approximation of solution of time fractional order...

Punjab UniversityJournal of Mathematics (ISSN 1016-2526)Vol.47(1)(2015) pp. 35-56

Approximation of Solution of Time Fractional Order Three-DimensionalHeat Conduction Problems with Jacobi Polynomials

Hammad KhalilDepartment of Mathematics,

University of Malakand , KPK, Pakistanemail: [email protected]

Rahmat Ali KhanDepartment of Mathematics,

University of Malakand , KPK, Pakistanemail: [email protected]

Mohammed H. Al-SmadiApplied Science Department, Ajloun College

Al- Balqa Applied University, Ajloun 26826, Jordanemail: [email protected]

Asad A. FreihatPioneer Center for Gifted Students,

Ministry of Education, Jerash 26110 Jordanemail: [email protected]

Received: 02 January, 2015 / Accepted: 13 February, 2015 / Published online: 25 February,2015

Abstract. In this paper, we extend the idea of pseudo spectral methodto approximate solution of time fractional order three-dimensional heatconduction equations on a cubic domain. We study shifted Jacobi poly-nomials and provide a simple scheme to approximate function of multivariables in terms of these polynomials. We develop new operational ma-trices for arbitrary order integrations as well as for arbitrary order deriv-atives. Based on these new matrices, we develop simple technique toobtain numerical solution of fractional order heat conduction equations.The new scheme is simple and can be easily simulated with any compu-tational software. We develop codes for our results using MatLab. Theresults are displayed graphically.

AMS (MOS) Subject Classification Codes: 65N35; 65M70; 35C11Key Words: Time fractional heat conduction equations; Jacobi polynomials; Operational

matrices; Numerical simulations; Approximation theory;35

36 H. Khalil

1. INTRODUCTION

In literature, diffusion equations play basic and important role in mathematical model-ing of large variety of engineering problems. Some of the important problems in whichdiffusion equation plays a basic role are: cyclic heating of the cylindrical surface of in-ternal combustion engines, heating and cooling of building structures, heating lakes andwater reservoirs by radiation, the heating of solid surfaces in materials processing, thecyclic heating of laminated steel during pickling, heating and cooling of vials containedDNA for polimerise-chain-reaction activation, the heating of electronics and many moresee for example [12, 11, 1, 35, 3, 25]. In this study we consider the following generalizedtime fractional heat conduction problem

χt∂σu(t, x, y, z)

∂tσ= λx

∂2u(t, x, y, z)∂x2

+ λy∂2u(t, x, y, z)

∂y2+

λz∂2u(t, x, y, z)

∂z2+ I(t, x, y, z), u(0, x, y, z) = f(x, y, z),

(1. 1)

whereχt is the volumetric heat capacity andλx, λy andλz are the thermal conductivitiesin thex, y andz directions,0 < σ ≤ 1 is the order of the derivative,t ∈ [0, τ ], x ∈ [0, a],y ∈ [0, b] andz ∈ [0, c]. I(t, x, y, z) is the internal source term andf(x, y, z) is the initialheat distribution in the space.

Exact analytical solution of time fractional order diffusion equations are generally verydifficult to obtain. The reason behind this difficulty is the higher computational complex-ities of fractional calculus involved in solving diffusion equations. This phenomena isrecently been reported by many authors. We refer to Poulikakos [28], Arpaci [4], O zisik[26], Kakac and Yener [15], and Carslaw and Jaeger [5] for some of the renowned resultsin this aspect. Different aspects of solution of the problem such as existence and unique-ness of positive solutions, analytical properties of solution and the correctness of initialand boundary conditions have already been studied by many authors. We refer to study[38, 34, 22, 23, 13, 14, 9, 21].

In the literature, many attempts were made to approximate solution of fractional diffu-sion equations. V. V. Kulish [20] provide a very efficient way for approximate solution ofsuch problems using laplace transform. M. Akbarzade [2] studied approximate solutionof integer order three dimensional transient state heat conduction equation by Homotopyanalysis method. However, for fractional order equations, the method used in [2] will re-sults very complex algorithms. Ting-Hui Ning [24] provided some results based on spheri-cal coordinates and the method of separation of variables for approximate solution of suchtype of problems. Y. Z. Povstenko [29] provided axisymmetric solutions to time-fractionalheat conduction equation in a half-space subject to Robin boundary conditions by the useof integral transform method. Recently G.C. Wu [36, 37] studied approximate solution tofractional diffusion equation by variational iteration technique.

One of the most powerful method for numerical solutions of differential equations is thewell known spectral method. This method has already been extensively used for numericalsolutions of fractional order differential equations and partial differential equations withdifferent types of boundary conditions, see for example [32, 27, 30, 31]. However, nogeneralized version of this method is available in the literature which can be used to dealwith higher dimensional problems.

