E-04 1.21

α = 0.5, β = 0.3, γ = 1.6 ||E(h, τ)||0 Order ||E(h, τ)||∞ Order

1/40 8.8331E-04 – 1.2492E-03 –1/80 3.0948E-04 1.51 4.3767E-04 1.511/160 1.0881E-04 1.51 1.5388E-04 1.511/320 3.8658E-05 1.49 5.4671E-05 1.491/640 1.4074E-05 1.46 1.9904E-05 1.46

Chapter 7 158

Then we apply the mixed L scheme (7.24) (Scheme II) to the equation. Table 7.2 displaysthe L2 error and L∞ error and convergence order of τ for different α, β, γ with h = 1/1000at t = 1. We can observe that the numerical results are in excellent agreement with theexact solution and the convergence order attains the expected min3−γ, 2−α, 2−β order.Compared to scheme I, the results of scheme II are more accurate. In addition, we presenta comparison of CPU time for two schemes in Table 7.3. Here the numerical computationswere carried out using MATLAB R2014b on a Dell desktop with configuration: Intel(R)Core(TM) i7-4790, 3.60 GHz and 16.0 GB RAM. We choose α = 0.7, β = 0.6, γ = 1.5and h = 1/1000 at t = 1 to observe the running time for different τ . We observe that therunning time of Scheme II is more than that of scheme I, which is due to the fact thatmore terms are added in Scheme II.

Table 7.3: The running time of Scheme I and II for different τ , with α = 0.7, β = 0.6, γ =1.5, h = 1/1000 at t = 1

τ Scheme I Scheme II

1/40 2.9279s 3.5410s1/80 9.5444s 10.6752s1/160 33.2557s 35.8488s1/320 123.4362s 130.4436s1/640 475.8209s 492.6424s

Example 7.5.2 Next, we consider the following unsteady MHD Couette flow of a gener-

alised Oldroyd-B fluid [163].(1 + λαDα

t )∂u(x,t)∂t = (1 + θβDβ

t )∂2u(x,t)∂x2

−K(1 + λαDαt )u(x, t), (x, t) ∈ (0, 1)× (0, 2],

u(x, 0) = 0, ut(x, 0) = 0, 0 ≤ x ≤ 1,

u(0, t) = 0, u(1, t) = 2tp, 0 ≤ t ≤ 2,

where 0 < α, β < 1, λ is the relaxation time, θ is the retardation time, K =σB2

0ρ , ρ is the

density of the fluid, B0 is the magnetic intensity and σ is the electrical conductivity, and

λ, θ,K ≥ 0.

This model describes the flow of an incompressible Olyroyd-B fluid bounded by two

infinite parallel rigid plates in a magnetic field. Initially, the whole system is at rest and

the lower plate is fixed. Then at time t = 0+, the upper plate starts to slide with some

velocity 2tp in the main flow direction, which is termed as the plane Couette flow. Due

to the influence of shear, the fluid is gradually in motion.

In the calculations, we choose h = 1/1000, τ = 1/100. In order to observe the effects of

different physical parameters on the velocity field, we plot some figures to demonstrate

the dynamic characteristics of the generalised Oldroyd-B fluid. Figure 7.2 shows the

effects of the power-law index p and constant K on the velocity. We can see that the

velocity increases with increasing the power-law index p and the magnetic body force is

favorable to the decay of the velocity. We can clearly find that the greater K is, the more

rapidly the velocity decays. Figure 7.3 exhibits the effects of the relaxation time λ and

the retardation time θ on the velocity. We can see that the smaller the λ, the more slowly

159 Chapter 7

the velocity decays. However, an opposite trend for the variation of θ can be seen. Figure

7.4 illustrates the change of the velocity with different parameters α and β. It is seen

that the larger the value of α, the more rapidly the velocity decays. The effect of β is

contrary to that of α. Figure 7.5 depicts the influence of time on the velocity and we can

observe that the flow velocity increases with increasing time.

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

1

2

3

4

5

6

7

8p=0.0p=0.5p=1.0p=2.0

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4

K=0K=2K=6K=10

Figure 7.2: Numerical solution profiles of velocity u(x, t) for different p (K = 2) and K(p = 1) with λ = 3, θ = 4, α = 0.5, β = 0.6 at t = 2

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4

λ=3λ=5λ=7λ=9

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4

θ=2θ=4θ=6θ=8

Figure 7.3: Numerical solution profiles of velocity u(x, t) for different λ (θ = 4) and θ(λ = 3) with p = 1, α = 0.5, β = 0.6, K = 2 at t = 2

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4

α=0.1α=0.3α=0.5α=0.7

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4

β=0.2β=0.4β=0.6β=0.8

Figure 7.4: Numerical solution profiles of velocity u(x, t) for different α (β = 0.6) and β(α = 0.5) with p = 1, λ = 3, θ = 4, K = 2 at t = 2

Chapter 7 160

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.5

1

1.5

2

2.5

3

3.5

4

t=0.5t=1.0t=1.5t=2.0

Figure 7.5: Numerical solution profiles of velocity u(x, t) at different t with p = 1, λ = 3,θ = 4, α = 0.5, β = 0.6, K = 2

7.6 Conclusions

In this chapter, we proposed the finite difference method to solve the novel multi-term

time fractional viscoelastic non-Newtonian fluid model. We not only presented an implicit

difference scheme with accuracy of O(τ + h2), but also give a higher order time scheme

with accuracy of O(τmin3−γs,2−αq ,2−β + h2). In addition, we established the stability

and convergence analysis of the finite difference schemes. Two numerical examples were

exhibited to verify the effectiveness and reliability of our method. We can conclude

that our numerical methods are robust and can be extended to other multi-term time

fractional diffusion equations, such as the generalized Oldroyd-B fluid in a rotating system,

the generalized Maxwell fluid model, the generalized second grade fluid model and the

generalized Burgers’ fluid model. In future work, we shall investigate the application of

these methods and techniques to the novel multi-term time fractional viscoelastic non-

Newtonian fluid model in high dimensional cases.

Chapter 8

Finite difference/finite element method for a novel 2D multi-term

time-fractional mixed sub-diffusion and diffusion-wave equation on

convex domains

8.1 Introduction

Fractional partial differential equations (FPDEs) have been widely used in many fields of

science because of their ability to model memory, non-local properties, anomalous diffusion

and spatial heterogeneity. To characterise some complex diffusion encountered in highly

heterogeneous fractured or disordered porous media, the concept of variable-order (VO)

fractional differential equations were proposed [54, 188, 189]. There are now well docu-

mented applications of FPDEs in physics [176, 230, 275], biology [23, 158, 174], chemistry

[273], hydrology [14, 148, 167], finance [220, 256], and the theory of fractional calculus is

growing rapidly [8, 60, 128, 139, 156, 203]. One class of fractional partial differential equa-

tions is the multi-term time-fractional diffusion equation, which mainly has three types:

the multi-term time-fractional sub-diffusion equation, whose fractional order belongs to

(0, 1); the multi-term time fractional diffusion-wave equation, whose fractional order be-

longs to (1, 2) and other multi-term time-fractional diffusion equations whose fractional

order belongs to (0, 2) or greater than 2. The analytical solutions of these equations have

been studied by some researchers. Daftardar-Gejji and Bhalekar [55] used the method

of separation of variables to deal with a multi-term fractional diffusion-wave equation

under homogeneous and non-homogeneous boundary conditions. Stojanovic [234] used

the Laguerre polynomials to approximate the solution of a one-dimensional diffusion-wave

problem. Based on a reasonable maximum principle, Luchko [171] established some prior

estimates and the uniqueness analysis for a multi-term time-fractional sub-diffusion equa-

tion. Jiang et al. [111] discussed the analytical solution of a multi-term time-fractional

diffusion-wave equation under three different kinds of boundary conditions. Jiang et al.

[112] also presented the analytical solution of a multi-term time-space fractional diffusion

equation. Based on a spectral representation of the Laplacian operator, Ye et al. [263]

obtained a series expansion solution for the 2D and 3D multi-term time-space fractional

diffusion equations. Li et al. [144] considered an initial-boundary value problem of a

multi-term time-fractional diffusion equation and gave the long-time asymptotic behavior

and the well-posedness analysis.

Although some analytical solutions of the multi-term time-fractional diffusion equation

can be derived, they are generally given in terms of multinomial Mittag-Leffler function-

s, which are extremely complex and difficult to evaluate. Therefore, applying numeri-

cal methods to deal with these equations would be more desirable, especially for those

161

Chapter 8 162

cases where analytical solutions are unavailable. For the multi-term time-fractional sub-

diffusion equation, the finite difference method [214], the fractional predictor-corrector

method [264], the finite element method [22, 57, 117, 209, 291], and the spectral method

[298] have been proposed as solution strategies. For the multi-term time-fractional diffusion-

wave equation, the spectral method [56], the finite difference method [215, 233, 278] and

the wavelet method [232] have been developed. For the solution of the third kind of

multi-term time-fractional diffusion equation, the decomposition scheme [87], the spline

collocation method [200], the fractional predictor-corrector method [154], the spectral

method [17, 41] and the finite element method [73] have been investigated.

In this article, a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-

wave equation on convex domains will be considered:

s∑p=1

a1,pDγpt u(y, z, t) + a2

∂u(y, z, t)

∂t+

w∑q=1

a3,qDαqt u(y, z, t) + a4u(y, z, t)

=a54u(y, z, t) + a6Dβt (4u(y, z, t)) + f(y, z, t), (y, z, t) ∈ Ω× J, (8.1)

subject to the initial conditions

u(y, z, 0) = φ1(y, z), ut(y, z, 0) = φ2(y, z), (y, z) ∈ Ω, (8.2)

the boundary conditions

u(y, z, t) = 0, (y, z) ∈ ∂Ω, t ∈ J , (8.3)

where a1,p ≥ 0, a3,q ≥ 0, ai ≥ 0, i = 2, 4, 5, 6, in which all the coefficients are not equal to

0 at the same time, 4u = ∂2u∂y2

+ ∂2u∂z2

, 1 < γ1 < γ2 < . . . < γs < 2, 0 < α1 < α2 < . . . <

αw < 1, 0 < β < 1, and J = (0, T ]. The Caputo time-fractional derivatives Dαt u(y, z, t),

Dβt u(y, z, t) (0 < α, β < 1) and Dγ

t u(y, z, t) (1 < γ < 2) are given by [203]

Dαt u(y, z, t) =

1

Γ(1− α)

∫ t

0(t− s)−α∂u(y, z, s)

∂sds,

Dβt u(y, z, t) =

1

Γ(1− β)

∫ t

0(t− s)−β ∂u(y, z, s)

∂sds,

Dγt u(y, z, t) =

1

Γ(2− γ)

∫ t

0(t− s)1−γ ∂

2u(y, z, s)

∂s2ds.

Remark 8.1.1 Eq.(8.1) originates from the generalized Oldroyd-B fluid model in non-

Newtonian fluids, which has been used to simulate the response of many dilute polymeric

liquids. Suppose that an incompressible Olyroyd-B fluid is bounded by a horizontal rect-

angular pipe and the whole system is at rest initially (See Figure 8.1). At time t = 0+,

the fluid begins to accelerate. Due to the shear effects, the fluid over the pipe is gradually

disturbed. Then the velocity u(y, z, t) of the main flow satisfies [85, 204, 223]

163 Chapter 8

Figure 8.1: The illustration of the MHD Olyroyd-B fluid

λD1+αt u(y, z, t) +

∂u(y, z, t)

∂t= ν(1 + θDβ

t )(4u(y, z, t)), (8.4)

where λ and θ are the relaxation time and the retardation time respectively, ν = µρ , µ is the

dynamic viscosity coefficient of the fluid, ρ is the density of the fluid. When a magnetic

field is imposed on the above flow and a low magnetic Reynolds number is supposed, the

following velocity equation can be derived [163]

λD1+αt u(y, z, t) +

∂u(y, z, t)

∂t= ν(1 + θDβ

t )(4u(y, z, t))−K(1 + λDαt )u(y, z, t), (8.5)

where K =σB2

0ρ and B0 is the magnetic intensity and σ is the electrical conductivity.

Remark 8.1.2 Eq.(8.1) also enables other different kinds of fractional diffusion equa-

tions to be considered, such as the time-fractional telegraph equation [191], the fractional

cable equation [101], the generalized Maxwell fluid model [293] and a heated generalized

second grade fluid model [39].

Remark 8.1.3 Compared with integer order models, the fractional viscoelastic models

(8.4) or (8.5) can accurately describe the behavior of materials using fewer parameters,

describe the fluids that have slight memory, and can simulate the damping behavior of

viscoelastic materials in larger frequency ranges [163]. For some complex diffusion models,

the single term fractional diffusion equation cannot describe the underlying processes,

however, these can be modelled using its generalization of a multi-term time-fractional

diffusion-wave and diffusion equation [154]. Since models (8.4) and (8.5) contain similar

terms, they can be expressed in a single generalised form (8.1).

The new model (8.1) is different from the general multi-term time-fractional diffusion

equation because it not only possesses the diffusion-wave and sub-diffusion terms simul-

taneously, but also has a special time-space coupled derivative. Although some literature

[163, 187] derives the exact solution of (8.1), the solution is typically given in a series form

involving special functions, such as the Fox H-function or the multivariate Mittag-Leffler

function, and both of these functions are challenging to calculate explicitly. Therefore,

seeking a numerical solution of (8.1) is of importance to provide insight into the behaviour

of the model. For the Eq.(8.4) in one dimension, Bazhlekova et al. [11] developed a finite

difference approximation utilising the Grunwald-Letnikov formula, of which the accuracy

Chapter 8 164

of the overall scheme was low and lacked a stability and convergence analysis. Recently,

Feng et al. [80] applied the finite difference method with a mixed L scheme and gave

a stability and convergence analysis, of which the convergence order of the method was

first order. To the best of our knowledge, there is no literature reported that treats the

numerical solution of (8.1). The main contributions of this paper are as follows:

• A novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equa-

tion is considered, which not only possesses the diffusion-wave and sub-diffusion

terms simultaneously, but also has a special time-space coupled derivative.

• The mixed L schemes are utilised to approximate the time-fractional sub-diffusion

term, diffusion-wave term and the coupled time-space derivative, respectively. Then

the variational formulation is derived and the finite element method is applied to the

equation, in which two unified numerical schemes with an accuracy of O(τ+h2) and

O(τmin3−γs,2−αw,2−β + h2) for (8.1) are constructed using the linear polynomials

on triangular elements.

• We prove that the numerical scheme is unconditionally stable and convergent under

the H1 norm and L2 norm.

• Three numerical examples illustrate that the method proposed is robust and effec-

tive, in which a 2D multi-term time fractional mixed diffusion equation on a circular

domain and a 2D generalized Oldroyd-B fluid in a magnetic field are solved. In ad-

dition, our method is flexible and can be used to deal with other fractional problems

such as the generalized Maxwell fluid model [293], the generalized Oldroyd-B fluid

model [85, 204, 223], the time-fractional telegraph equation [191] and others.

The outline of this chapter is as follows. In Section 8.2, we present some useful lemmas

and discrete schemes for the time-fractional derivatives. In Section 8.3, the fully discrete

scheme of the equation is derived and the matrix form of the scheme using the finite

element method is constructed. In Section 8.4, the stability and convergence analysis of

the scheme are established. In Section 8.5, to illustrate the efficacy of our method, three

numerical examples are considered and some conclusions are summarised in Section 8.6.

8.2 Preliminary knowledge

Firstly, we assume that C,C1, C2, . . . are positive constants, whose values will be implicitly

determined by the surrounding context in the subsequent sections. Let τ = TN be the

time step and tn = nτ , n = 0, 1, 2, ..., N . Define the grid function un = u(y, z, tn),

∆un = ∆u(y, z, tn) and fn = f(y, z, tn). Using the notations

∇tun =un − un−1

τ, un−

12 =

un + un−1

2,

we now introduce the following lemma.

165 Chapter 8

Lemma 8.2.1 If u(y, z, t) ∈ C0,0,3y,z,t (Ω× J), then we have [43]

u(y, z, tn− 12) =

u(y, z, tn) + u(y, z, tn−1)

2+ rn1 ,

∂

∂tu(y, z, tn− 1

2) =

u(y, z, tn)− u(y, z, tn−1)

τ+ rn2 ,

∂

∂tu(y, z, tn) =

u(y, z, tn)− u(y, z, tn−1)

τ+ rn3 ,

where |rn1 | ≤ Cτ2, |rn2 | ≤ Cτ2, |rn3 | ≤ Cτ .

To discrete the time-fractional derivative Dγt u(y, z, t) (1 < γ < 2) at (y, z, tn), we have

the following formula [83]

Dγt u(y, z, tn) =

τ1−γ

Γ(3− γ)

[a

(γ)0 ∇tu

n −n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

k − a(γ)n−1

∂u(y, z, 0)

∂t

]+Rn,γ0

=∇(γ)t un +Rn,γ0 , (8.6)

where a(γ)k = (k + 1)2−γ − k2−γ , k = 0, 1, 2, . . . , n, |Rn,γ0 | ≤ Cτ . Using the L2 formula, at

(y, z, tn− 12), the time-fractional derivative Dγ

t u(y, z, t) (1 < γ < 2) can be discretised as

[238]

Dγt u(y, z, tn− 1

2) =

τ1−γ

Γ(3− γ)

[a

(γ)0 ∇tu

n −n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)∇tu

k − a(γ)n−1

∂u(y, z, 0)

∂t

]+Rn,γ1

=∇(γ)t un +Rn,γ1 , (8.7)

where |Rn,γ1 | ≤ Cτ3−γ .

Lemma 8.2.2 For 1 < γ < 2, define a(γ)k = (k + 1)2−γ − k2−γ, k = 0, 1, 2, . . . , n and

vector S = [S1, S2, S3, . . . , SN ]T and constant P , then it holds that

τ1−γ

Γ(3− γ)

N∑n=1

[a

(γ)0 Sn −

n−1∑k=1

(a(γ)n−k−1 − a

(γ)n−k)Sk − a

(γ)n−1P

]Sn

≥ T 1−γ

2Γ(2− γ)

N∑n=1

S2n −

T 2−γ

2τΓ(3− γ)P 2, N = 1, 2, 3, . . .

Proof. See [238].

Utilising the following L1 formula, at (y, z, tn), the time-fractional derivatives Dαt u(y, z, t),

Dβt u(y, z, t) (0 < α, β < 1) can be discretised as [83, 238]

Chapter 8 166

Dαt u(y, z, tn) =

τ−α

Γ(2− α)

[d

(α)0 un −

n−1∑k=1

(d(α)n−k−1 − d

(α)n−k)u

k − d(α)n−1u

0]

+Rn,α2

=τ1−α

Γ(2− α)

n∑k=1

d(α)n−k∇tu

k +Rn,α2 = ∇(α)t un +Rn,α2 , (8.8)

Dβt u(y, z, tn) =

τ−β

Γ(2− β)

[d

(β)0 un −

n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)u

k − d(β)n−1u

0]

+Rn,β3

=τ1−β

Γ(2− β)

n∑k=1

d(β)n−k∇tu

k +Rn,β3 = ∇(β)t un +Rn,β3 , (8.9)

where d(α)k = (k+1)1−α−k1−α, d

(β)k = (k+1)1−β−k1−β, k = 0, 1, 2, . . . , n, |Rn,α2 | ≤ Cτ2−α

and |Rn,β3 | ≤ Cτ2−β.