We provide generalized version of the method to find numerical solutions of higherdimensional fractional order partial differential equations. The methods is based on oper-ational matrices of integrations and differentiations. Operational matrices in case of single

Approximate Solution of Three Dimensional Heat Equation 37

variable are available for different orthogonal polynomials such as Sine-Cosine, Legendre,Jacobi, Lagurree and hermite polynomials, we refer to [33, 6, 7, 10, 8, 16, 17, 18].

In this paper, we use two parametric shifted Jacobi polynomials and generalize theoperational matrices of fractional order integrations and differentiations. We use theseoperational matrices to reduce the differential equation under consideration to a system ofeasily solvable algebraic equations.

The rest of the paper is organized as follow, in section2 we provide some basic prop-erties of fractional calculus and orthogonal polynomials and some relations for approx-imation of multivariate function. In section3, we develop new operational matrices ofintegrations and differentiations, in section4 these operational matrices are used to convertthe corresponding differential equation to a system of algebraic equations. In section5 theproposed algorithms are applied to several test problems and finally in section6 a shortconclusion is made.

2. PRELIMINARIES

For convenience, this section summarizes some concepts, definitions and basic resultsfrom fractional calculus in the sequel.

DEFINITION 1. Given an interval[0, a] ⊂ R, the Riemann-Liouville fractional orderintegral of orderσ ∈ R+ of a functionφ ∈ (L1[0, a],R) is defined by

Iσ0+φ(t) =

1Γ (σ)

∫ t

0

(t− s)σ−1φ(s)ds,

provided that the integral on right hand side exists.

DEFINITION 2. For a given functionφ(t) ∈ Cn[0, a], the Caputo fractional orderderivative of orderσ is defined as

Dσφ(t) =1

Γ(n− σ)

∫ x

0

φ(n)(t)(x− t)σ+1−n

dt, n− 1 ≤ σ < n , n ∈ N,

provided that the right side is pointwise defined on(0,∞), wheren = [σ] + 1.

Hence, it follows that

Dσtk =Γ(1 + k)

Γ(1 + k − σ)tk−σ, Iσtk =

Γ(1 + k)Γ(1 + k + σ)

tk+σ and DσC = 0, for a constantC.

(2. 2)

2.1. The shifted Jacobi polynomials: [19] The well known two parametric Jacobi poly-nomials defined on[0, τ ] with parameterξ, ζ is given by the following relation

J(ξ,ζ)(τ,i) (t) =

i∑

l=0

fτ(l,i)t

l, i = 0, 1, 2, 3..., (2. 3)

wherefτ(l,i) is defined by

fτ(l,i) =

(−1)i−lΓ(i + ζ + 1)Γ(i + l + ξ + ζ + 1)Γ(l + ζ + 1)Γ(i + ξ + ζ + 1)(i− l)!l!τ l

. (2. 4)

These polynomials are orthogonal with respect to the weight function

ω(ξ,ζ)τ (t) = (τ − t)ξtζ .

38 H. Khalil

The orthogonality condition of these polynomials are given as under∫ τ

0

ω(ξ,ζ)τ (t)J (ξ,ζ)

(τ,i) (t)J (ξ,ζ)(τ,j) (t) = γ

(ξ,ζ)(τ,j)δi,j , (2. 5)

whereδi,j is the Kroncker delta andγ(ξ,ζ)(τ,j) is defined as

γ(ξ,ζ)(τ,j) =

τ ξ+ζ+1Γ(j + ξ + 1)Γ(j + ζ + 1)(2j + ξ + ζ + 1)Γ(j + 1)Γ(j + ξ + ζ + 1)

. (2. 6)

The orthogonality relation allows us to approximateu(t) ∈ C([0, τ ] in the form of Jacobiseries as follows

u(t) =∞∑

i=0

ciJ(ξ,ζ)(τ,i) (t), (2. 7)

whereci can be easily calculated by using the orthogonality relation, that is,

ci =1

γ(ξ,ζ)(τ,j)

∫ τ

0

u(t)ω(ξ,ζ)τ (t)J (ξ,ζ)

(τ,i) (t)dt.