Then we have

Dβt (4u(y, z, tn)) =

τ−β

Γ(2− β)

[d

(β)0 4u

n −n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)4u

k − d(β)n−14u

0]

+ Rn,β3

=τ1−β

Γ(2− β)

n∑k=1

d(β)n−k∇t(4u

k) + Rn,β3 , (8.10)

where |Rn,β3 | ≤ Cτ2−β.

Now we consider the discretization of Dαt u(y, z, t), Dβ

t u(y, z, t) at grid points (y, z, tn− 12).

From (8.8), (8.9) and Lemma 8.2.1, we have

Dαt u(y, z, tn− 1

2) =

1

2

[Dαt u(y, z, tn) +Dα

t u(y, z, tn−1)]

+ rn1

=τ1−α

2Γ(2− α)

[ n∑k=1

d(α)n−k∇tu

k +n−1∑k=1

d(α)n−1−k∇tu

k]

+Rn,α2 +Rn−1,α

2

2+ rn1 ,

Dβt u(y, z, tn− 1

2) =

1

2

[Dβt u(y, z, tn) +Dβ

t u(y, z, tn−1)]

+ rn2

=τ1−β

2Γ(2− β)

[ n∑k=1

d(β)n−k∇tu

k +

n−1∑k=1

d(β)n−1−k∇tu

k]

+Rn,β3 +Rn−1,β

3

2+ rn2 .

Since |rn1 | ≤ Cτ2, |rn2 | ≤ Cτ2, then we have

Dαt u(y, z, tn− 1

2) =

τ1−α

2Γ(2− α)

[ n∑k=1

d(α)n−k∇tu

k +n−1∑k=1

d(α)n−1−k∇tu

k]

+Rn,α4 , (8.11)

Dβt u(y, z, tn− 1

2) =

τ1−β

2Γ(2− β)

[ n∑k=1

d(β)n−k∇tu

k +

n−1∑k=1

d(β)n−1−k∇tu

k]

+Rn,β5 , (8.12)

167 Chapter 8

where |Rn,α4 | ≤ Cτ2−α, |Rn,β5 | ≤ Cτ2−β. Furthermore, we have

Dβt (4u(y, z, tn− 1

2))

=τ1−β

2Γ(2− β)

[ n∑k=1

d(β)n−k∇t(4u

k) +

n−1∑k=1

d(β)n−1−k∇t(4u

k)]

+Rn,β6 , (8.13)

where |Rn,β6 | ≤ Cτ2−β.

Denote V = H10 (Ω). We use triangular elements to partition the domain Ω. Denote Th

as the triangulation and let h be the maximum diameter of the triangles. Then the finite

element subspace can be defined as:

Vh :=vh|vh ∈ C(Ω) ∩ V, vh|K is linear for all K ∈ Th.

For any u, v ∈ Vh, we denote the following inner products and induced norms:

(u, v) =

∫∫uvdΩ, ||u||0 =

√(u, u), 〈u, v〉 =

(∂u∂y,∂v

∂y

)+(∂u∂z,∂v

∂z

),

|u|1 =√〈u, u〉, ||u||1 =

√a4||u||20 + a5|u|21.

In the following, some important lemmas will be presented.

Lemma 8.2.3 For any u, v ∈ Vh and 0 < β < 1, it holds that

(4uk, vn) = −〈uk, vn〉,( n∑k=1

d(β)n−k∇t(4u

k), v)

= −⟨ n∑k=1

d(β)n−k∇tu

k, v⟩.

Proof. Integration by parts in two dimensions gives

(4uk, vn) =

∫∫Ω

∂2uk

∂y2vndΩ +

∫∫Ω

∂2uk

∂z2vndΩ

=

∫Γuk(∂vn∂y· n)dΓ−

∫∫Ω

∂uk

∂y

∂vn

∂ydΩ +

∫Γuk(∂vn∂z· n)dΓ−

∫∫Ω

∂uk

∂z

∂vn

∂zdΩ

= −(∂uk∂y

,∂vn

∂y

)−(∂uk∂z

,∂vn

∂z

)= −〈uk, vn〉.

Combining (8.10), we obtain

( n∑k=1

d(β)n−k∇t(4u

k), v)

=[d

(β)0

(4un, v

)−n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)

(4uk, v

)− d(β)

n−1

(4u0, v

)]=−

[d

(β)0

⟨un, v

⟩−n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)

⟨uk, v

⟩− d(β)

n−1

⟨u0, v

⟩]

Chapter 8 168

=−⟨d

(β)0 un −

n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)u

k − d(β)n−1u

0, v⟩

=−⟨ n∑k=1

d(β)n−k∇tu

k, v⟩.

Lemma 8.2.4 For 0 < β < 1, define d(β)k = (k + 1)1−β − k1−β, k = 0, 1, 2, . . . , n, and

vector Q = [v1, v2, . . . , vN−1, vN ] ∈ RN , where N is a positive integer, then we have

N∑n=1

n∑k=1

d(β)n−k(v

k, vn) ≥ 0.

Proof. According to the Lemma 5 in [83], we have

N∑n=1

n∑k=1

d(β)n−k v

kvn ≥ 0,

and integration over Ω gives the result

N∑n=1

n∑k=1

d(β)n−k(v

k, vn) ≥ 0.

8.3 The finite element method

8.3.1 Finite element scheme I

Assume that u(y, z, t) ∈ C2,2,3y,z,t (Ω× J). According to (8.1), at (y, z, tn), we have

s∑p=1

a1,pDγpt u(y, z, tn) + a2

∂u(y, z, tn)

∂t+

w∑q=1

a3,qDαqt u(y, z, tn) + a4u(y, z, tn)

=a54u(y, z, tn) + a6Dβt (4u(y, z, tn)) + f(y, z, tn). (8.14)

Assume that unh ∈ Vh is the approximate solution of u(y, z, t) at t = tn. Then, according

to Lemma 8.2.3, the fully discrete finite element scheme of Eq.(8.1) can be written as:

find unh ∈ Vh such that

s∑p=1

a1,p(∇(γp)t unh, vh) + a2(∇tunh, vh) +

w∑q=1

a3,q(∇(αq)t unh, vh)

+a4(unh, vh) + a5A(unh, vh) + a6A(∇(β)t unh, vh) = (fn, vh), ∀vh ∈ Vh, (8.15)

where

A(u, v) = 〈u, v〉 =(∂u∂y,∂v

∂y

)+(∂u∂z,∂v

∂z

).

169 Chapter 8

According to (8.6), (8.8) and (8.10), Eq.(8.15) can be recast into the form

s∑p=1

a1,pµ1,p

[(∇tunh, vh)−

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇tu

kh, vh)− a(γp)

n−1(φ2, vh)]

+a2(∇tunh, vh) +w∑q=1

a3,qµ2,q

n∑k=1

d(αq)n−k(∇tu

kh, vh) + a4(unh, vh)

+a5A(unh, vh) + a6µ3

n∑k=1

d(β)n−kA(∇tukh, vh) = (fn, vh), (8.16)

where µ1,p = τ1−γp

Γ(3−γp) , µ2,q = τ1−αq

Γ(2−αq) , µ3 = τ1−β

Γ(2−β) . Then (8.16) can be arranged as

ω1,1(unh, vh) + ω1,2A(unh, vh) = ω1,3(un−1h , vh)

+s∑

p=1

a1,pµ1,p

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇tu

kh, vh) +

s∑p=1

a1,pµ1,pa(γp)n−1(φ2, vh)

+

w∑q=1

a3,qµ2,q

τ

n−1∑k=1

(d(αq)n−k−1 − d

(αq)n−k)(u

kh, vh) +

w∑q=1

a3,qµ2,q

τd

(αq)n−1(φ1, vh)

+a6µ3

τ

n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)A(ukh, vh) +

a6µ3

τd

(β)n−1A(u0

h, vh) + (fn, vh), (8.17)

where

ω1,1 =s∑

p=1

a1,pµ1,p

τ+a2

τ+

w∑q=1

a3,qµ2,q

τ+ a4, ω1,2 = a5 +

a6µ3

τ, ω1,3 =

s∑p=1

a1,pµ1,p

τ+a2

τ.

According to (8.6), (8.8), (8.10) and Lemma 8.2.1, the accuracy of scheme (8.16) is O(τ +

h2).

8.3.2 Implementation of the finite element method

Here we choose piecewise linear polynomials on the triangle ep, p = 1, 2, ..., Ne to proceed

with the computation, where Ne is the total number of triangular elements. On a single

element ep, we can write the field function up(y, z) as

up(y, z) =3∑j=1

ujϕj(y, z),

in which we number the triangle vertices in a counter-clockwise direction as 1, 2, 3 and

define the basis function ϕj(y, z) as

ϕj(y, z)∣∣∣(y,z)∈ep

=1

2∆ep

(qjy + bjz + cj), ϕj(y, z)∣∣∣(y,z)/∈ep

= 0,

q1 = z2 − z3, q2 = z3 − z1, q3 = z1 − z2,

Chapter 8 170

b1 = y3 − y2, b2 = y1 − y3, b3 = y2 − y1,

c1 = y2z3 − y3z2, c2 = y3z1 − y1z3, c3 = y1z2 − y2z1,

where ∆ep is the area of triangular element p. We have that ϕj(yi, zi) = δij , i, j = 1, 2, 3,

where δ is the Kronecker delta function. Now, we rewrite unh in the form

unh =

Np∑i=1

uni li(y, z), u(y, z, tn) = unh + rn4 , (8.18)

where uni are the coefficients that are to be solved for, |rn4 | ≤ Ch2 and Np denotes the total

number of nodes on the domain Ω. Substituting (8.18) into (8.17) with vh = lj(y, z), j =

1, 2, . . . , Np gives

ω1,1

Np∑i=1

uni (li, lj) + ω1,2

Np∑i=1

uni A(li, lj) = ω1,3

Np∑i=1

un−1i (li, lj)

+

s∑p=1

a1,pµ1,p

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)

Np∑i=1

∇tuki (li, lj) +

s∑p=1

a1,pµ1,pa(γp)n−1(φ2, lj)

+w∑q=1

a3,qµ2,q

τ

n−1∑k=1

(d(αq)n−k−1 − d

(αq)n−k)

Np∑i=1

unk(li, lj) +w∑q=1

a3,qµ2,q

τd

(αq)n−1(φ1, lj)

+a6µ3

τ

n−1∑k=1

(d(β)n−k−1 − d

(β)n−k)

Np∑i=1

unkA(li, lj) +a6µ3

τd

(β)n−1u

n0A(li, lj) + (fn, lj). (8.19)

Then, (8.19) can be expressed in matrix form as

(ω1,1M + ω1,2K)Un = ω1,3MUn−1 + Sn1,1 + Sn1,2 + Sn1,3

+s∑

p=1

a1,pµ1,pa(γp)n−1P +

w∑q=1

a3,qµ2,q

τd

(αq)n−kG +

a6µ3

τd

(β)n−1KU0 + Fn

1 , n ≥ 2,

(ω1,1M + ω1,2K)Un =(ω1,3 +

w∑q=1

a3,qµ2,q

τ

)G +

s∑p=1

a1,pµ1,pP +a6µ3

τKU0 + Fn

1 , n = 1,

where the mass matrix M has elements Mij = (lj , li), the stiffness matrix K has el-

ements Kij = A(lj , li), Un = [un1 , un2 , ..., u

nNp

]T , G = [(φ1, l1), (φ1, l2), ..., (φ1, lNp)]T ,

P = [(φ2, l1), (φ2, l2), ..., (φ2, lNp)]T , Fn

1 = [(fn, l1), (fn, l2), ..., (fn, lNp)]T and

Sn1,1 =

s∑p=1

a1,pµ1,p

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)M

Uk −Uk−1

τ,

Sn1,2 =w∑q=1

a3,qµ2,q

τ

n−1∑k=1

(d(αq)n−k−1 − d

(αq)n−k)MUk,

Sn1,3 =a6µ3

τ

n−1∑k=1

(d(β)n−k − d

(β)n−k−1)KUk.

171 Chapter 8

8.3.3 Finite element scheme II

Here we will present the Crank-Nicolson scheme of (8.1) to improve the temporal order.

Assume that u(y, z, t) ∈ C2,2,3y,z,t (Ω× J). According to (8.1), at (y, z, tn− 1

2), we have

s∑p=1

a1,pDγpt u(y, z, tn− 1

2) + a2

∂u(y, z, tn− 12)

∂t+

w∑q=1

a3,qDαqt u(y, z, tn− 1

2) + a4u(y, z, tn− 1

2)

=a54u(y, z, tn− 12) + a6D

βt (4u(y, z, tn− 1

2)) + f(y, z, tn− 1

2). (8.20)

Then we obtain the fully discrete finite element scheme of Eq.(8.1): find unh ∈ Vh such

that

s∑p=1

a1,p(∇(γp)t unh, vh) + a2(∇tunh, vh) +

w∑q=1

a3,q(∇(αq)t u

n− 12

h , vh) + a4(un− 1

2h , vh)

+a5A(un− 1

2h , vh) + a6A(∇(β)

t un− 1

2h , vh) = (fn−

12 , vh), ∀vh ∈ Vh. (8.21)

According to (8.7), (8.11) and (8.13), Eq.(8.21) can be recast into the form

s∑p=1

a1,pµ1,p

[(∇tunh, vh)−

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇tu

kh, vh)− a(γp)

n−1(φ2, vh)]

+ a2(∇tunh, vh)

+w∑q=1

a3,qµ2,q

2

[ n∑k=1

d(αq)n−k(∇tu

kh, vh) +

n−1∑k=1

d(αq)n−1−k(∇tu

kh, vh)

]+ a4(u

n− 12

h , vh) + a5A(un− 1

2h , vh)

+a6µ3

2

[ n∑k=1

d(β)n−kA(∇tukh, vh) +

n−1∑k=1

d(β)n−1−kA(∇tukh, vh)

]= (fn−

12 , vh). (8.22)

Then Eq.(8.22) can be arranged as

ω2,1(unh, vh) + ω2,2A(unh, vh) = ω2,3(un−1h , vh) + ω2,4A(un−1

h , vh)

+

s∑p=1

a1,pµ1,p

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇tu

kh, vh)−

w∑q=1

a3,qµ2,q

2

n−1∑k=1

(d(αq)n−k + d

(αq)n−k−1)(∇tukh, vh)

−a6µ3

2

n−1∑k=1

(d(β)n−k + d

(β)n−k−1)A(∇tukh, vh) +

s∑p=1

a1,pµ1,pa(γp)n−1(φ2, vh) + (fn−

12 , vh), (8.23)

where

ω2,1 =s∑

p=1

a1,pµ1,p

τ+a2

τ+

w∑q=1

a3,qµ2,q

2τ+a4

2, ω2,2 =

a6µ3

2τ+a5

2,

ω2,3 =s∑

p=1

a1,pµ1,p

τ+a2

τ+

w∑q=1

a3,qµ2,q

2τ− a4

2, ω2,4 =

a6µ3

2τ− a5

2.

Chapter 8 172

Similarly, we can obtain the matrix form of (8.23) as

(ω2,1M + ω2,2K)Un = (ω2,3M + ω2,4K)Un−1 + Sn2,1 − Sn2,2 − Sn2,3

+s∑

p=1

a1,pµ1,pa(γp)n−1P + Fn

2 , n ≥ 2,

(ω2,1M + ω2,2K)Un = ω2,4KUn−1 + ω2,3G +s∑

p=1

a1,pµ1,pP + Fn2 , n = 1,

where

Sn2,1 =s∑

p=1

a1,pµ1,p

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)M

Uk −Uk−1

τ,

Sn2,2 =w∑q=1

a3,qµ2,q

2

n−1∑k=1

(d(αq)n−k + d

(αq)n−k−1)M

Uk −Uk−1

τ,

Sn2,3 =a6µ3

2

n−1∑k=1

(d(β)n−k + d

(β)n−k−1)K

Uk −Uk−1

τ,

Fn2 = [(fn−

12 , l1), (fn−

12 , l2), ..., (fn−

12 , lNp)]

T .

According to (8.7), (8.11), (8.13) and Lemma 8.2.1, the accuracy of scheme (8.23) is

O(τmin3−γs,2−αw,2−β + h2).

8.4 Stability and convergence

Here, we discuss the stability and the convergence analysis of the scheme (8.15).

Theorem 8.4.1 The fully discrete finite element scheme (8.15) is unconditionally stable

and it holds that

||ulh||21 ≤ ||u0h||21 +

s∑p=1

a1,pT2−γp

Γ(3− γp)||φ2||20 +

T

2ε0max

1≤n≤l||fn||20,

where ε0 =s∑

p=1

a1,pT1−γp

2Γ(2−γp) + a2 and 1 ≤ l ≤ N .

Proof. In (8.15), let vh = ∇tunh, we sum n from 1 to l, 1 ≤ l ≤ N and obtain

s∑p=1

a1,pµ1,p

l∑n=1

(∇tunh −n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)∇tu

kh − a

(γp)n−1φ2,∇tunh) + a2

l∑n=1

(∇tunh,∇tunh)

+

w∑q=1

a3,qµ2,q

l∑n=1

n∑k=1

d(αq)n−k(∇tu

kh,∇tunh) + a4

l∑n=1

(unh,∇tunh) + a5

l∑n=1

A(unh,∇tunh)

+a6µ3

l∑n=1

n∑k=1

d(β)n−kA(∇tukh,∇tunh) =

l∑n=1

(fn,∇tunh).

173 Chapter 8

According to Lemma 8.2.2, we have

s∑p=1

a1,pµ1,p

l∑n=1

(∇tunh −n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)∇tu

kh − a

(γp)n−1φ2,∇tunh)

≥s∑

p=1

a1,pT1−γp

2Γ(2− γp)

l∑n=1

||∇tunh||20 −s∑

p=1

a1,pT2−γp

2τΓ(3− γp)||φ2||20. (8.24)

Using Lemma 8.2.4, we have

w∑q=1

a3,qµ2,q

l∑n=1

n∑k=1

d(αq)n−k(∇tu

kh,∇tunh) ≥ 0. (8.25)

Similarly, we have

a6µ3

l∑n=1

n∑k=1

d(β)n−kA(∇tukh,∇tunh) ≥ 0. (8.26)

For the fourth term, utilising the inequality a(a− b) ≥ a2−b22 , we have

a4

l∑n=1

(unh,∇tunh) ≥ a4

2τ(||ulh||20 − ||u0

h||20). (8.27)

Similarly, for the fifth term, we have

a5

l∑n=1

A(unh,∇tunh) = a5

l∑n=1

〈unh,∇tunh〉 ≥a5

2τ(|ulh|21 − |u0

h|21). (8.28)

According to (8.24)-(8.28), we have

2τ(s∑

p=1

a1,pa1,pT

1−γp

2Γ(2− γp)+ a2)

l∑n=1

||∇tunh||20 + a4||ulh||20 + a5|ulh|21

≤a4||u0h||20 + a5|u0

h|21 +

s∑p=1

a1,pT2−γp

Γ(3− γp)||φ2||20 + 2τ

l∑n=1

(fn,∇tunh).

Using the important inequality ab ≤ εa2 + b2

4ε (ε > 0), we have

2τ

l∑n=1

(fn,∇tunh) ≤ 2τε0

l∑n=1

||∇tunh||20 +τ

2ε0

l∑n=1

||fn||20

≤ 2τε0

l∑n=1

||∇tunh||20 +T

2ε0max

1≤n≤l||fn||20,

Chapter 8 174

where ε0 =s∑

p=1a1,p

a1,pT1−γp

2Γ(2−γp) + a2. Then we obtain

a4||ulh||20 + a5|ulh|21 ≤ a4||u0h||20 + a5|u0

h|21 +s∑

p=1

a1,pT2−γp

Γ(3− γp)||φ2||20 +

T

2ε0max

1≤n≤l||fn||20,

namely,

||ulh||21 ≤ ||u0h||21 +

s∑p=1

a1,pT2−γp

Γ(3− γp)||φ2||20 +

T

2ε0max

1≤n≤l||fn||20.

Therefore the scheme (8.15) is unconditionally stable.

Define the projection operator Ph: V → Vh

A(Phu, vh) = A(u, vh), ∀ vh ∈ Vh.

It is well known that Ph has the approximation property [53]

||v − Phv||Hµ(Ω) ≤ Chr−µ||v||Hr(Ω), ∀ v ∈ Hµ(Ω) ∩Hr(Ω), µ = 0, 1, r = 1, 2.

Then we have

||v − Phv||0 ≤ Ch2||v||H2(Ω). (8.29)

Theorem 8.4.2 Assume that unh, u(tn) are the numerical solution and exact solution of

problem (8.1)-(8.3) at t = tn respectively and u, ut,C0 D

αt u, C

0 Dγt u ∈ L∞(0, T ;H2(Ω)),

when we choose the triangular linear basis function, the error satisfies

||unh − u(tn)||20 ≤ C[τ2 + h4 + h4

( s∑p=1

a1,p max1≤n≤N

||Dγpt u(tn)||22 + max

1≤n≤N||∂u(tn)

∂t||22

+w∑q=1

a3,q max1≤n≤N

||Dαqt u(tn)||22 + max

1≤n≤N||u(tn)||22

)].

Proof. Let en = unh − u(tn), combining (8.14) and (8.15), we have

s∑p=1

a1,pµ1,p

[(∇ten, vh)−

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇te

k, vh)]−

s∑p=1

a1,p(Rn,γp0 , vh) + a2(∇ten, vh)

−a2(rn3 , vh) +

w∑q=1

a3,qµ2,q

n∑k=1

d(αq)n−k(∇te

k, vh)−w∑q=1

a3,q(Rn,αq2 , vh) + a4(en, vh)− a4(rn4 , vh)

+a5A(en, vh) + a6µ3

n∑k=1

d(β)n−kA(∇tek, vh)− a6(Rn,β3 , vh) = 0.

175 Chapter 8

Define en = ρn + θn, ρn = Phu(tn)−u(tn), θn = unh−Phu(tn), taking vh = ∇tθn, we have

s∑p=1

a1,pµ1,p

[(∇tθn,∇tθn)−

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇tθ

k,∇tθn)]

+ a2(∇tθn,∇tθn)

+w∑q=1

a3,qµ2,q

n∑k=1

d(αq)n−k(∇tθ

k,∇tθn) + a4(θn,∇tθn) + a5A(θn,∇tθn) + a6µ3

n∑k=1

d(β)n−kA(∇tθk,∇tθn)

=s∑

p=1

a1,p(Rn,γp0 ,∇tθn)−

s∑p=1

a1,pµ1,p

[(∇tρn,∇tθn)−

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇tρ

k,∇tθn)]

−a2(∇tρn,∇tθn) + a2(rn3 ,∇tθn)−w∑q=1

a3,qµ2,q

n∑k=1

d(αq)n−k(∇tρ

k,∇tθn)

+a3

w∑q=1

a3,q(Rn,αq2 ,∇tθn)− a4(ρn,∇tθn) + a4(rn4 ,∇tθn) + a6(Rn,β3 ,∇tθn). (8.30)

Summing n from 1 to N , for the left hand side (L.H.S.) of (8.30), we have

s∑p=1

a1,pµ1,p

N∑n=1

[(∇tθn,∇tθn)−

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇tθ

k,∇tθn)]≥

s∑p=1

a1,pT1−γp

2Γ(2− γp)

N∑n=1

||∇tθn||20,

w∑q=1

a3,qµ2,q

N∑n=1

n∑k=1

d(αq)n−k(∇tθ

k,∇tθn) ≥ 0, a4

N∑n=1

(θn,∇tθn) ≥ a4

2τ(||θN ||20 − ||θ0||20),

a5

N∑n=1

A(θn,∇tθn) ≥ a5

2τ(|θN |21 − |θ0|21), a6µ3

n∑k=1

d(β)n−kA(∇tθk,∇tθn) ≥ 0.

Then we have

L.H.S. ≥ (

s∑p=1

a1,pT1−γp

2Γ(2− γp)+ a2)

N∑n=1

||∇tθn||20 +a4

2τ(||θN ||20 − ||θ0||20) +

a5

2τ(|θN |21 − |θ0|21).

For the right hand side (R.H.S.) of (8.30), using again the important inequality ab ≤εa2 + b2

4ε(ε > 0), we have

s∑p=1

a1,p

N∑n=1

(Rn,γp0 ,∇tθn) ≤

s∑p=1

a1,pε1,p

N∑n=1

||Rn,γp0 ||20 +s∑

p=1

a1,p

4ε1,p

N∑n=1

||∇tθn||20

≤ C1

s∑p=1

a1,pε1,pNτ2 +

s∑p=1

a1,p

4ε1,p

N∑n=1

||∇tθn||20,

−a2

N∑n=1

(∇tρn,∇tθn) ≤ C3a2ε3N(τ2 + h4 max

1≤n≤N||∂u(tn)

∂t||22)

+a2

4ε3

N∑n=1

||∇tθn||20,

a2

N∑n=1

(rn3 ,∇tθn) ≤ C3a2ε3Nτ2 +

a2

4ε3

N∑n=1

||∇tθn||20,

Chapter 8 176

−s∑

p=1

a1,pµ1,p

N∑n=1

[(∇tρn,∇tθn)−

n−1∑k=1

(a(γp)n−k−1 − a

(γp)n−k)(∇tρ

k,∇tθn)]

≤s∑

p=1

a1,pµ1,pε2,p

N∑n=1

(||∇(γp)t ρn −Dγp

t ρn||20 + ||Dγp

t ρn||20) +

s∑p=1

a1,pµ1,p

4ε2,p

N∑n=1

||∇tθn||20

≤C2

s∑p=1

a1,pµ1,pε2,pN(τ2 + h4 max1≤n≤N

||Dγpt u(tn)||22) +

s∑p=1

a1,pµ1,p

4ε2,p

N∑n=1

||∇tθn||20,

−w∑q=1

a3,qµ2,q

N∑n=1

n∑k=1

d(αq)n−k(∇tρ

k,∇tθn)

≤C4

w∑q=1

a3,qµ2,qε4,pN(τ4−2αq + h4 max1≤n≤N

||Dαqt u(tn)||22) +

w∑q=1

a3,qµ2,q

4ε4,p

N∑n=1

||∇tθn||20,

w∑q=1

a3,q

N∑n=1

(Rn,αq2 ,∇tθn) ≤ C5

w∑q=1

a3,qε5,pNτ4−2αq +

w∑q=1

a3,q

4ε5,p

N∑n=1

||∇tθn||20,

−a4

N∑n=1

(ρn,∇tθn) ≤ C6a4ε6Nh4 max

1≤n≤N||u(tn)||22 +

a4

4ε6

N∑n=1

||∇tθn||20,

a4

N∑n=1

(rn4 ,∇tθn) ≤ C6a4ε6Nh4 +

a4

4ε6

N∑n=1

||∇tθn||20,

a6

N∑n=1

(Rn,β3 ,∇tθn) ≤ C7a6ε7Nτ4−2β +

a6

4ε7

N∑n=1

||∇tθn||20,

where ε1,p =Γ(2−γp)

T 1−γp , ε2,p =µ1,pΓ(2−γp)

T 1−γp , ε3 = 52 ,∑w

q=1a3,qµ2,q

4ε4,p= a2

5 ,∑w

q=1a3,q4ε5,p

= a25 ,

ε6 = 5a42a2

, ε7 = 5a64a2

. Then we have

a4

2τ(||θN ||20 − ||θ0||20) +

a5

2τ(|θN |21 − |θ0|21) ≤ CN

[τ2 + h4 + h4

( s∑p=1

a1,p max1≤n≤N

||Dγpt u(tn)||22

+ max1≤n≤N

||∂u(tn)

∂t||22 +

w∑q=1

a3,q max1≤n≤N

||Dαqt u(tn)||22 + max

1≤n≤N||u(tn)||22

)].

Furthermore, we have

a4||θN ||20 + a5|θN |21 ≤ a4||θ0||20 + a5|θ0|21 + 2CT[τ2 + h4 + h4

( s∑p=1

a1,p max1≤n≤N

||Dγpt u(tn)||22

+ max1≤n≤N

||∂u(tn)

∂t||22 +

w∑q=1

a3,q max1≤n≤N

||Dαqt u(tn)||22 + max

1≤n≤N||u(tn)||22

)].

177 Chapter 8

When we choose u0h = Phu(t0), we can obtain

||θN ||20 ≤ C[τ2 + h4 + h4

( s∑p=1

a1,p max1≤n≤N

||Dγpt u(tn)||22 + max

1≤n≤N||∂u(tn)

∂t||22

+w∑q=1

a3,q max1≤n≤N

||Dαqt u(tn)||22 + max

1≤n≤N||u(tn)||22

)].

As ||ρn||20 = ||Phu(tn)− u(tn)||20 ≤ Ch4||u(tn)||22, then we have

||eN ||20 ≤||θN ||20 + ||ρN ||20 ≤ C[τ2 + h4 + h4

( s∑p=1

a1,p max1≤n≤N

||Dγpt u(tn)||22

+ max1≤n≤N

||∂u(tn)

∂t||22 +

w∑q=1

a3,q max1≤n≤N

||Dαqt u(tn)||22 + max

1≤n≤N||u(tn)||22

)].

The proof is completed.

8.5 Numerical examples

In this section, we will present three numerical examples to verify the effectiveness of our

method. Here, the numerical computations were carried out using MATLAB R2014b on

a Dell desktop with configuration: Intel(R) Core(TM) i7-4790, 3.60 GHz and 16.0 GB

RAM. To calculate the convergence order, we utilise the following formula:

Order =

log(||E(h1)||0/||E(h2)||0)

log(h1/h2) , in space,log(||E(τ1)||0/||E(τ2)||0)

log(τ1/τ2) , in time.

Example 8.5.1 Firstly, we consider the following multi-term time-fractional mixed d-

iffusion equation on a rectangular domaina1D

γt u(y, z, t) + a2

∂u(y,z,t)∂t + a3D

αt u(y, z, t) + a4u(y, z, t) = a54u(y, z, t)

+a6Dβt (4u(y, z, t)) + f(y, z, t), (y, z, t) ∈ Ω× J,

u(y, z, 0) = sinπy sinπz, ut(y, z, 0) = 0, (y, z) ∈ Ω,

u(y, z, t) = 0, (y, z) ∈ ∂Ω, t ∈ J ,

where 1 < γ < 2, 0 < α, β < 1, Ω = (0, 1)× (0, 1), J = (0, 1],

f(y, z, t) = sinπy sinπz[a1Γ(4)t3−γ

Γ(4− γ)+ 3a2t

2 +a3Γ(4)t3−α

Γ(4− α)

+(a4 + 2a5π2)(t3 + 1) + 2a6π

2 Γ(4)t3−β

Γ(4− β)

].

The exact solution to this equation is u(y, z, t) = (t3 + 1) sinπy sinπz.

Here, for simplicity, we choose a1 = a2 = a3 = a4 = a5 = a6 = 1. We apply the finiteelement schemes I and II to solve the equation and obtain the corresponding numericalresults, which are presented in Tables 8.1 to 8.3.

Chapter 8 178

Table 8.1: The L2 error and convergence order of h for schemes I and II with γ = 1.6,α = 0.7, β = 0.8, τ = 1

1000 at t = 1

scheme I scheme II

h L2 error Order CPU time h L2 error Order CPU time

1/4 1.0036E-01 – 40.08 s 1/4 1.0109E-01 – 31.10 s1/8 2.4708E-02 2.02 1.11 min 1/8 2.4974E-02 2.02 1.83 min1/16 6.0427E-03 2.03 4.58 min 1/16 6.1518E-03 2.02 8.85 min1/32 1.4076E-03 2.10 1.14 h 1/32 1.4727E-03 2.06 1.41 h

Table 8.2: The L2 error and convergence order of τ for scheme II with τ = h, γ = 1.6,α = 0.7, β = 0.8 at t = 1

τ L2 error Order CPU time

1/16 1.1746E-02 – 3.84 s1/32 6.7573E-03 0.80 33.58 s1/64 3.4399E-03 0.97 7.69 min1/128 1.6546E-03 1.06 6.13 h

Table 8.3: The L2 error and convergence order of τ for scheme II at t = 1

τ3−γ ≈ h2 τ L2 error Order CPU time

1/47 2.0250E-02 – 8.07 sα = 0.6 1/102 7.6712E-03 1.25 44.37 sβ = 0.4 1/148 4.7723E-03 1.28 1.89 minγ = 1.8 1/323 1.6738E-03 1.34 17.37 min

τ2−α ≈ h2 τ L2 error Order CPU time

1/27 2.0737E-02 – 4.60 sα = 0.6 1/53 8.1512E-03 1.38 23.20 sβ = 0.4 1/73 5.2213E-03 1.39 52.42 sγ = 1.5 1/142 2.0355E-03 1.42 5.60 min

τ2−β ≈ h2 τ L2 error Order CPU time

1/47 1.7921E-02 – 8.39 sα = 0.7 1/102 6.7742E-03 1.26 44.93 sβ = 0.8 1/148 4.2360E-03 1.26 1.90 minγ = 1.6 1/323 1.5356E-03 1.30 17.11 min

Table 8.1 displays the L2 error and convergence order of h for the two schemes withγ = 1.6, α = 0.7, β = 0.8, τ = 1

1000 at t = 1. We can see that the convergence order of his second order. We can also observe that scheme II is more time-consuming than schemeI, which is due to the fact that it involves two time level calculations while scheme I justinvolves one time level calculation. Table 8.2 shows the L2 error and the convergenceorder of τ for scheme I with τ = h, γ = 1.6, α = 0.7, β = 0.8 at t = 1, in whichwe can see that the convergence order is first order. Table 8.3 shows the L2 error andthe convergence order of τ for scheme II at t = 1, in which three cases are considered:γ = 1.8, α = 0.6, β = 0.4 with τ3−γ ≈ h2, γ = 1.5, α = 0.6, β = 0.4 with τ2−α ≈ h2,γ = 1.6, α = 0.7, β = 0.8 with τ2−β ≈ h2. We can observe that the convergence order

179 Chapter 8

of τ is min3 − γ, 2 − α, 2 − β, which is in agreement with the theoretical analysis. Inaddition, we present a figure that compares the exact solution with the numerical solutionin Figure 8.2 for γ = 1.6, α = 0.7, β = 0.8 with τ = h = 1

50 at t = 1. We can concludethat the numerical results agree with the exact solution very well, which demonstratesthe effectiveness of our method.

10.8

0.6

x

0.4

Numerical solution

0.200

0.2

0.4

y

0.6

0.8

0

1.5

0.5

1

2

1

uh(x,y,t=

1)

10.8

0.6

x

0.4

Exact solution

0.200

0.2

0.4

y

0.6

0.8

0

1.5

0.5

1

2

1

u(x,y,t=

1)

Figure 8.2: The comparison of exact solution with numerical solution for γ = 1.6, α = 0.7,β = 0.8 with τ = h = 1

50 at t = 1

Example 8.5.2 Next, we study a multi-term time-fractional mixed diffusion problem

on a circular domainDγt u(y, z, t) + ∂u(y,z,t)

∂t +Dαt u(y, z, t) + u(y, z, t) = 4u(y, z, t)

+Dβt (4u(y, z, t)) + f(y, z, t), (y, z, t) ∈ Ω× J,

u(y, z, 0) = 1− y2 − z2, ut(y, z, 0) = 0, (y, z) ∈ Ω,

u(y, z, t) = 0, (y, z) ∈ ∂Ω, t ∈ J ,

where 1 < γ < 2, 0 < α, β < 1, Ω = (y, z)|y2 + z2 < 1, J = (0, 1],

f(y, z, t) = (1− y2 − z2)[Γ(4)t3−γ

Γ(4− γ)+ 3t2 +

Γ(4)t3−α

Γ(4− α)+ (t3 + 1)

]+ 4(t3 + 1) +

4Γ(4)t3−β

Γ(4− β).

The corresponding solution is u(y, z, t) = (t3 + 1)(1 − y2 − z2). Firstly, we show the

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 8.3: The triangulation for the circular domain (h ≈ 1.6444 × 10−1 and h ≈8.6550× 10−2)

Chapter 8 180

triangulation used for the circular domain in Figure 8.3, in which the maximum diameter

h of the triangular elements is h ≈ 1.6444 × 10−1 and h ≈ 8.6550 × 10−2, respectively.

Then we apply the finite element scheme II to solve the problem on the circular domain

and observe the error and the convergence. Table 8.4 shows the L2 error and convergence

of h for γ = 1.6, α = 0.8, β = 0.7 with τ = 11000 at t = 1. We can see that the

expected second order is obtained and the multi-term time fractional problem on a circular

domain for fine grids can become extremely time-consuming. Table 8.5 displays the L2

error and convergence of τ for τ ≈ h with γ = 1.6, α = 0.8, β = 0.7 at t = 1. If

τ ≈ h, then O(τmin3−γ,2−α,2−β + h2) ≈ O(τ2−α). From Table 8.5, we can observe that

the convergence order of τ is approximate to 2 − α order. Furthermore, we present a

figure comparison between the exact solution and the numerical solution in Figure 8.4

for γ = 1.6, α = 0.8, β = 0.7 with τ = 1100 , h ≈ 8.6550 × 10−2 at t = 1. From these

numerical results, we can conclude that our method is also effective for the multi-term

time-fractional mixed diffusion problem problem on a circular domain.