It is clear from Lemma2.2.1 in [19] that the coefficientsci decay faster. In practice, we areconcerned with the truncated series of (2. 7 ). Them terms truncated series can be writtenin vector form as

u(t) ' KTMΛM (t), (2. 8)

whereM = m + 1, KM is the coefficient column vector andΛM (t) is M terms columnvector function defined by

ΛM (t) =[

J(ξ,ζ)(τ,0) (t) J

(ξ,ζ)(τ,1) (t) · · · J

(ξ,ζ)(τ,i) (t) · · · J

(ξ,ζ)(τ,m)(t)

]T

. (2. 9)

2.2. Error Estimate. For sufficiently smooth functionu(t) ∈ ∆, where∆ = [a, b], themaximum amount of error in the approximation of a function withm terms of Jacobipolynomials is given as

‖g(x)− g(M)(x)‖2 ≤ (C11

MM+1), (2. 10)

where

C1 =14

maxt∈[0,τ ]

| ∂M+1

∂tM+1u(x)|. (2. 11)

For the proof of this relation, we refer the reader to study [19]. In our current paper, we areconcerned with four dimensional problems. Therefore, we must first establish a suitableapproximation method of a function of four variable with Jacobi polynomials. Using thesame procedure as developed in [16], we extend the notion to three-dimensional space anddefine three-dimensional Jacobi polynomials of orderM on the domain[0, a]×[0, b]×[0, c]as a product function of three Jacobi polynomials

J(a,b,c)(q,r,s) = J

(ξ,ζ)(a,q) (x)J (ξ,ζ)

(b,r) (y)J (ξ,ζ)(c,s) (z). (2. 12)

The orthogonality condition ofJ (a,b,c)(q,r,s) is found to be

∫ c

0

∫ b

0

∫ a

0

J(a,b,c)(q,r,s) J

(a,b,c)(q′,r′,s′)ω

(ξ,ζ)a,b,cdxdydz

= δ(q,q′)δ(r,r′)δ(s,s′)γ(ξ,ζ)(a,q)γ

(ξ,ζ)(b,r) γ

(ξ,ζ)(c,s) ,


whereω(ξ,ζ)a,b,c = ω

(ξ,ζ)a (x)ω(ξ,ζ)

b (y)ω(ξ,ζ)c (z) is the weight function regarding three dimen-

sional Jacobi polynomials. Hence, anyu(x, y, z) ∈ C([0, a]× [0, b]× [0, c]) can be easilyapproximated with three dimensional Jacobi polynomialsJ

(a,b,c)(q,r,s) as follows

u(x, y, z) =m∑

q=0

m∑r=0

m∑s=0

cqrsJ(a,b,c)(q,r,s) , (2. 13)

wherecqrs can be obtained by using the following relation

cqrs =1

γ(ξ,ζ)(a,q)γ

(ξ,ζ)(b,r) γ

(ξ,ζ)(c,s)

∫ c

0

∫ b

0

∫ a

0

u(x, y, z)ω(ξ,ζ)a,b,cJ

(a,b,c)(q,r,s) dxdydz. (2. 14)

For simplicity, use the notationcn = cqrs wheren = M2q + Mr + s + 1, and rewrite(2. 13 ) in vector notation, as follows

u(x, y, z) =M3∑n=1

cnJ(a,b,c)(n) (x, y, z) = CT

M3Λ(a,b,c)(x, y, z).

WhereCM3 is coefficient column vector of orderM3 andΛ(a,b,c)(x, y, z) is column vectorof functions defined by

Λ(a,b,c)(x, y, z) =[

J(a,b,c)(1) J

(a,b,c)(2) · · · J (a,b,c)

(n) · · · J(a,b,c)(M3)

]T

. (2. 15)

2.3. Four-dimensional Jacobi polynomials.Now, extend the idea to four dimensionalspace defined on domain[0, τ ] × [0, a] × [0, b] × [0, c] by the product function of Jacobipolynomials of orderM as

J(τ,a,b,c)(p,q,r,s) = J

(ξ,ζ)(τ,p) (t)J

(ξ,ζ)(a,q) (x)J (ξ,ζ)

(b,r) (y)J (ξ,ζ)(c,s) (z). (2. 16)

The orthogonality condition ofJ (τ,a,b,c)(p,q,r,s) is found to be

∫ c

0

∫ b

0

∫ a

0

∫ τ

0

J(τ,a,b,c)(p,q,r,s) J

(τ,a,b,c)(p′,q′,r′,s′)ω

(ξ,ζ)τ,a,b,cdtdxdydz

= δ(p,p′)δ(q,q′)δ(r,r′)δ(s,s′)γ(ξ,ζ)(τ,p)γ

(ξ,ζ)(a,q)γ

(ξ,ζ)(b,r) γ

(ξ,ζ)(c,s) ,

whereω(ξ,ζ)τ,a,b,c = ω

(ξ,ζ)τ (t)ω(ξ,ζ)

a (x)ω(ξ,ζ)b (y)ω(ξ,ζ)

c (z) is the weight function regardingfour-dimensional Jacobi polynomials. Anyu(t, x, y, z) ∈ C([0, τ ]× [0, a]× [0, b]× [0, c])can be easily approximated with four dimensional Jacobi polynomialsJ

(τ,a,b,c)(p,q,r,s) as follows

u(t, x, y, z) =m∑

p=0

m∑q=0

m∑r=0

m∑s=0

dpqrsJ(τ,a,b,c)(p,q,r,s) , (2. 17)

wheredpqrs can be obtained by the relation

dpqrs =1

γ(ξ,ζ)(τ,p)γ

(ξ,ζ)(a,q)γ

(ξ,ζ)(b,r) γ

(ξ,ζ)(c,s)

∫ c

0

∫ b

0

∫ a

0

∫ τ

0

u(t, x, y, z)ω(ξ,ζ)τ,a,b,cJ

(τ,a,b,c)(p,q,r,s) dtdxdydz.