Table 8.4: The L2 error and convergence order of h for the problem on a circular domain

h L2 error Order CPU time

2.8917E-01 2.7944E-02 – 2.00 min1.6444E-01 9.1158E-03 1.98 7.33 min8.6550E-02 2.2614E-03 2.17 1.75 h4.5873E-02 6.5809E-04 1.94 27.91 h

Table 8.5: The L2 error and convergence order of τ for the problem on a circular domain

τ L2 error Order CPU time

1/8 4.6368E-02 – 4.06 s1/12 2.9069E-02 1.15 25.20 s1/15 2.2152E-02 1.22 47.88 s1/22 1.3930E-02 1.21 3.76 min

1

0.5

x

0

-0.5

Exact solution

-1-1

-0.5

0

y

0.5

2

1.5

1

0

0.5

1

u(x,y,t=

1)

1

0.5

x

0

Numerical solution

-0.5

-1-1

-0.5

0

y

0.5

2

1.5

1

0

0.5

1

uh(x,y,t=

1)

Figure 8.4: The comparison of exact solution with numerical solution for the problem ona circular domain

181 Chapter 8

Example 8.5.3 Finally, we investigate the following 2D unsteady generalized Oldroyd-

B fluid in a magnetic field [83, 163]λD1+α

t u(y, z, t) + ∂u(y,z,t)∂t = ν(1 + θDβ

t )(4u(y, z, t))

−K(1 + λDαt )u(y, z, t), (y, z, t) ∈ Ω× J,

u(y, z, 0) = 0, ut(y, z, 0) = 0, (y, z) ∈ Ω,

u(0, z, t) = z(1− z)t2, u(1, z, t) = 0, u(y, 0, t) = 0, u(y, 1, t) = 0, t ∈ J ,

(8.31)

where 0 < α, β < 1, Ω = (0, 1)× (0, 1), J = (0, 1], λ and θ are the relaxation time and the

retardation time respectively, ν = µρ , µ is the dynamic viscosity coefficient of the fluid, ρ

is the density of the fluid, K =σB2

0ρ , B0 is the magnetic intensity and σ is the electrical

conductivity, and λ, θ,K ≥ 0. This model depicts an incompressible Olyroyd-B fluid flow

bounded by a horizontal rectangular pipe in a magnetic field. Initially, the whole system

is at rest. Then at time t = 0+, the fluid starts to flow with some velocity t2 in one flow

direction. Due to the effect of the shear stress, the whole fluid moves gradually.

To implement the finite element method, we need to transform the non-zero boundary

condition u(0, z, t) = z(1− z)t2 in (8.31) into a homogeneous boundary conditions for use

in the numerical scheme. Let

u(y, z, t) = W (y, z, t) + z(1− z)(1− y)t2, (8.32)

then W (y, z, t) satisfies the following system with homogeneous boundary conditionsλD1+α

t W (y, z, t) + ∂W (y,z,t)∂t = ν(1 + θDβ

t )(4W (y, z, t))

−K(1 + λDαt )W (y, z, t) + f(y, z, t), (y, z, t) ∈ Ω× J,

W (y, z, 0) = 0, Wt(y, z, 0) = 0, (y, z) ∈ Ω,

W (y, z, t) = 0, (y, z) ∈ ∂Ω, t ∈ J ,

(8.33)

where

f(y, z, t) =− 2ν(1− y)(t2 +

θΓ(3)

Γ(3− β)t2−β

)− 2z(1− z)(1− y)

(t+

λ

Γ(2− α)t1−α

)−Kz(1− z)(1− y)

(t2 +

λΓ(3)

Γ(3− α)t2−α

).

Then the finite element scheme can be applied to solve (8.33). Furthermore, combining

(8.32), we can obtain u(y, z, t).

Here we choose λ = ν = θ = K = 1, α = 0.8, β = 0.7. Since the exact solution for this

problem is difficult to calculate explicitly, we adopt the approximate solution unh of a fine

mesh (h = 164 and τ = 1

1000) as the exact solution to observe the convergence behaviour.

The corresponding numerical results of the L2 error for different h are presented in Table

8.6. Again, the numerical results exhibit a convergence order of O(h2), which implies that

our method is effective. We also find that the 2D unsteady generalized Oldroyd-B fluid on

a rectangular domain is less time-consuming than when computing on a circular domain.

Chapter 8 182

Figure 8.5 depicts the profile of velocity u(y, z, t) at different times. We can observe that

the flow velocity propagates along the y-direction and increases with increasing time.

Table 8.6: The L2 error and convergence order of h for α = 0.8, β = 0.7 with τ = 11000 at

t = 1

h L2 error Order CPU time

1/4 1.4716E-03 – 29.63 s1/8 3.9907E-04 1.88 1.81 min1/16 9.7914E-05 2.03 9.00 min1/32 1.9774E-05 2.31 1.37 h

10.8

0.6

y

0.40.2

t=0.1

00

0.2

0.4

z

0.6

0.8

0.25

0.2

0.15

0.1

0.05

01

u(y,z,t)

10.8

0.6

y

0.40.2

t=0.5

00

0.2

0.4

z

0.6

0.8

0.25

0.2

0.15

0.1

0.05

01

u(y,z,t)

10.8

0.6

y

0.40.2

t=1.0

00

0.2

0.4

z

0.6

0.8

0.25

0.2

0.15

0.1

0.05

01

u(y,z,t)

Figure 8.5: Numerical solution profiles of velocity u(y, z, t) at different t for α = 0.8,β = 0.7 with h = 1

40 , τ = 1100

8.6 Conclusions

In this chapter, we considered the finite element method for a novel 2D multi-term time-

fractional mixed sub-diffusion and diffusion-wave equation. Utilising the mixed L schemes

to approximate the time-fractional derivatives and combining the finite element method,

the fully finite element scheme of the equation was derived. Then two unified numerical

schemes with accuracy of O(τ + h2) and O(τmin3−γ,2−α,2−β + h2) for the equation were

constructed. Furthermore, the stability and convergence of scheme I was established. It is

183 Chapter 8

straightforward to conclude that the numerical method proposed is flexible and robust and

can be applied directly to deal with other multi-term time-fractional diffusion equations,

such as the generalized Maxwell fluid model, the generalized Oldroyd-B fluid model and

other fractional models. In future work, we shall explore the application of these methods

and techniques to the three-dimensional multi-term time-fractional diffusion equation and

the stability and convergence analysis of scheme II.

Chapter 9

Conclusions

In this chapter, the main objectives of the thesis are revisited and the contributions,

highlights and conclusions of the research are summarised.

The research objectives were to:

• Develop new numerical methods and analytical techniques for simulating complex

fractional dynamical models to reduce computational cost, which was achieved by

utilising high-order numerical methods (Chapter 2) and fast algorithms (Chapter

3);

• Develop numerical methods for complex fractional dynamical models with the Riesz

fractional operator on irregular domains, which was achieved by using the finite

element method (Chapter 4) and finite volume method (Chapter 5) combining the

unstructured mesh to approximate the space fractional derivative;

• Develop numerical methods for viscoelastic non-Newtonian fluid models, such as

the generalised Maxwell fluid model, the generalised Oldroyd-B fluid model and the

generalised Burgers’ fluid model, which was achieved by applying mixed difference

schemes to discretise the different time fractional derivatives and finite difference

method (Chapter 6 and 7) and finite element method (Chapter 8).

These objectives were addressed via a series of six published papers and one paper under

review presented on the high-order numerical methods for the Riesz space fractional

advection-dispersion equation, a fast second-order accurate method for a two-sided space-

fractional diffusion equation with variable coefficients, unstructured mesh finite element

method for the 2D time-space Riesz fractional diffusion equation on irregular convex

domains, unstructured mesh control volume method for two-dimensional space fractional

diffusion equations with variable coefficients on convex domains, finite difference method

for the generalised Oldroyd-B fluid and MHD Couette flow of a generalised Oldroyd-B fluid

and finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion

and diffusion-wave equation on convex domains, respectively. These papers presented in

Chapters 2 to 8 of the thesis respectively form the backbone of the numerical investigation

and application of fractional dynamical systems to viscoelastic non-Newtonian fluids. In

the following section, the contributions of this thesis will be discussed in detail. The

chapter ends with some recommendations for future research.

184

185 Chapter 9

9.1 Summary and Discussion

To summarise, one original contribution of this thesis is the treatment of the Riesz space

fractional derivative on irregular convex domains. We proposed a novel numerical tech-

nique based on the Galerkin finite element method with an unstructured mesh to deal

with the Riesz space fractional derivative on arbitrarily shaped convex and non-convex

domains and present the theoretical analysis, which was more flexible compared to the

finite difference method. We also developed a novel unstructured mesh control volume

method to deal with space fractional derivative on arbitrarily shaped convex domains,

which could reduce CPU time significantly (almost reduced 23 of the computational time

for h = 4.5873× 10−2) while retaining the same accuracy and approximation property as

the unstructured mesh finite element method.

Another important contribution of this thesis is that we presented a unified numerical

scheme to solve a class of novel multi-term time fractional diffusion-wave and sub-diffusion

equations and established the rigorous stability and convergence analysis, which not only

could be extended to solve the generalised Maxwell fluid model, the generalised Oldroyd-B

fluid model and the generalised Burgers’ fluid model but also to solve the general multi-

term time fractional diffusion-wave or sub-diffusion equation, the time fractional telegraph

equation, the fractional cable equation and the fractional Cattaneo diffusion equation.

9.2 Recommendations for Future Research

According to the mathematical models and numerical methods discussed in this thesis,

several potential directions can be pursued for the future research. Here, we list three

main recommendations.

F Fast algorithms or numerical methods for space or time fractional diffusion equation

For the space fractional diffusion equation, due to the the nonlocal property of the frac-

tional operators, a full coefficient matrix will be generated when the general finite differ-

ence method is applied, which require storage of O(M2) and computational cost of O(M2)

where M is the number of spatial grid points. According to the structure of the coefficient

matrix, it can be decomposed into a sum of matrices involving Toeplitz matrices, which

reduced the total memory requirement from O(M2) to O(M). Then, the Toeplitz matrix

can be embedded into a circulant matrix and evaluated using the fast Fourier transform

and inverse fast Fourier transform. This method is effective for the problems from one

dimension to three dimensions [248, 249, 252]. For the two-dimensional space fractional

diffusion equation on the rectangular domain, a slightly sparse and special block matrix

is generated that is a dense matrix [194]. Then the backward differentiation formulas and

Jacobian-free Newton-Krylov methods with a fast preconditioner are feasible [194, 195].

However, when it comes to the problems on an irregular domain, the pattern of the coeffi-

cient matrix is not regular and the matrix is sparse. Except for the Bi-CGSTAB iterative

Chapter 9 186

method, effective decomposition of the coefficient matrix and preconditioner needs fur-

ther exploration accordingly to reduce the computational cost. Due to the nonlocality

and singularity of the fractional operator, the calculation for the time-fractional equations

is more difficult and time-consuming, which has been a worldwide research focus as well

recently [6, 258, 279].

F Numerical methods and analysis for the fractional diffusion equation involving the

Riesz fractional operator or fractional Laplacian operator on irregular domains

In Chapter 4, we proposed the FEM with unstructured triangular meshes to deal with

space fractional derivatives on an irregular convex domain, which needs fewer grid nodes

to generate the meshes and has high accuracy compared to the finite difference method

[157, 208]. However, we found that the calculation of the fractional derivative of the basis

functions is extremely time-consuming due to the nonlocality of the fractional operators.

Although we developed an unstructured mesh control volume method with fast iterative

solver, of which the computational cost was reduced significantly, the theoretical analysis

of the control volume method is more complex than that of FEM, which needs further

study. The fractional differential equation involving the Laplacian operator has many

applications [23, 28, 59, 75, 146, 261] and is more complex than the Riesz fractional

differential equation due to the different definitions of fractional Laplacian operator [146].

The numerical methods for solving the fractional Laplacian equation on a general domain

is sparse [24, 66, 106, 108] and on an irregular domain is more sparse [28, 146, 260]. It

is still an important direction to explore effective numerical methods and analysis for the

fractional diffusion equation involving the Riesz fractional operator or fractional Laplacian

operator on irregular domains.

F High-order algorithm or numerical methods for the novel multi-term time fractional

diffusion equation of order from 0 to 2

For the time fractional operator Dαt (0 < α < 1), the general L1 scheme can be applied to

discretise it [145, 149, 238] and some high order numerical schemes were also proposed [91,

276, 299]. For the time fractional operator Dγt (1 < γ < 2), the general L2 scheme can be

applied to discretise it [238], which has the accuracy of O(τ2−γ). When the two operators

exist at the same time, some method can be applied [154], however, the theoretical analysis

is lacking. For our new multi-term time fractional diffusion equation, one more term

Dβt ( ∂2

∂x2) (0 < β < 1) is added. Although we presented a first-order scheme to the equation

and established the stability and convergence of the scheme, it is of low accuracy. As

time fractional operators are more time-consuming than space fractional operators in the

calculation, high-order algorithms or numerical methods are needed. It is easy to apply

the general high-order scheme to each time fractional operator. However, the difficulty

is to balance the three time fractional operators in a uniform scheme and establish the

theoretical analysis. Besides the matrix analysis presented in Chapter 7, the energy

method may be alternative way to prove the stability and convergence of the scheme. In

187 Chapter 9

addition, other numerical methods such as finite volume method or spectral method may

also be applied to solve the novel multi-term time fractional diffusion equation of order

from 0 to 2.

Appendix A

A novel finite volume method for the Riesz space distributed-order

diffusion equation

A.1 Introduction

In the past centuries, many scientists thought the classical diffusion equation:

∂u(x, t)

∂t= K

∂2u(x, t)

∂x2

can describe many phenomena in physics, environmental science, hydrodynamics, econo-

my, medical science, geography and so on. However, it is found that the above equation

cannot model the abnormal diffusion processes in complex media because of the long-

tailed profile in the spatial distribution or the memorability. To overcome this, fractional

partial differential equations are introduced [98, 128, 148, 149, 156, 203, 269, 270]. For

example, time-fractional derivatives can be used to describe Markovian processes and

space-fractional derivatives can be used to describe complicated processes in fractional

kinetic systems [231]. Recently it was found that the single order of differentiation is

not suitable to represent phenomena where the order of differentiation varies in a given

range. Distributed-order fractional differential equations were first put forward by Caputo

[31, 32]. Time distributed-order fractional equations can model processes lacking power-

law scaling over the whole time-domain [37, 115, 169, 196, 245], such as the ultraslow

diffusion where a plume of particles spread at a logarithmic rate. Space distributed-order

fractional equations are more flexible to represent the effect of the medium and its spatial

interactions with the fluid, such as accelerated superdiffusion [231].

To date, some papers have appeared on how to solve distributed-order fractional equation-

s. Meerschaert et al. [184] investigated explicit strong solutions and stochastic analogues

for time distributed-order fractional diffusion equations on bounded domains with Dirich-

let boundary data. Gorenflo and his co-workers [97] provided the fundamental solution of

the Cauchy problem for time distributed-order fractional equations by employing Laplace

and Fourier transforms and interpreted the fundamental solution as a probability density.

Luchko [170] showed the uniqueness and continuous dependence on the initial data for

the generalised time distributed-order fractional diffusion equation on a bounded domain.

For more fundamental solutions of time distributed-order fractional diffusion equations,

the readers can refer to [4, 37, 231]. A number of papers have appeared on the numer-

ical solution of time distributed-order fractional equations. Ye et al. [265, 266] applied

an implicit numerical method and compact difference scheme for time distributed-order

fractional equations and obtained their convergence. Hu et al. [104, 105] discussed a time

distributed-order two-sided space-fractional advection-dispersion equation and obtained

188

189 Chapter A

the stability and convergence for the implicit numerical method. Gao and Sun [92, 93]

derived some high order difference schemes for one and two-dimensional time distributed-

order diffusion equations. Ford et. al [89] considered an implicit numerical method for the

time distributed-order diffusion equation. Alikhanov [2] studied numerical methods for

the multi-term variable distributed-order diffusion equation. Morgado and Rebelo [193]

proposed an implicit scheme for the numerical approximation of the distributed-order

time-fractional reaction-diffusion equation with a nonlinear source term. Wang and Liu

[254] used a finite difference method to solve the Riesz space distributed-order equation

and obtained second-order accuracy. However, most of these numerical methods only used

finite difference methods, which makes it difficult to generalise the numerical method to

deal with irregularly shaped solution domains. Some finite volume methods to solve a

fractional differential equation with a constant dispersion coefficient have been proposed.

Zhang et al. [283] presented a finite volume approach to solving a fractional advection-

dispersion equation with a constant dispersion coefficient. Hejazi et al. [99] proposed a

finite volume method to solving the time-space, two-sided fractional advection-dispersion

equation on a one-dimensional domain. The spatial discretisation employs fractionally-

shifted Grunwald formulae to discretise the Riemann-Liouville fractional derivatives at

the control volume faces in terms of function values at the nodes. However, these finite

volume methods are not extended to two-dimensional and three-dimensional problems in

a natural manner.

In this paper, we propose a novel finite volume method (FVM) with a nonlocal operator

(using nodal basis functions) for solving a distributed-order space-fractional diffusion

equation (FDE). The finite volume method with a nonlocal operator (using nodal basis

functions) can be extended to two-dimensional and three-dimensional problems in any

irregular region. We focus on the finite volume method for the following space distributed-

order fractional equation:

∂u(x, t)

∂t=

∫ 2

1P (α)

∂αu(x, t)

∂|x|αdα+ f(x, t), (x, t) ∈ (0, L)× (0, T ], (A.1)

with homogeneous Dirichlet boundary conditions

u(0, t) = 0, u(L, t) = 0, t ∈ [0, T ], (A.2)

and initial condition

u(x, 0) = ψ(x), x ∈ [0, L], (A.3)

where f(x, t) can be used to represent sources and sinks. P (α) is a non-negative weight

function satisfying the conditions

P (α) ≥ 0, P (α) 6≡ 0, α ∈ (1, 2), 0 <

∫ 2

1P (α)dα <∞.

Chapter A 190

According to [231], in the general case, P (α) can be represented as

P (α) = lα−2K[A1δ(α− α1) +A2δ(α− α2)],

where l and K are dimensional positive constants, A1 > 0, A2 > 0, 0 < α1 < α2 ≤ 2.

If α1 6= 1 and α2 6= 1, then P (1) = 0. Therefore, for convenience, throughout the

paper we assume that P (1) = 0 in Eq.(A.1), which is reasonable. The Reisz fractional

derivative ∂αu(x,t)∂|x|α (we adopt here the notation introduced in [218, 231]) is defined for

a ‘sufficiently well-behaved’ function u(x, t) ∈ L1(−∞,∞; 0, T ) through the Riemann-

Leouville derivatives [217]. When 1 < α ≤ 2 the Riesz fractional derivative is defined as

given in [231]:

∂αu(x, t)

∂|x|α=

− 1

2 cos(απ/2)

(∂αu(x,t)∂xα + ∂αu(x,t)

∂(−x)α

), α 6= 1,

− ∂∂xHu(x, t), α = 1,

where∂αu(x, t)

∂xα=

1

Γ(2− α)

∂2

∂x2

∫ x

−∞

u(s, t)

(x− s)α−1ds,

∂αu(x, t)

∂(−x)α=

1

Γ(2− α)

∂2

∂x2

∫ ∞x

u(s, t)

(s− x)α−1ds,

and H is the Hilbert transfer operator [231]. Thus, the integrand function is defined as

F (α) :=

0, α = 1,

P (α)∂αu(x,t)∂|x|α , 1 < α ≤ 2.

Thus, we can ensure that F ∈ C2(1, 2]. Therefore, the second order accuracy of the

midpoint rule can be ensured. We will assume that the Riesz space distributed-order

diffusion Eq.(A.1) has a unique and sufficiently smooth solution under the boundary

conditions (A.2) and the initial condition (A.3) (the reader is referred to [69] where results

on existence and uniqueness are developed).

In this paper, we propose a novel finite volume method (FVM) for a distributed order

space-fractional diffusion equation (FDE). Firstly, we use the mid-point quadrature rule

to transform the space distributed-order diffusion Eq.(A.1) into a multi-term fractional

equation [111, 154]. Then we use the finite volume method [77, 140, 155, 302] to solve the

multi-term fractional equation and derive the Crank-Nicolson scheme. Furthermore, we

prove that the finite volume method for the space distributed-order fractional equation

is unconditionally stable and convergent with the accuracy of O(σ2 + τ2 + h2). In this

paper, we suppose the solution is smooth enough to guarantee the proposed numerical

method is effective for implementing high order numerical schemes.

The structure of this chapter is as follows. In Section A.2, we discretise the space

distributed-order fractional equation into a multi-term fractional equation by using the

finite volume method and derive the Crank-Nicolson scheme. We prove the stability and

191 Chapter A

convergence of the Crank-Nicolson scheme in Section A.3. Finally, we present two nu-

merical experiments to show the effectiveness of our finite volume method in Section A.4

and give the conclusions in Section A.5.

A.2 Finite volume method for the distributed-order diffusion equation

In Eq.(A.1), if P (1) = 0, then we have∫ 2

1P (α)

∂αu(x, t)

∂|x|αdα =

∫ 2

1+P (α)

∂αu(x, t)

∂|x|αdα.

In the calculation, we define 1+ = 1 + ε, ε → 0. As now the integrand function F ∈C2(1, 2], we discretise the interval [1+, 2] by the grid 1 + ε = ξ0 < ξ1 < · · · < ξS = 2 and

denote ∆ξk = ξk − ξk−1 = 1S = σ, k = 1, 2, · · · , S, αk =

ξk+ξk−1

2 = 1 + 2k−12S . Using the

mid-point quadrature rule for (A.1), we obtain

∫ 2

1+P (α)

∂αu(x, t)

∂|x|αdα =

S∑k=1

P (αk)∂αku(x, t)

∂|x|αk∆ξk +O(σ2). (A.4)

Next, the computing domain [0, L] × [0, T ] is discretised by xi = ih, i = 0, 1, · · · ,Mand tn = nτ, n = 0, 1, · · · , N , where h = L/M, τ = T/N are the space and time steps,

respectively. Assuming that the solution u(x, t) is smooth enough (for example, here in

time u ∈ C2[0, T ]) and note that

∂u(x, tn− 12)

∂t=u(x, tn)− u(x, tn−1)

τ+O(τ2).

Combining (A.4) with the above equation, we obtain

u(x, tn)− u(x, tn−1)

τ=σ

2

S∑k=1

P (αk)

[∂αku(x, tn)

∂|x|αk+∂αku(x, tn−1)

∂|x|αk

]+

1

2[f(x, tn) + f(x, tn−1)] +O(σ2 + τ2)

=σ

2

S∑k=1

akP (αk)∂

∂x

[∂βku(x, tn)

∂xβk− ∂βku(x, tn)

∂(−x)βk

]

+σ

2

S∑k=1

akP (αk)∂

∂x

[∂βku(x, tn−1)

∂xβk− ∂βku(x, tn−1)

∂(−x)βk

]+

1

2[f(x, tn) + f(x, tn−1)] +O(σ2 + τ2), (A.5)

where 0 < βk = αk − 1 < 1, ak = − 12 cos(π(1+βk)/2) > 0.

Furthermore, let xi− 12

= xi+xi−1

2 , i = 1, 2, · · · ,M be the mid-point of the interval [xi−1, xi].

Then, we take the integration of (A.5) over a control volume [xi− 12, xi+ 1

2] for i = 1, 2, · · · ,M−

Chapter A 192

1, which leads to

∫ xi+1

2

xi− 1

2

u(x, tn)dx− τσ

2

S∑k=1

akP (αk)

[∂βku(x, tn)

∂xβk− ∂βku(x, tn)

∂(−x)βk

]xi+1

2

xi− 1

2

=

∫ xi+1

2

xi− 1

2

u(x, tn−1)dx+τσ

2

S∑k=1

akP (αk)

[∂βku(x, tn−1)

∂xβk− ∂βku(x, tn−1)

∂(−x)βk

]xi+1

2

xi− 1

2

+τ

2

∫ xi+1

2

xi− 1

2

[f(x, tn) + f(x, tn−1)] dx+O(τh(σ2 + τ2)). (A.6)

Now, we define the space Vh as the set of piecewise-linear polynomials on the mesh

[xi−1, xi]i=1,2,···M , the nodal based functions φ0, φ1, · · · , φM of Vh can be expressed in

the form

φi(x) =

x−xi−1

h , x ∈ [xi−1, xi],xi+1−x

h , x ∈ [xi, xi+1], i = 1, 2, · · · ,M − 1 and

0, else,

φ0(x) =

x1−xh , x ∈ [x0, x1],

0, else,φM (x) =

x−xM−1

h , x ∈ [xM−1, xM ],

0, else.

Then the approximate solution uh(x, tn) ∈ P (0, 1) with piecewise polynomials can be

expressed as

uh(x, tn) =

M−1∑j=1

unj φj(x). (A.7)

Thus, we obtain the following Crank-Nicolson scheme:

M−1∑j=1

unj

∫ xi+1

2

xi− 1

2

φj(x)dx− τσ

2

M−1∑j=1

unj

S∑k=1

akP (αk)

[∂βkφj(x)

∂xβk− ∂βkφj(x)

∂(−x)βk

]xi+1

2

xi− 1

2

=

M−1∑j=1

un−1j

∫ xi+1

2

xi− 1

2

φj(x)dx+τσ

2

M−1∑j=1

un−1j

S∑k=1

akP (αk)

[∂βkφj(x)

∂xβk− ∂βkφj(x))

∂(−x)βk

]xi+1

2

xi− 1

2

+τ

2

∫ xi+1

2

xi− 1

2

[f(x, tn) + f(x, tn−1)] dx. (A.8)

Note that by direct calculations, it can be concluded that

∫ xi+1

2

xi− 1

2

φj(x)dx =

h/8, |i− j| = 1,

3h/4, i = j,

0, else,

and

∂βkφj(xi+ 12)

∂xβk=

1

Γ(2− βk)hβk

0, j > i+ 1,

2βk−1, j = i+ 1,

(3/2)1−βk − 2βk , j = i,

cki−j+1, j < i,

193 Chapter A

∂βkφj(xi− 12)

∂xβk=

1

Γ(2− βk)hβk

0, j > i,

2βk−1, j = i,

(3/2)1−βk − 2βk , j = i− 1,

cki−j , j < i− 1,

∂βkφj(xi+ 12)

∂(−x)βk=

1

Γ(2− βk)hβk

ckj−i, j > i+ 1,

(3/2)1−βk − 2βk , j = i+ 1,

2βk−1, j = i,

0, j < i,

∂βkφj(xi− 12)

∂(−x)βk=

1

Γ(2− βk)hβk

ckj−i+1, j > i,

(3/2)1−βk − 2βk , j = i,

2βk−1, j = i− 1,

0, j < i− 1,

where, cki =(i− 3

2

)1−βk − 2(i− 12)1−βk + (i+ 1

2)1−βk , i = 2, 3, · · · .Therefore, (A.8) can be rewritten as:

h

8(uni−1 + 6uni + uni+1)−

M−1∑j=1

unjGij

=h

8(un−1i−1 + 6un−1

i + un−1i+1 ) +

M−1∑j=1

un−1j Gij + (Fn)i, (A.9)

where (Fn)i = τ2

∫ xi+12

xi− 1

2

[f(x, tn) + f(x, tn−1)] dx, Gij = G1,ij −G2,ij with

G1,ij =τσ

2

S∑k=1

akP (αk)

[∂βkφj(xi+ 1

2)

∂xβk−∂βkφj(xi− 1

2)

∂xβk

],

G2,ij =τσ

2

S∑k=1

akP (αk)

[∂βkφj(xi+ 1

2)

∂(−x)βk−∂βkφj(xi− 1

2)

∂(−x)βk

].

For i, j = 1, 2, · · · ,M−1, denote Aij =∫ xi+1

2xi− 1

2

φj(x)dx, Un =[un1 , u

n2 , · · · , unM−1

]T. Then,

we can express the system (A.9) in matrix form as

(A−G)Un = (A+G)Un−1 + Fn. (A.10)

The boundary and initial conditions are discretised as ψ0i = ψ(ih), U0 =

[u0

1, u02, · · · , u0

M−1

]Tfor i = 1, 2, · · · ,M − 1, respectively.

A.3 Stability and convergence

In this section, we will analyse the stability and convergence of the finite volume method

for the space distributed-order fractional equation.

Chapter A 194

Lemma A.3.1 Assume that 0 < βk < 1, cki =(i− 3

2

)1−βk − 2(i− 1

2

)1−βk +(i+ 3

2

)1−βk ,

i = 2, 3, · · · then the following hold [77]:

(1) cki is increasing monotonically as i increases, and cki < 0, i = 2, 3, · · · ;(2) lim

i→+∞cki = 0;

(3)+∞∑i=2

(cki+1 − cki ) = −ck2.

By following a similar strategy of proof as outlined in Theorem 1 [77], we propose the

following theorem.

Theorem A.3.1 For 0 < βk < 1, the coefficients Gij satisfy

|Gii| >M−1∑j=1,j 6=i

|Gij |, i = 1, 2, · · · ,M − 1,

i.e., G is strictly diagonally dominant.

Theorem A.3.2 For 0 < βk < 1, B = A−G, then B is also strictly diagonally dominant

and the spectral radius of B−1 fulfills

ρ(B−1) <2

h.

Proof. It is straightforward to show that

Bij =

τσ2

S∑k=1

akP (αk)

Γ(2−βk)hβk

(ckj−i − ckj−i+1

), j > i+ 1,

h8 −

τσ2

S∑k=1

akP (αk)

Γ(2−βk)hβk

[3(

12

)1−βk − (32

)1−βk + ck2

], j = i+ 1,

3h4 −

τσ2

S∑k=1

akP (αk)

Γ(2−βk)hβk· 2[(

32

)1−βk − 3(

12

)1−βk] , j = i,

h8 −

τσ2

S∑k=1

akP (αk)

Γ(2−βk)hβk

[3(

12

)1−βk − (32

)1−βk + ck2

], j = i− 1,

τσ2

S∑k=1

akP (αk)

Γ(2−βk)hβk

(cki−j − cki−j+1

), j < i− 1.

(A.11)

Following the same line of Theorem 2 in [77], we can obtain the assertion.

Setting B = A−G, then according to the above theorem, we know B is strictly diagonally

dominant, so B is nonsingular and invertible. The discrete scheme (A.10) can be written

as

Un = B−1(A+G)Un−1 +B−1F. (A.12)

195 Chapter A

Theorem A.3.3 Let D = (λ− 1)A− (λ+ 1)G, for 0 < βk < 1, if λ > 1 or λ < −1, we

can conclude that D is strictly diagonally dominant, i.e.,

|Dii| >M−1∑j=1,j 6=i

|Dij |, i = 1, 2, · · · ,M − 1.

Proof.

Dij =

(λ+ 1) · τσ2S∑k=1

akP (αk)

Γ(2−βk)hβk

(ckj−i − ckj−i+1

), j > i+ 1,

(λ− 1) · h8 − (λ+ 1) · τσ2S∑k=1

akP (αk)

Γ(2−βk)hβk

[3(

12

)1−βk − (32

)1−βk + ck2

], j = i+ 1,

(λ− 1) · 3h4 − (λ+ 1) · τσ2

S∑k=1

akP (αk)

Γ(2−βk)hβk· 2[(

32

)1−βk − 3(

12

)1−βk] , j = i,

(λ− 1)h8 − (λ+ 1) · τσ2S∑k=1

akP (αk)

Γ(2−βk)hβk

[3(

12

)1−βk − (32

)1−βk + ck2

], j = i− 1,

(λ+ 1) τσ2

S∑k=1

akP (αk)

Γ(2−βk)hβk

(cki−j − cki−j+1

), j < i− 1.

(1) For λ > 1, from Lemma A.3.1, it is obvious that Dij < 0 for j > i+ 1 and j < i− 1.

Since(

32

)1−βk − 3(

12

)1−βk =(

12

)1−βk(31−βk − 3) < 0, then Dii > 0.

Now we turn to discuss the sign of Di,i−1 and Di,i+1.

(i) IfDi,i−1 = Di,i+1 = (λ−1)h8−(λ+1)· τσ2S∑k=1

akP (αk)

Γ(2−βk)hβk

[3(

12

)1−βk − (32

)1−βk + ck2

]≥

0, then

M−1∑j=1,j 6=i

|Dij | =i−2∑j=1

|Dij |+M−1∑j=i+2

|Dij |+ |Di,i−1|+ |Di,i+1|

<(λ+ 1) · τσ2

i−2∑j=−∞

S∑k=1

akP (αk)

Γ(2− βk)hβk(cki−j+1 − cki−j

)+(λ+ 1) · τσ

2

+∞∑j=i+2

S∑k=1

akP (αk)

Γ(2− βk)hβk(ckj−i+1 − ckj−i

)+(λ− 1)

h

4− 2(λ+ 1) · τσ

2

S∑k=1

akP (αk)

Γ(2− βk)hβk

[3(1

2

)1−βk − (3

2

)1−βk + ck2

]

=(λ− 1)h

4− 2(λ+ 1) · τσ

2

S∑k=1

akP (αk)

Γ(2− βk)hβk

[5(1

2

)1−βk − 5(3

2

)1−βk + 2(5

2

)1−βk]

<(λ− 1)3h

4+ 2(λ+ 1) · τσ

2

S∑k=1

akP (αk)

Γ(2− βk)hβk·[3(1

2

)1−βk − (3

2

)1−βk] = Dii

Chapter A 196

as

2(λ+ 1) · τσ2

S∑k=1

akP (αk)

Γ(2− βk)hβk

[8(1

2

)1−βk − 6(3

2

)1−βk + 2(5

2

)1−βk]

=4(λ+ 1) · τσ2

S∑k=1

akP (αk)

Γ(2− βk)hβk(1

2

)1−βk [4− 3 · 31−βk + 51−βk]

>0 > −(λ− 1)h

2.

Here, we use the assertion that 51−βk − 3 · 31−βk + 4 > 0 for 0 < 1 − βk < 1.

Therefore,

|Dii| >M−1∑j=1,j 6=i

|Dij |.

(ii) IfDi,i−1 = Di,i+1 = (λ−1)h8−(λ+1)· τσ2S∑k=1

akP (αk)

Γ(2−βk)hβk

[3(

12

)1−βk − (32

)1−βk + ck2

]<

0, then

M−1∑j=1,j 6=i

|Dij | =i−2∑j=1

|Dij |+M−1∑j=i+2

|Dij |+ |Di,i−1|+ |Di,i+1|

<(λ+ 1) · τσ2

i−2∑j=−∞

S∑k=1

akP (αk)

Γ(2− βk)hβk(cki−j+1 − cki−j

)+(λ+ 1) · τσ

2

+∞∑j=i+2

S∑k=1

akP (αk)

Γ(2− βk)hβk(ckj−i+1 − ckj−i

)−(λ− 1)

h

4+ 2(λ+ 1) · τσ

2

S∑k=1

akP (αk)

Γ(2− βk)hβk·[3(1

2

)1−βk − (3

2

)1−βk + ck2

]

<(λ− 1)3h

4+ 2(λ+ 1) · τσ

2

S∑k=1

akP (αk)

Γ(2− βk)hβk·[3(1

2

)1−βk − (3

2

)1−βk] = Dii.

Thus, |Dii| >∑M−1

j=1,j 6=i |Dij |.(2) If λ < −1, it can be easily seen that Dii < 0 and Dij > 0 for j > i+ 1 and j < i−1.

Following the same line of proof for the case λ > 1, we also can conclude that

|Dii| >∑M−1

j=1,j 6=i |Dij |.

This completes the proof.

Theorem A.3.4 The spectral radius of B−1(A+G) satisfies ρ(B−1(A+G)) < 1, hence

the scheme (A.10) is unconditionally stable.

Proof. Since B,A and G are symmetric positive definite, it is obvious to assert that

B−1 is symmetric positive definite and

B−1(A+G) = B−12(B−

12 (A+G)B−

12)B

12 ,

197 Chapter A

which means that B−1(A + G) is similar to B−12 (A + G)B−

12 . Thus, B−1(A + G) and

B−12 (A + G)B−

12 have the same eigenvalues. As a result of the symmetric positive defi-

niteness of A,G and B, we know that B−12 (A+G)B−

12 is also symmetric positive definite.

Hence, all the eigenvalues of B−12 (A+G)B−

12 and B−1(A+G) are real. Suppose λ is an

eigenvalue of B−1(A+G), and

det(λI −B−1(A+G)) = det(B−1) · det(λB − (A+G)) = 0. (A.13)

Note that B is nonsingular and invertible, then det(B−1) 6= 0, thus det(λB−(A+G)) = 0.

Let D = λB − (A+G), then

D = (λ− 1)A− (λ+ 1)G.

(1) Clearly if λ = ±1 or 0, then D is diagonally dominant;

(2) If λ > 1 or λ < −1, Theorem A.3.3 asserts that D is also diagonally dominant.

Therefore, det(D) 6= 0 for all λ ≥ 1 or λ ≤ −1 or λ = 0. According to the above analysis,

as the roots of the equation det(λB − (A + G)) = 0 exist, λ must satisfy −1 < λ < 0 or

0 < λ < 1, so the eigenvalue of B−1(A+G) satisfies |λ| < 1. Thus, ρ(B−1(A+G)) < 1,

which completes the proof.

Combining Lemma 3 in [77] with the integral mean value theorem, we can conclude the

following lemmas.

Lemma A.3.2 We suppose the solution u(x, t) is smooth enough, then∫ xi+1

2

xi− 1

2

u(x, tn)dx−∫ x

i+12

xi− 1

2

u(x, tn−1)dx

=

∫ xi+1

2

xi− 1

2

M−1∑j=1

u(xj , tn)φj(x)dx−∫ x

i+12

xi− 1

2

M−1∑j=1

u(xj , tn−1)φj(x)dx+O(τh3). (A.14)

Lemma A.3.3 For 0 < βk < 1, then [77]

∂βku(x, tn)

∂xβk

∣∣∣∣xi+12

xi− 1

2

=M−1∑j=1

unj

(∂βkφj(xi+ 1

2)

∂xβk−∂βkφj(xi− 1

2)

∂xβk

)+O(h3), (A.15)

∂βku(x, tn)

∂(−x)βk

∣∣∣∣xi+12

xi− 1

2

=M−1∑j=1

unj

(∂βkφj(xi+ 1

2)

∂(−x)βk−∂βkφj(xi− 1

2)

∂(−x)βk

)+O(h3). (A.16)

Theorem A.3.5 Let un be the exact solution of the problem (A.1)-(A.3). Then the

numerical solution Un unconditionally converges to the exact solution un as h, τ and σ

tend to zero. Moreover,

||un − Un|| ≤ C(σ2 + τ2 + h2).

Here, un = (u(x1, tn), u(x2, tn), · · · , u(xM−1, tn)), Un = (un1 , un2 , · · · , unM−1).

Chapter A 198

Proof. Let eni denote the error at the point (xi, tn). Substituting eni = u(xi, tn)−uni into

(A.9) and combining (A.5) with (A.14)-(A.16), we obtain

h

8(eni−1 + 6eni + eni+1)−

M−1∑j=1

enjGij

=h

8(en−1i−1 + 6en−1

i + en−1i+1 ) +

M−1∑j=1

en−1j Gij +O(τh(σ2 + τ2 + h2)).

Note that en0 = enM = 0 and e0i = 0 for i = 1, 2, · · · ,M − 1. Thus,

(A−G)En = (A+G)En−1 +O(τh(σ2 + τ2 + h2))χ,

where χ = (1, 1, · · · , 1)T , En = (en1 , en2 , · · · , enM−1)T . Setting Q = (A−G)−1(A+G) and

b = O(τh(σ2 + τ2 + h2))(A−G)−1, by iteration, one has

En =(Qn−1 +Qn−2 + · · ·+ I

)b.

Since by Theorem A.3.2, we know that ρ((A−G)−1

)< 2

h and ρ(Q) < 1, then there exists

a vector norm and induced matrix norm ||·|| such that ||Q|| < 1 and ||(A−G)−1|| < Ch−1.

Then by deduction, it yields

||En|| ≤(||Qn−1||+ ||Qn−2||+ · · ·+ 1

)||b|| ≤ n||b|| ≤ O(σ2 + τ2 + h2).

Therefore,

||En|| ≤ C(σ2 + τ2 + h2).

A.4 Numerical examples

In order to demonstrate the effectiveness of the numerical method derived in the previous

section, two examples are presented.

Example A.4.1 Firstly, we consider the following distributed-order equation:

∂u(x, t)

∂t=

∫ 2

1P (α)

∂αu(x, t)

∂|x|αdα+ f(x, t), (x, t) ∈ (0, 1)× (0, 1],

with boundary conditions

u(0, t) = 0, u(1, t) = 0, t ∈ [0, 1],

and initial condition

u(x, 0) = x2(1− x)2, x ∈ [0, 1],

where

P (α) = −2Γ(5− α) cos(πα

2

).

199 Chapter A

It is readily to check P (1) = 0. The exact solution of the above equation is u(x, t) =

etx2(1−x)2. Table A.1 shows the error and convergence order of the finite volume method

with respect to τ and h for σ = 1/500 at t = 1. We can see that with τ = h decreasing,

the convergence order of τ and h attain second order. Table A.2 shows the error and

convergence order with respect to σ for τ = h = 1/400 at t = 1. It can be seen that

with σ decreasing, the convergence order of σ is also second order. According to the

error and convergence rate in the above tables, we can conclude that the finite volume

method for the Riesz-space distributed-order equations is effective and stable as expected.

In addition, we give a comparison of the exact and numerical solutions in Figure A.1, in

which we see that the numerical solution is in excellent agreement with the analytical

solution.

Table A.1: The error and the convergence order of τ and h for σ = 1/500 at t = 1

τ = h Error Order

1/8 1.4126E-04 –1/16 3.6200E-05 1.961/32 9.1862E-06 1.981/64 2.3298E-06 1.981/128 6.0233E-07 1.95

Table A.2: The error and the convergence order of σ for τ = h = 1/400 at t = 1

σ Error Order

1/4 3.2299E-04 –1/8 7.7350E-05 2.061/16 1.9201E-05 2.011/32 4.8387E-06 1.991/64 1.2565E-06 1.95

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18Exact solutionNumerical solution

Figure A.1: Comparison of the exact solution and numerical solution with σ = τ = h =1/80 at t = 1.0

Chapter A 200

Example A.4.2 Next, we consider the following Riesz space distributed-order equation

[37, 231]:∂u

∂t=

∫ 2

1P (α)

∂αu

∂|x|αdα, (x, t) ∈ (0, 1)× (0, 1],

with initial condition

u(x, 0) = δ(x− 0.5),

and boundary conditions

u(0, t) = 0, u(1, t) = 0.

According to [37, 231], in the general case, P (α) can be represented as

P (α) = lα−2K[A1δ(α− α1) +A2δ(α− α2)],

where l and K are dimensional positive constants, [l] = cm, [K] = cm2/sec 0 < α1 < α2 ≤2 and A1 > 0, A2 > 0. Here, we choose K = A1 = A2 = 1 α1 6= 1 and α2 6= 1. To givethe error estimate, we have chosen to use the numerical solution u(x, tn) =

∑m−1i=1 uni φi(x)

on a fine grid (h = 1/500) as the ‘target’ exact solution. Then, we adopt a set of pointsto calculate the discrete L2 error on the coarse grids, which is given in Table A.3. Wecan see that second order convergence is attained, which again shows the stability andreliability of our method.

Table A.3: The error and the convergence order of τ = h for σ = 1/100, l = 2, α1 = 1.255,α1 = 1.755 at t = 1

τ = h Error Order

1/160 6.7803E-03 –1/200 4.7873E-03 1.561/300 2.1277E-03 2.00

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08t=0.1t=0.5t=1.0t=2.0

Figure A.2: Numerical solution profile of u(x, t) at different t with σ = 1/100, τ = h =1/200, l = 2, α1 = 1.255, α1 = 1.755

Now we observe the diffusion behaviour of u(x, t). Figure A.2 displays the evolution ofu(x, t) at different times, which decays with increasing time. Figure A.3 illustrates theimpact of α1 and α2 on the diffusion behaviour of u(x, t). As α1 and α2 increase, there isa decrease in amplitude and more diffusive behaviour in the profiles. Finally, Figure A.4

201 Chapter A

exhibits the effect of different l on the numerical solution profile of u(x, t), which increasesin amplitude with increasing l.

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045(α

1, α

2)=(1.255, 1.355)

(α1, α

2)=(1.555, 1.655)

(α1, α

2)=(1.855, 1.955)

Figure A.3: Numerical solution profile of u(x, t) for different α1 and α2 with σ = 1/100,τ = h = 1/200, l = 2, at t = 1

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.005

0.01

0.015

0.02

0.025

0.03l=2l=5l=10

Figure A.4: Numerical solution profile of u(x, t) for different l with σ = 1/100, τ = h =1/200, α1 = 1.255, α1 = 1.755 at t = 1

A.5 Conclusions

In this chapter, based on the finite volume method, we have investigated the Crank-

Nicolson scheme for the Riesz space distributed-order diffusion equation. We prove that

the Crank-Nicolson scheme is unconditionally stable and convergent. Two numerical ex-

amples are presented to show the effectiveness of our computational method. Throughout

this paper, we only consider the space distributed-order derivatives, while in many prac-

tical problems, the equation may also involve time distributed-order derivatives, we plan

to develop the finite volume method to solve time and space distributed-order diffusion

equations in future research.

Appendix B

A novel finite volume method for the Riesz space distributed-order

advection-diffusion equation

B.1 Introduction

In the past few decades, there has been considerable interest in modelling complex physical

phenomena arising in areas such as the natural sciences, biology, geological sciences,

medicine and signal processing. It is often the case that important effects such as spatial

heterogeneity and memory give rise to anomalous transport processes that require the use

of fractional transport models that are based on operators derived from fractional calculus

[98, 115, 128, 169, 196, 203, 218, 245]. In general, these models take the form of single or

multi-term time-, space-, or time-space-fractional differential equations. However, both

single and multi-term fractional equations are not suitable for simulating the diffusion

processes in multi-fractal media because they have no fixed scaling exponent. However,

distributed-order diffusion equations are shown to be useful tools for describing anomalous

diffusion characterized by two or more scaling exponents in the mean squared displacement

(MSD), or even by logarithmic time dependency of the MSD.

Caputo [30] first proposed the use of differential equations with distributed-order deriva-

tives for generalizing stress-strain relations of unelastic media. Later, Caputo [31, 32]

discussed distributed-order time fractional differential equations and distributed-order s-

pace fractional differential equations, respectively, and derived the solutions with closed

form formulae of the classical problems. He found that one of the major differences be-

tween distributed-order time fractional differential equations and distributed-order space

fractional differential equations is that the former represents local variations and is par-

ticularly valid when considering local phenomena, while in an infinite medium it is more

appropriate to introduce the space fractional order derivative to represent the effect of the

medium and its space interaction with the fluid. Following on from this work, Chechkin

and Sokolov et al. [37, 231] proposed diffusion-like equations with distributed-order time

and space fractional derivatives for the kinetic description of anomalous diffusion and

relaxation phenomena. They showed that the equations with distributed-order deriva-

tives on the ‘proper’ side describe processes becoming more anomalous over the course

of time (accelerating superdiffusion and decelerating subdiffusion), while equations with

the additional distributed-order on the ‘wrong’ side describe the situations becoming less

anomalous (decelerating superdiffusion and accelerating subdiffusion). In 2006, Meer-

schaert and Scheffler [182] developed a stochastic model based on random walks with a

random waiting time between jumps. Scaling limits of these random walks were sub-

ordinated to random processes whose density functions solved the ultraslow diffusion

202

203 Chapter B

equation. Umarov and Steinberg [244] constructed multi-dimensional random walk mod-

els governed by distributed fractional order differential equations. In addition, they used

distributed-order differential equations to model the input-output relationship of a linear

time-variant system, some ultraslow and lateral diffusion processes, and to study the rhe-

ological properties of composite materials. Kochubei [130] also applied distributed-order

diffusion equation to discuss ultraslow and lateral diffusion processes. Soon after, Caputo

and Carcione [33] developed and solved a dissipative model for the propagation and atten-

uation of two-dimensional dilatational waves, using a new modeling algorithm based on

distributed-order fractional time derivatives. Li et al. [142] applied the distributed-order

filtering technique to model signal processing. Atanackovic et al. [5] studied waves in a

viscoelastic rod of finite length which was described by a constitutive equation of fractional

distributed-order with a special choice of weight functions. Eab and Lim [67] introduced

the distributed-order fractional Langevin-like equations and applied them to describe

anomalous diffusion without unique diffusion, or a scaling exponent. The distributed-

order equations [177] were also used to describe a variety of memory mechanisms and to

represent the dispersion acting with several different relaxations (e.g. Anelastic relaxation

mechanisms, or spectral lines in the case of dielectric media). Bulavatskya and Krivonosa

[26] on the basis of a sub-diffusion model described by a distributed-order system of e-

quations performed mathematical modeling of the dynamics of a local nonequilibrium

(in time) geomigration process in a geoporous medium saturated with a salt solution.

Recently, Sandev et al. [219] studied distributed-order time fractional diffusion equations

characterized by multifractal memory kernels, in contrast to the simple power-law kernel

of common time fractional diffusion equations. Su et al. [235] presented a distributed-

order fractional diffusion-wave equation (dofDWE) to describe radial groundwater flow

to or from a well, and give three sets of solutions of the dofDWE for aquifer pumping and

slug tests, which were useful for gaining further insights into groundwater flow properties.

To date, there are several papers that have focused on how to solve distributed-order frac-

tional equations. Meerschaert [184] investigated explicit strong solutions and stochastic

analogues for time distributed-order fractional diffusion equations on bounded domains

with Dirichlet boundary data. Gorenflo [97] and his co-workers provided the fundamen-

tal solution of the Cauchy problem for time distributed-order fractional equations by

employing Laplace and Fourier transform and interpreted the fundamental solution as a

probability density. Luchko et al. [170] showed the uniqueness and continuous dependence

on the initial data for the generalized distributed-order fractional diffusion equations on

bounded domains. There are also a few papers that discuss the numerical solutions of

distributed-order fractional equations. Ye et al. [156, 265, 266] applied an implicit numer-

ical method and compact difference scheme for time distributed-order fractional equations

and obtained their convergence. Hu et al. [104, 105] used an implicit numerical method to

discuss a time distributed-order two-sided space-fractional advection-dispersion equation

and obtained stability and convergence criteria. Li et al. [141] applied a reproducing

kernel method to solve time distributed-order diffusion equations. Morgado et al. [193]

proposed an implicit scheme for time distributed-order reaction-diffusion equations with

Chapter B 204

a nonlinear source term. Gao and Sun [92, 93, 240] focused on finite difference method-

s to solve one-dimensional and two-dimensional distributed-order differential equations

and derived two alternating direction implicit difference schemes. They also used an

extrapolation method to improve the accuracy order and obtained a high-order conver-

gence rate. Wang and Liu [254] used the shifted Grunwald−Letnikov method to obtain

a second-order accurate implicit numerical method for the Riesz space distributed-order

advection-dispersion equations. Du, Hao and Sun [64] studied some high-order difference

schemes for the distributed-order time-fractional equations in both one and two spatial

dimensions. Based on the composite Simpson formula and Lubich second-order operator,

they derived stable numerical solutions with a higher order convergence rate in space.

However, to our best knowledge, there are only few works on the solution of distributed-

order space fractional equations. In Section 5 of [231], Sokolov et al. discussed the

distributed-order space fractional diffusion equation:

∂u(x, t)

∂t=

∫ 2

0P (α)

∂αu(x, t)

∂|x|αdα, (B.1)

where P (α) is a dimensional function of the order of the derivative α, and ∂α

∂|x|α denotes

the Riesz space functional derivative. In the general case P (α) = lα−2Kw(α), l and K

are dimensional positive constants, [l] = cm, [K] = cm2/sec and w(α) = A1δ(α − α1) +

A2δ(α − α2), 0 < α1 < α2 ≤ 2, A1 > 0, A2 > 0. The equation for the characteristic

function of Eq.(B.1) has the solution

g(k, t) = exp−a1|k|α1t− a2|k|α2t, (B.2)

with a1 = A1K/l2−α1 , a2 = A2K/l

2−α2 . Eq.(B.2) is a product of two characteristic func-

tions of Levy stable PDFs with Levy indices α1, α2 and scale parameters a1/α1

1 and a1/α2

2 ,

respectively. Through a series of analyse, Solokov et al. [231] concluded that at small

times the characteristic displacement grew as t1/α2 , whereas at large times it grew as

t1/α1 . This meant that the process was an accelerated superdiffusion.

Based on this model, in this paper, we derive a numerical method for the following more

general space distributed-order fractional advection-diffusion equation:

∂u(x, t)

∂t=λ1

∫ 2

1P (α)

∂αu(x, t)

∂|x|αdα+ λ2

∫ 1

0Q(γ)

∂γu(x, t)

∂|x|γdγ

+f(x, t), (x, t) ∈ (0, 1)× (0, 1], (B.3)

with boundary condition

u(0, t) = 0, u(1, t) = 0, t ∈ [0, 1] (B.4)

and initial condition

u(x, 0) = ψ(x), x ∈ [0, 1]. (B.5)

205 Chapter B

In Eq.(B.3), λ1 > 0, λ2 > 0, P (α), Q(γ) are non-negative weight functions that satisfy the

conditions

0 <

∫ 2

1P (α)dα <∞, 0 <

∫ 1

0Q(γ)dγ <∞, 1 < α < 2, 0 < γ < 1.

We can see that when λ1 = λ2 = 1, P (α) = Q(γ) and f(x, t) = 0, Eq.(B.3) can be

reduced to Eq.(B.1). Noting the advection orders are always close to 1, here we suppose

that Q(γ) vanish outside the interval (12 , 1) [14, 15].

First, we use the mid-point quadrature rule to transform the space distributed-order

diffusion Eq.(B.3) into a multi-term fractional equation [113, 154]. There appears to

be little research published in the literature on how to treat the convection term, and

most existing methods are finite difference methods [78, 227, 254].To the best of our

knowledge, there is no finite volume or finite element method reported. As the order of

the fractional derivative of the convection term is 0 < γ < 1, it is difficult to utilise a finite

volume method for the convection term directly. Based on the definition of the fractional

derivative, we rewrite the convection term as a type of integral form, which is suitable

for applying the finite volume method. Then, combining the nodal basis functions, the

discrete form of the convection term is obtained. It is worth noticing that the finite

element method is also available to deal with the convection term using this technique,

which is encouraging and promising. Therefore, we believe that the use of nodal basis

functions to derive the discrete form of our model is the most significant contribution

of this paper. We use the finite volume method (FVM) [77, 100, 155] to discrete the

multi-term fractional equation and then obtain the Crank-Nicolson scheme. Furthermore,

we prove that the Crank-Nicolson scheme is unconditionally stable and convergent with

second order accuracy.

The structure of this paper is as follows. In Section B.2, we discretize the space distributed-

order fractional equation into a multi-term fractional equation and then use the finite

volume method to derive the Crank-Nicolson scheme for the transformed multi-term frac-

tional equation. We prove the stability and convergence of the Crank-Nicolson iteration

scheme in Section B.3. Finally, two examples are presented to show the effectiveness of

our finite volume method in Section B.4 and conclusions of our method are presented in

Section B.5.

B.2 The Crank-Nicolson scheme with the finite volume method

We first introduce some preliminary definitions of the Riesz fractional derivatives, for

0 < α < 2, α 6= 1, as follows:

∂αu(x, t)

∂|x|α= − 1

2 cos(απ/2)

(∂αu(x, t)

∂xα+∂αu(x, t)

∂(−x)α

)

Chapter B 206

with

∂αu(x, t)

∂xα=

1

Γ(n− α)

(∂

∂x

)n ∫ x

0

u(s, t)

(x− s)α−n+1ds, n = [α] + 1,

∂αu(x, t)

∂(−x)α=

1

Γ(n− α)

(− ∂

∂x

)n ∫ 1

x

u(s, t)

(s− x)α−n+1ds, n = [α] + 1.

Note that when α = n (n = 1, 2), ∂αu(x,t)∂|x|α = ∂nu(x,t)

∂xn . First, we discretize the integral

interval (1, 2) in Eq.(B.3) for α by the grid 1 = ξ0 < ξ1 < · · · < ξS = 2 and denote

∆ξk = ξk − ξk−1 = 1S = σ, k = 1, 2, · · · , S, αk =

ξk+ξk−1

2 = 1 + 2k−12S . The integral

interval (0, 1) for γ is discretized using the grid 0 = η0 < η1 < · · · < ηS = 1. Denote

∆ηl = ηl − ηl−1 = 1S

= %, l = 1, 2, · · · , S, γl =ηl+ηl−1

2 = 2l−12S

. Applying we the mid-point

quadrature rule, we obtain that

∫ 2

1P (α)

∂αu(x, t)

∂|x|αdα =

S∑k=1

P (αk)∂αku(x, t)

∂|x|αk∆ξk +O(σ2), (B.6)

∫ 1

0Q(γ)

∂γu(x, t)

∂|x|γdγ =

S∑l=1

Q(γl)∂γlu(x, t)

∂|x|γl∆ηl +O(%2). (B.7)

Additionally, discretize the time domain [0, 1] by tn = nτ, n = 0, 1, · · · , N with τ = 1/N.

Let Sh be a uniform partition of the space domain [0, 1], which is given by xi = ih, i =

0, 1, · · · ,M with h = 1/M . Assume that u(x, ·) ∈ C3([0, 1]), u(·, t) ∈ C2([0, 1]) and let

tn− 12

= tn+tn−1

2 , then

∂u(x, tn− 12)

∂t=u(x, tn)− u(x, tn−1)

τ+O(τ2). (B.8)

Combining Eqs.(B.6)-(B.8), we obtain

u(x, tn)− u(x, tn−1)

τ=λ1σ

2

S∑k=1

P (αk)

[∂αku(x, tn)

∂|x|αk+∂αku(x, tn−1)

∂|x|αk

]

+λ2%

2

S∑l=1

Q(γl)

[∂γlu(x, tn)

∂|x|γl+∂γlu(x, tn−1)

∂|x|γl

]+

1

2

[f(x, tn) + f(x, tn−1)

]+O(σ2 + %2 + τ2). (B.9)

Note that

∂αku(x, t)

∂|x|αk= − 1

2 cos(αkπ

2

) [∂αku(x, t)

∂xαk+∂αku(x, t)

∂(−x)αk

]= − 1

2 cos(αkπ

2

) ∂∂x

[∂βku(x, t)

∂xβk− ∂βku(x, t)

∂(−x)βk

], (B.10)

207 Chapter B

with βk = αk − 1 and

∂γlu(x, t)

∂|x|γl= − 1

2 cos(γlπ

2

) [∂γlu(x, t)

∂xγl+∂γlu(x, t)

∂(−x)γl

]= − 1

2 cos(γlπ

2

) ∂∂x

[I1−γl

0+u(x, t)− I1−γl

1− u(x, t)], (B.11)

where

I1−γl0+

u(x, t) =1

Γ(1− γl)

∫ x

0

u(ζ, t)

(x− ζ)γldζ, I1−γl

1− u(x, t) =1

Γ(1− γl)

∫ 1

x

u(ζ, t)

(ζ − x)γldζ.

Then, let xi− 12

= xi+xi−1

2 , i = 1, 2, · · · ,M be the mid-point of the interval [xi−1, xi].

Take the integration of the governing Eq.(B.9) over a control volume [xi− 12, xi+ 1

2] for

i = 1, 2, · · · ,M − 1 to obtain

∫ xi+1

2

xi− 1

2

u(x, tn)dx−S∑k=1

ak

[∂βku(x, tn)

∂xβk− ∂βku(x, tn)

∂(−x)βk

]xi+1

2

xi− 1

2

−S∑l=1

bl

[I1−γl

0+u(x, tn)− I1−γl

1− u(x, tn)]x

i+12

xi− 1

2

=

∫ xi+1

2

xi− 1

2

u(x, tn−1)dx+

S∑k=1

ak

[∂βku(x, tn−1)

∂xβk− ∂βku(x, tn−1)

∂(−x)βk

]xi+1

2

xi− 1

2

+

S∑l=1

bl

[I1−γl

0+u(x, tn−1)− I1−γl

1− u(x, tn−1)]x

i+12

xi− 1

2

+τ

2

∫ xi+1

2

xi− 1

2

[f(x, tn) + f(x, tn−1)

]dx+O(τh(σ2 + %2 + τ2)), (B.12)

where ak = −λ1στP (αk)

4 cos(αkπ

2)> 0, bl = −λ2%τQ(γl)

4 cos(γlπ

2)≤ 0.

Now, we define the space Vh as the set of piecewise-linear polynomials on the mesh Sh.

Then, the approximate solution uh(x, tn) ∈ P (0, 1) with piecewise polynomials can be

expressed as

uh(x, tn) =

M−1∑j=1

unj φj(x), (B.13)

with φi, 0 ≤ i ≤M being the nodal based functions of Vh. For more details, one can refer

to [77]. Therefore, we obtain the subsequent Crank-Nicolson scheme:

M−1∑j=1

unj

∫ xi+1

2

xi− 1

2

φj(x)dx−M−1∑j=1

unj

S∑k=1

ak

[dβkφj(x)

dxβk− dβkφj(x)

d(−x)βk

]xi+1

2

xi− 1

2

−M−1∑j=1

unj

S∑l=1

bl

[I1−γl

0+φj(x)− I1−γl

1− φj(x)]x

i+12

xi− 1

2

Chapter B 208

=

M−1∑j=1

un−1j

∫ xi+1

2

xi− 1

2

φj(x)dx+

M−1∑j=1

un−1j

S∑k=1

ak

[dβkφj(x)

dxβk− dβkφj(x))

d(−x)βk

]xi+1

2

xi− 1

2

+M−1∑j=1

un−1j

S∑l=1

bl

[I1−γl

0+φj(x)− I1−γl

1− φj(x)]x

i+12

xi− 1

2

+τ

2

∫ xi+1

2

xi− 1

2

[f(x, tn) + f(x, tn−1)

]dx.

By direct calculations, it follows that for 1 ≤ i, j ≤M − 1,

∫ xi+1

2

xi− 1

2

φj(x)dx =

h/8, |i− j| = 1,

3h/4, i = j,

0, else,

and

dβkφj(xi+ 12)

dxβk=

1

Γ(2− βk)hβk

0, j > i+ 1,

2βk−1, j = i+ 1,

(3/2)1−βk − 2βk , j = i,

cki−j+1, j < i,

dβkφj(xi− 12)

dxβk=

1

Γ(2− βk)hβk

0, j > i,

2βk−1, j = i,

(3/2)1−βk − 2βk , j = i− 1,

cki−j , j < i− 1,

dβkφj(xi+ 12)

d(−x)βk=

1

Γ(2− βk)hβk

ckj−i, j > i+ 1,

(3/2)1−βk − 2βk , j = i+ 1,

2βk−1, j = i,

0, j < i,

dβkφj(xi− 12)

d(−x)βk=

1

Γ(2− βk)hβk

ckj−i+1, j > i,

(3/2)1−βk − 2βk , j = i,

2βk−1, j = i− 1,

0, j < i− 1,

where, cki =(i− 3

2

)1−βk − 2(i− 12)1−βk + (i+ 1

2)1−βk , i = 2, 3, · · · In addition,

I1−γl0+

φj(xi+ 12) =

h1−γl

Γ(3− γl)

0, j > i+ 1,(1/2)2−γl , j = i+ 1,

(3/2)2−γl − 2(1/2)2−γl , j = i,

dli−j+1, j < i,

I1−γl0+

φj(xi− 12) =

h1−γl

Γ(3− γl)

0, j > i,(1/2)2−γl , j = i,

(3/2)2−γl − 2(1/2)2−γl , j = i− 1,

dli−j , j < i− 1,

209 Chapter B

I1−γl1− φj(xi+ 1

2) =

h1−γl

Γ(3− γl)

dlj−i, j > i+ 1,

(3/2)2−γl − 2(1/2)2−γl , j = i+ 1,(

1/2)2−γl , j = i,

0, j < i,

I1−γl1− φj(xi− 1

2) =

h1−γl

Γ(3− γl)

dlj−i+1, j > i,

(3/2)2−γl − 2(1/2)2−γl , j = i,(

1/2)2−γl , j = i− 1,

0, j < i− 1

with dli =(i− 3

2

)2−γl − 2(i− 1

2

)2−γl +(i+ 1

2

)2−γl , i = 2, 3, · · · . Henceforth, we obtain the

discrete form of (B.12) as :

h

8(uni−1 + 6uni + uni+1)−

M−1∑j=1

unjGij −M−1∑j=1

unjDij

=h

8(un−1i−1 + 6un−1

i ) + un−1i+1 ) +

M−1∑j=1

un−1j Gij +

M−1∑j=1

un−1j Dij

+τ

2

∫ xi+1

2

xi− 1

2

[f(x, tn) + f(x, tn−1)

]dx, (B.14)

where Gij = G1,ij −G2,ij , Dij = D1,ij −D2,ij and

G1,ij =S∑k=1

ak[dβkφj(xi+ 1

2)

dxβk−dβkφj(xi− 1

2)

dxβk

],

G2,ij =S∑k=1

ak[dβkφj(xi+ 1

2)

d(−x)βk−dβkφj(xi− 1

2)

d(−x)βk

],

D1,ij =

S∑l=1

bl[I1−γl

0+φj(xi+ 1

2)− I1−γl

0+φj(xi− 1

2)],

D2,ij =S∑l=1

bl[I1−γl

1− φj(xi+ 12)− I1−γl

1− φj(xi− 12)].

For i = 1, 2, · · · ,M − 1, denote (Fn)i = τ2

∫ xi+12

xi− 1

2

[f(x, tn) + f(x, tn−1)

]dx and Aij =∫ xi+1

2xi− 1

2

φj(x)dx, Un =[un1 , u

n2 , · · · , unM−1

]T. Then, Eq.(B.14) can be expressed in matrix

form as

(A−G−D)Un = (A+G+D)Un−1 + Fn. (B.15)

The initial condition is discretized as ψi = ψ(ih) for i = 0, 1, 2, · · · ,M and U0 =[u0

1, u02, · · · , u0

M−1

]T= [ψ(h), ψ(2h), · · · , ψ((M − 1)h)]T .

B.3 Stability and convergence of the Crank-Nicolson scheme

Along the same lines as the proof of Lemma 1 in [77], we obtain the following two lemmas.

Chapter B 210

Lemma B.3.1 For 0 < βk = αk − 1 < 1, cki =(i− 3

2

)1−βk − 2(i− 1

2

)1−βk +(i+ 1

2

)1−βk ,

i = 2, 3, · · · , the following hold:

(1) cki is increasing monotonically as i increases, and cki < 0, i = 2, 3, · · · ;(2) lim

i→+∞cki = 0;

(3)+∞∑i=2

(cki+1 − cki ) = −ck2.

Lemma B.3.2 For 0 < γl < 1, dli =(i− 3

2

)2−γl − 2(i− 1

2

)2−γl +(i+ 1

2

)2−γl, i = 2, 3, · · · ,the following hold:

(1) dli is decreasing monotonically as i increases, and dli > 0, i = 2, 3, · · · ;(2) lim

i→+∞dli = 0;

(3)+∞∑i=2

(dli+1 − dli) = −dl2.

Also, following a proof similar to Theorem 1 in [77], we can obtain the following theorem.

Theorem B.3.1 For 0 < βk < 1, the coefficients Gij satisfy

|Gii| >M−1∑j=1,j 6=i

|Gij |, i = 1, 2, · · · ,M − 1,

i.e., G is strictly diagonally dominant.

Theorem B.3.2 For 0 < βk < 1, 12 < γl < 1, B = A − G − D, then B is also strictly

diagonally dominant and the spectral radius of B−1 fulfills

ρ(B−1) <2

h. (B.16)

Proof. Since ak > 0, bl < 0, from Lemma B.3.1 and Lemma B.3.2, we know Bij < 0 for

j > i+ 1 and j < i− 1. Note that(3

2

)1−βk− 3

(1

2

)1−βk=

(1

2

)1−βk(31−βk − 3) < 0, (B.17)(

3

2

)2−γl− 3

(1

2

)2−γl=

(1

2

)2−γl(32−γl − 3) > 0, (B.18)

which asserts Bii > 0. In addition,

3

(1

2

)1−βk−(

3

2

)1−βk+ ck2 =

(1

2

)1−βk(4− 3 · 31−βk + 51−βk), (B.19)

3

(1

2

)2−γl−(

3

2

)2−γl+ dl2 =

(1

2

)2−γl(4− 3 · 32−γl + 52−γl). (B.20)

211 Chapter B

Let g(x) = 4−3 ·3x+5x, we know that g(x) > 0 for x ∈ (0, 1) and g(x) < 0 for x ∈ (1, 32).

Thus, for 0 < βk < 1 and 12 < γl < 1, we have 3 · (1

2)1−βk − (32)1−βk + ck2 > 0 and

3 · (12)2−γl − (3

2)2−γl + dl2 < 0. Hence,

M−1∑j=1,j 6=i

|Bij | =i−2∑j=1

|Bij |+M−1∑j=i+2

|Bij |+ |Bi,i−1|+ |Bi,i+1|

<

S∑k=1

akΓ(2− βk)hβk

i−2∑j=−∞

(cki−j+1 − cki−j

)+

S∑l=1

blh1−γl

Γ(3− γl)

i−2∑j=−∞

(dli−j+1 − dli−j

)+

S∑k=1

akΓ(2− βk)hβk

+∞∑j=i+2

(ckj−i+1 − ckj−i

)+

S∑l=1

blh1−γl

Γ(3− γl)

+∞∑j=i+2

(dlj−i+1 − dlj−i

)+h

4

+S∑k=1

2akΓ(2− βk)hβk

[3(

1

2)1−βk − (

3

2)1−βk + ck2

]+

S∑l=1

2blh1−γl

Γ(3− γl)[3(

1

2)2−γl − (

3

2)2−γl + dl2

]=

S∑k=1

akΓ(2− βk)hβk

· (−2ck2) +

S∑l=1

blh1−γl

Γ(3− γl)· (−2dl2) +

h

4

+

S∑k=1

2akΓ(2− βk)hβk

[3(

1

2)1−βk − (

3

2)1−βk + ck2

]+

S∑l=1

2blh1−γl

Γ(3− γl)[3(

1

2)2−γl − (

3

2)2−γl + dl2

]=h

4+

S∑k=1

2ak[3(1

2)1−βk − (32)1−βk

]Γ(2− βk)hβk

+S∑l=1

2blh1−γl

[3(1

2)2−γl − (32)2−γl

]Γ(3− γl)

<3h

4+

S∑k=1

2ak[3(1

2)1−βk − (32)1−βk

]Γ(2− βk)hβk

+S∑l=1

2blh1−γl

[3(1

2)2−γl − (32)2−γl

]Γ(3− γl)

=Bii.

The proof of Eq.(B.16) follows part b of Theorem 2 in [77].

Since B = A−G−D is strictly diagonally dominant, then B is nonsingular and invertible.

The Crank-Nicolson scheme can be rewritten as

Un = B−1(A+G+D)Un−1 +B−1F. (B.21)

Theorem B.3.3 Define W = (λ− 1)A− (λ+ 1)(G+D), for 0 < βk < 1, 12 < γl < 1, if

λ > 1 or λ < −1, we can conclude that W is strictly diagonally dominant, i.e.,

|Wii| >M−1∑j=1,j 6=i

|Wij |, i = 1, 2, · · · ,M − 1.

Proof.

Chapter B 212

(1) If λ > 1, since ak > 0, bl < 0, (B.17) and (B.18) yield that Wii > 0. From Lemma

B.3.1 and Lemma B.3.2, it is obvious that Wii > 0 and Wij < 0 for j > i+1 and j < i−1.

Now, we focus on the sign of Wi,i−1 and Wi,i+1.

(i) If Wi,i−1 = Wi,i+1 = (λ− 1)h8 − (λ+ 1)

S∑k=1

akΓ(2−βk)hβk

[3(

12

)1−βk − (32

)1−βk + ck2]

+

S∑l=1

blh1−γl

Γ(3−γl)[3(1

2)2−γl − (32)2−γl + dl2

]< 0, then

M−1∑j=1,j 6=i

|Wij | =i−2∑j=1

|Wij |+M−1∑j=i+2

|Wij |+ |Wi,i−1|+ |Wi,i+1|

<(λ+ 1)

[ i−2∑j=−∞

S∑k=1

akΓ(2− βk)hβk

(cki−j+1 − cki−j

)+

i−2∑j=−∞

S∑l=1

blh1−γl

Γ(3− γl)

(dli−j+1 − dli−j

)

++∞∑j=i+2

S∑k=1

akΓ(2− βk)hβk

(ckj−i+1 − ckj−i

)+

+∞∑j=i+2

S∑l=1

blh1−γl

Γ(3− γl)

(dlj−i+1 − dlj−i

)]

−(λ− 1)h

4+ 2(λ+ 1)

S∑k=1

akΓ(2− βk)hβk

[3

(1

2

)1−βk−(

3

2

)1−βk+ ck2

]

+S∑l=1

blh1−γl

Γ(3− γl)

[3

(1

2

)2−γl−(

3

2

)2−γl+ dl2

]

<(λ− 1)3h

4+ 2(λ+ 1)

S∑k=1

akΓ(2− βk)hβk

[3

(1

2

)1−βk−(

3

2

)1−βk ]

+

S∑l=1

blh1−γl

Γ(3− γl)

[3

(1

2

)2−γl−(

3

2

)2−γl ]= Wii.

Therefore,

|Wii| >M−1∑j=1,j 6=i

|Wij |.

(ii) If Wi,i−1 = Wi,i+1 = (λ− 1)h8 − (λ+ 1)

S∑k=1

akΓ(2−βk)hβk

[3(

12

)1−βk − (32

)1−βk + ck2]

+

S∑l=1

blh1−γl

Γ(3−γl)[3(1

2)2−γl − (32)2−γl + dl2

]≥ 0, then

M−1∑j=1,j 6=i

|Wij | =i−2∑j=1

|Wij |+M−1∑j=i+2

|Wij |+ |Wi,i−1|+ |Wi,i+1|

<(λ+ 1)

[ i−2∑j=−∞

S∑k=1

akΓ(2− βk)hβk

(cki−j+1 − cki−j

)

+i−2∑

j=−∞

S∑l=1

blh1−γl

Γ(3− γl)

(dli−j+1 − dli−j

)+

+∞∑j=i+2

S∑k=1

akΓ(2− βk)hβk

(ckj−i+1 − ckj−i

)

213 Chapter B

++∞∑j=i+2

S∑l=1

blh1−γl

Γ(3− γl)

(dlj−i+1 − dlj−i

) ]+ (λ− 1)

h

4

−2(λ+ 1)

S∑k=1

akΓ(2− βk)hβk

[3

(1

2

)1−βk−(

3

2

)1−βk+ ck2

]

+S∑l=1

blh1−γl

Γ(3− γl)

[3

(1

2

)2−γl−(

3

2

)2−γl+ dl2

]

=(λ− 1)h

4− 2(λ+ 1)

S∑k=1

akΓ(2− βk)hβk

[5

(1

2

)1−βk− 5

(3

2

)1−βk+ 2

(5

2

)1−βk ]

+

S∑l=1

blh1−γl

Γ(3− γl)

[5

(1

2

)2−γl− 5

(3

2

)2−γl+ 2

(5

2

)2−γl ]

<(λ− 1)3h

4+ 2(λ+ 1)

S∑k=1

akΓ(2− βk)hβk

[3

(1

2

)1−βk−(

3

2

)1−βk ]

+

S∑l=1

blh1−γl

Γ(3− γl)

[3

(1

2

)2−γl−(

3

2

)2−γl ]= Wii

as

2(λ+ 1)

S∑k=1

akΓ(2− βk)hβk

[8

(1

2

)1−βk− 6

(3

2

)1−βk+ 2

(5

2

)1−βk ]

+S∑l=1

blh1−γl

Γ(3− γl)

[8

(1

2

)2−γl− 6

(3

2

)2−γl+ 2

(5

2

)2−γl ]

=4(λ+ 1)

S∑k=1

akΓ(2− βk)hβk

(1

2

)1−βk [4− 3 · 31−βk + 51−βk

]

+

S∑l=1

blh1−γl

Γ(3− γl)

(1

2

)2−γl [4− 3 · 32−γl + 52−γl

]>0 > −(λ− 1)

h

2.

Henceforth, |Wii| >M−1∑j=1,j 6=i

|Wij |.

(2) If λ < −1, it can be easily seen that Wii < 0 and Wij > 0 for j > i+ 1 and j < i− 1.

Along the same line of the proof for the case λ > 1, we also can obtain |Wii| >M−1∑j=1,j 6=i

|Wij |.

Thus, the proof is complete.

Theorem B.3.4 The spectral radius of B−1(A+G+D) satisfies ρ(B−1(A+G+D)) < 1,

hence the Crank-Nicolson scheme (B.21) is unconditionally stable.

Chapter B 214

Proof. Since B,A,G and D are symmetric positive definite, it is straightforward to

conclude that B−1 is symmetric positive definite and

B−1(A+G+D) = B−12

(B−

12 (A+G+D)B−

12

)B

12 ,

which means that B−1(A+G+D) ∼ B−12 (A+G+D)B−

12 . Thus, B−1(A+G+D) and

B−12 (A+G+D)B−

12 have the same eigenvalues. The symmetric positive definiteness of

A,G,D and B gives that B−12 (A+G+D)B−

12 is symmetric positive definite. Thus, all

the eigenvalues of B−12 (A+G+D)B−

12 are real and so are B−1(A+G+D). Suppose λ is

an eigenvalue of B−1(A+G+D), and det(λI −B−1(A+G+D))= det(B−1) · det(λB −(A+G+D)) = 0. Note that if B is nonsingular and invertible, then det(B−1) 6= 0, thus

det(λB − (A+G+D)) = 0. Let W = λB − (A+G+D), then

W = (λ− 1)A− (λ+ 1)(G+D)

(i) If λ = ±1 or 0, it is obvious that W is diagonally dominant;

(ii) If λ > 1 or λ < −1, Theorem B.3.3 asserts that W is also diagonally dominant.

Therefore, det(W ) 6= 0 for all λ ≥ 1 or λ ≤ −1 or λ = 0. According to the above analysis,

as the roots of the equation det(λB− (A+G+D)) = 0 exist, λ must satisfy −1 < λ < 0

or 0 < λ < 1, which means the eigenvalue of B−1(A + G + D) satisfy |λ| < 1. Thus,

ρ(B−1(A+G+D)) < 1, which completes the proof.

By following the proof in Corollary 1 [77], it follows that

Lemma B.3.3 For 0 < βk < 1, 12 < γl < 1, if ∂βk+1u(x,t)

∂xβk+1 ∈ L1(R), then

∂βku(x, tn)

∂xβk

∣∣∣xi+12

xi− 1

2

=M−1∑j=1

unj

(dβkφj(xi+ 1

2)

dxβk−dβkφj(xi− 1

2)

dxβk

)+O(h3), (B.22)

∂βku(x, tn)

∂(−x)βk

∣∣∣xi+12

xi− 1

2

=M−1∑j=1

unj

(dβkφj(xi+ 1

2)

d(−x)βk−dβkφj(xi− 1

2)

d(−x)βk

)+O(h3), (B.23)

I1−γl0+

u(x, tn)∣∣∣xi+1

2

xi− 1

2

=M−1∑j=1

unj

(I1−γl

0+φj(xi+ 1

2)− I1−γl

0+φj(xi− 1

2))

+O(h3), (B.24)

I1−γl1− u(x, tn)

∣∣∣xi+12

xi− 1

2

=

M−1∑j=1

unj

(I1−γl

1− φj(xi+ 12)− I1−γl

1− φj(xi− 12))

+O(h3). (B.25)

Theorem B.3.5 Let un be the exact solution of the problem (B.3). Then the numerical

solution Un unconditionally converges to the exact solution un as h, τ and σ, % tend to

zero. Moreover,

‖un − Un‖ ≤ C(σ2 + %2 + τ2 + h2),

where un = [u(x1, tn), u(x2, tn), · · · , u(xM−1, tn)]T and Un = [un1 , un2 , · · · , unM−1]T .

215 Chapter B

Proof. Let eni denote the error at the point (xi, tn). Substituting eni = u(xi, tn)−uni into

(B.14) and combining (B.12) with Eqs.(B.22)-(B.25) yields that

h

8(eni−1 + 6eni + eni+1)−

M−1∑j=1

enjGij −M−1∑j=1

enjDij

=h

8(en−1i−1 + 6en−1

i + en−1i+1 ) +

M−1∑j=1

en−1j Gij +

M−1∑j=1

en−1j Dij +O(τh(σ2 + %2 + τ2 + h2)).

Note that en0 = enM = 0 and e0i = 0 for i = 1, 2, · · · ,M − 1. Thus,

BEn = (A+G+D)En−1 +O(τh(σ2 + %2 + τ2 + h2))χ,

here χ = [1, 1, · · · , 1]T , En = [en1 , en2 , · · · , enM−1]T . Setting Q = B−1(A + G + D) and

b = O(τh(σ2 + %2 + τ2 + h2))B−1, by iteration, one has

En =(Qn−1 +Qn−2 + · · ·+ I

)b.

According to Theorem B.3.2, we know that ρ(B−1

)< 2

h and ρ(Q) < 1. Hence, there

exists a vector norm and induced matrix norm ‖·‖ such that ‖Q‖ < 1 and ‖B−1‖ < Ch−1.

Then upon taking norms, we have

‖En‖ ≤(‖Qn−1‖+ ‖Qn−2‖+ · · ·+ 1

)‖b‖ ≤ n‖b‖ ≤ O(σ2 + %2 + τ2 + h2).

Therefore, ‖En‖ ≤ C(σ2 + %2 + τ2 + h2).

B.4 Numerical examples

In order to illustrate the behavior of our numerical method and demonstrate the effec-

tiveness of our theoretical analysis, some examples are given. In the first example, an

exact solution is available to assess the accuracy of our numerical scheme. Example B.4.2

highlights the potential of our method to provide a solution of challenging distributed-

order diffusion problems and Example B.4.3 is used to further the understanding of a

distributed-order diffusion equation.

Example B.4.1 Firstly, we consider the following distributed-order equation [254]:

∂u(x, t)

∂t=

∫ 2

1P (α)

∂αu(x, t)

∂|x|αdα+

∫ 1

0Q(γ)

∂γu(x, t)

∂|x|γdγ + f(x, t), (x, t) ∈ (0, 1)× (0, 1],

with boundary condition

u(0, t) = 0, u(1, t) = 0, t ∈ [0, 1]

and initial condition

u(x, 0) = x2(1− x)2, x ∈ [0, 1],

Chapter B 216

where

P (α) = −2Γ(5− α) cos(πα

2

), Q(γ) =

0, 0 < γ ≤ 1

2 ,

2Γ(5− γ) cos(πγ

2

), 1

2 < γ < 1.

f(x, t) = etx2(1− x)2 −∫ 2

1P (α)

∂αu(x, t)

∂|x|αdα−

∫ 1

0Q(γ)

∂γu(x, t)

∂|x|γdγ

= etx2(1− x)2 − et[R1(x) +R1(1− x)−R2(x)−R2(1− x)],

where,

R1(x) =Γ(5)

lnx(x3 − x2)− 2Γ(4)

[1

lnx(3x2 − 2x)− 1

(lnx)2(x2 − x)

]+

Γ(3)

lnx

[6x− 2− 5x

lnx+

3

lnx+

2x

(lnx)2− 2

(lnx)2

],

R2(x) =Γ(5)

lnx

(x

72 − x3

)− 2Γ(4)

[1

lnx

(7

2x

52 − 3x2

)− 1

(lnx)2

(x

52 − x2

)]+

Γ(3)

lnx

[35

4x

32 − 6x− 1

lnx

(6x

32 − 5x− 2x

32

lnx+

2x

lnx

)].

The exact solution is u(x, t) = etx2(1 − x)2. Figure B.1 exhibits a comparison of the

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Numerical solutionExact solution

Figure B.1: Exact solution and numerical solution with σ = % = τ = h = 1/80 at t = 1.0

exact and numerical solutions for this example. We can see that the numerical solution

is in excellent agreement with the analytical solution. Table B.1 shows the error and

convergence orders for our method with respect to τ and h. For different σ, % (σ = % =

1/40 and σ = % = 1/80), with decreasing τ = h, the convergence orders of τ and h reach

second order. Table B.2 shows the error and convergence orders with respect to σ and

%. For different τ and h (τ = h = 1/100 and τ = h = 1/200), with decreasing σ = %,

we observe that the convergence orders of σ and % are also second order. According to

the errors and convergence rates in the first two tables, the finite volume method for the

Riesz-space distributed-order equations is effective and stable as expected. In Table B.3,

we present a comparison of the errors and convergence orders between our finite volume

217 Chapter B

method and the finite difference method in [254]. Compared to [254], our numerical error

is much smaller. Therefore, our method is more effective for the one-dimensional case.

However, for higher dimensional problems, an alternating direction method was applied

in [254] to reduce the CPU computation time, which is a key advantage.

Table B.1: The errors and convergence orders with respect to τ and h

σ = % = 1/40 σ = % = 1/80

τ = h error order error order

1/8 2.9352E-03 – 2.9370E-03 –1/16 7.5119E-04 1.97 7.5301E-04 1.961/32 1.8832E-04 2.00 1.9014E-04 1.991/64 4.5576E-05 2.05 4.7381E-05 2.001/128 9.6818E-06 2.23 1.1439E-05 2.05

Table B.2: The errors and convergence orders with respect to σ and %

τ = h = 1/200 τ = h = 1/400

σ = % error order error order1/2 9.9864E-04 – 1.0031E-03 –1/4 2.4910E-04 2.00 2.5273E-04 1.991/8 5.9049E-05 2.08 6.2526E-05 2.021/16 1.1457E-05 2.37 1.4798E-05 2.081/32 1.8889E-06 2.60 2.8853E-06 2.36

Table B.3: The errors and convergence orders comparison of FVM and FDM in [254] forσ = % = 1/1000 at t = 1

FVM FDM

τ = h error order error order1/20 4.8471E-04 – 3.592E-02 –1/40 1.2238E-04 1.986 8.916E-03 2.0101/80 3.0745E-05 1.993 2.204E-03 2.0161/160 7.7034E-06 1.997 5.388E-04 2.032

Example B.4.2 Next, we consider the following distributed-order equation [37, 231]:

∂u(x, t)

∂t=

∫ 2

0P (α)

∂αu(x, t)

∂|x|αdα. (B.26)

According to [37, 231], if u(x, 0) = δ(x) and P (α) = Klα−2[A1δ(α− α1) + A2δ(α− α2)],

then the solution of Eq.(B.26) can be expressed as a convolution of two stable PDFs,

u(x, t) = a− 1α1

1 a− 1α2

2 t− 1α1− 1α2

∫ +∞

−∞Lα1,1

(x− x′

(a1t)1α1

)Lα2,1

(x′

(a2t)1α2

)dx′, (B.27)

Chapter B 218

where a1 = A1K/l2−α1 , a2 = A2K/l

2−α2 and Lα,1 is the PDF of a symmetric Levy stable

law given by its characteristic function

Lα,1(ξ) = exp(−|ξ|α).

As the exact solution involves convolution and an inverse Fourier transform, it is challeng-

ing to observe the behavior of u(x, t) directly from Eq.(B.27). Therefore, the numerical

solution proposed here becomes a promising tool. Without loss of generality, here we

consider the numerical solution of Eq.(B.26) with initial condition u(x, 0) = δ(x − 0.5),

x ∈ (0, 1) and K = 2, l = 3, A1 = A2 = 1. To give the error estimate, here we have chosen

the numerical solution u(x, tn) =∑m−1

i=1 uni φi(x) on a fine grid (h = 1/500) as the ‘target’

exact solution. Then, we adopt a set of points to calculate the discrete L2 error on the

coarse grids, which is given in the Table B.4. We can see that second order convergence is

attained, which again shows the stability and reliability of our method. Now we observe

the diffusion behaviour of u(x, t). Figure B.2 displays the evolution of u(x, t) at differ-

ent times t = 1, 5, 10, 20, which decays with increasing time. Figure B.3 and Figure B.4

illustrate the impact of α1 and α2 on the diffusion behaviour of u(x, t). We can observe

that with increasing α1 or α2, u(x, t) decays and the effect of α2 on the diffusion is more

significant. Although we give the numerical scheme of Eq.(B.26), exactly how to apply

the scheme to solve the actual problem and establish the connection between the kinetics

equation and multifractality still needs further investigation.

Table B.4: The errors and the convergence orders with σ = % = 1/100, α1 = 0.955,α2 = 1.255 at t = 1

τ = h error order

1/160 1.0414E-02 –1/200 7.3644E-03 1.551/320 2.7608E-03 2.09

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.01

0.02

0.03

0.04

0.05

0.06

t=1t=5t=10t=20

Figure B.2: Numerical solution profile of u(x, t) at different t with σ = % = 1/100,τ = h = 1/200, α1 = 0.955, α2 = 1.255

219 Chapter B

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

α1=0.555

α1=0.755

α1=0.855

α1=0.955

Figure B.3: Numerical solution profile of u(x, t) for different α1 with σ = % = 1/100,τ = h = 1/200, α2 = 1.255 at t = 1

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(x,t)

0

0.01

0.02

0.03

0.04

0.05

0.06

α2=1.255

α2=1.555

α2=1.755

α2=1.955

Figure B.4: Numerical solution profile of u(x, t) for different α2 with σ = % = 1/100,τ = h = 1/200, α1 = 0.955 at t = 1

Example B.4.3 Finally, we consider the following distributed-order equation:

∂u(x, t)

∂t=

∫ 2

1P (α)

∂αu(x, t)

∂|x|αdα+

∫ 1

0Q(γ)

∂γu(x, t)

∂|x|γdγ, (x, t) ∈ (0, 1)× (0, 1],

with boundary condition

u(0, t) = 0, u(1, t) = 0, t ∈ [0, 1],

and initial condition

u(x, 0) = sin(πx) x ∈ [0, 1].

Here we choose

P (α) = lα−2KA1δ(α− α1), Q(γ) = lγ−2KA2δ(γ − γ1),

Chapter B 220

σ = % = τ = h = 1/100, l = 3, K = 1, A1 = 8 and A2 = 2. Firstly, in FigureB.5 we exhibit the diffusion behavior of u(x, t) at the different times T = 0.5, 0.7, 1.0with α1 = 1.255, γ1 = 0.755, which decays with increasing time. Next we consider thediffusion behavior of u(x, t) by choosing a distinct α1 at T = 1.0 with γ1 = 0.755, whichis shown in Figure B.6. Similarly, in Figure B.7, we observe the diffusion behavior ofu(x, t) by adopting a distinct γ1 at T = 1.0 with a magnified view of the plot for theregion [0.45, 0.55] × [0.852, 0.875]. We can similarly see that with increasing α1 or γ1,u(x, t) decays and the effect of α1 on the diffusion is more significant. Finally, we exhibita comparison of the diffusive behavior of u(x, t) by selecting different P (α) or Q(γ) atT = 0.5 in Figure B.8 and Figure B.9 at time T = 0.5, respectively, in which P0(α) =8·3α−2δ(α−1.255), P1(α) = 1

α , P2(α) = 1α2 , P3(α) = 3α−2 and Q(γ) = 2·3γ−2δ(γ−0.755)

in Figure B.8, while Q0(γ) = 2 · 3γ−2δ(γ− 0.755), Q1(γ) = 1γ , Q2(γ) = 1

γ2, Q3(γ) = 3γ−2

and P (α) = 8 · 3α−2δ(α− 1.255) in Figure B.9. We can conclude that the both P (α) andQ(γ) have effects on the diffusion behavior of u(x, t).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u(x,

t)

T=0.5T=0.7T=1.0

Figure B.5: Numerical solution at different times

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u(x,

t)

α1=1.255

α1=1.455

α1=1.655

α1=1.855

Figure B.6: Numerical solution for different α1

221 Chapter B

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u(x,t)

γ1=0.555

γ1=0.755

γ1=0.955

0.45 0.5 0.55

0.855

0.86

0.865

0.87

0.875

Figure B.7: Numerical solution for different γ1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u(x,

t)

P0(α)

P1(α)

P2(α)

P3(α)

Figure B.8: Numerical solution for different P (α)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u(x,

t)

Q0(γ)

Q1(γ)

Q2(γ)

Q3(γ)

Figure B.9: Numerical solution for different Q(γ)

Chapter 9 222

B.5 Conclusions

In this chapter, we have investigated a second order in both space and time numerical

scheme for the Riesz space distributed-order advection-diffusion equation. We prove that

the Crank-Nicolson scheme is unconditionally stable and convergent with second order

accuracy O(σ2 + %2 + τ2 + h2). Three numerical examples are presented to show the

effectiveness of our computational method. In the future, we would like to develop the

finite volume method to solve time distributed-order and time-space distributed-order

advection-diffusion equations in one- and two-dimensional space. Moreover, we shall

consider other computational methods to improve the convergence rate of our method.

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