40 H. Khalil

For simplicity, we use the notationdp,n = dpqrs wheren = M2q + Mr + s + 1, andrewrite (2. 17 ) as follows

u(t, x, y, z) =m∑

p=0

M3∑n=1

dp,nJτp (t)J (a,b,c)

n (x, y, z)

= ΛTM (t)DM×M3Λ(a,b,c)(x, y, z),

whereΛTM (t) is the function vector related to variablet and is as defined in (2. 9 ) and

Λ(a,b,c)(x, y, z) is the function vector related to variablex, y, z and is defined in (2. 15 ).

3. OPERATIONAL MATRICES OFINTEGRATION AND DIFFERENTIATIONS

The operational matrices of derivatives and integrations play the role of building blocksin the establishment of the new pseudo spectral method. In literature, these are used onlyfor solutions of differential equations including fractional differential with only one vari-able and partial differential equations with two and three variables [16]. Here we constructnew operational matrices for three variables and use them to convert a generalized class ofPDEs with four variable to a system of easily solvable algebraic equations.

LEMMA 3.1. Let Λ(a,b,c)(x, y, z) be the function vector as defined in(2. 15 ), then thefractional order partial derivative of orderσ of Λ(a,b,c)(x, y, z) w.r.t x is given by

∂σ

∂xσΛ(a,b,c)(x, y, z) = xA

(σ,a,b,c)M3×M3Λ(a,b,c)(x, y, z),

wherexA(σ,a,b,c)M3×M3 is the operational matrix of differentiation of orderσ, and is defined as

xA(σ,a,b,c)M3×M3 =

Ω1,1 Ω1,2 · · · Ω1,n′ · · · Ω1,M3

Ω2,1 Ω2,2 · · · Ω2,n′ · · · Ω2,M3

......

......

......

Ωn,1 Ωn,2 · · · Ωn,n′ · · · Ωn,M3

......

......

......

ΩM3,1 ΩM3,2 · · · ΩM3,n′ · · · ΩM3,M3

, (3. 18)

wheren′ = M2h + Mi + j + 1, n = M2q + Mr + s + 1,

i, j, k, q, r, s = 0, 1, 2, ...,m,

and

Ωn,n′ = hnn′ = hqrs

ijk =q∑

l=dσehqrs

ijk

︷︸︸︷fa

(l,q), (3. 19)

hqrsijk =

δ(j,r)δ(k,s)

γ(ξ,ζ)(a,i)

i∑

l′=0

fa(l′,i)

Γ(l′ + ζ + l − σ + 1)Γ(ξ + 1)a(l′+ζ+l−σ+ξ+1)

Γ(l′ + ζ + l − σ + ξ + 1),

︷︸︸︷fa

(l,q) = fa(l,q)

1+l1+l−σ , andfa

(.,.) is as defined in(2. 4 ).

Proof. Consider the general term of the function vector (2. 15 ). Then, in view of (2. 12 ),we can write

∂σ

∂xσJ

(a,b,c)(n) (x, y, z) =

∂σ

∂xσJ

(a,b,c)(q,r,s) , (3. 20)


wheren = M2q + Mr + s + 1. After expansion of the left side we get

∂σ

∂xσJ

(a,b,c)(n) (x, y, z) = J

(ξ,ζ)(b,r) (y)J (ξ,ζ)

(c,s) (z)∂σ

∂xσJ

(ξ,ζ)(a,q) (x). (3. 21)

Using (2. 2 ) and (2. 3 ), and after simplification, we obtain

∂σ

∂xσJ

(a,b,c)(n) (x, y, z) = J

(ξ,ζ)(b,r) (y)J (ξ,ζ)

(c,s) (z)q∑

l=dσefa

(l,q)

1 + l

1 + l − σxl−σ, (3. 22)

which can be written in the following form

∂σ

∂xσJ

(a,b,c)(n) (x, y, z) =

q∑

l=dσe

︷︸︸︷fa

(l,q) J(ξ,ζ)(b,r) (y)J (ξ,ζ)

(c,s) (z)xl−σ, (3. 23)

where︷︸︸︷fa

(l,q) = fa(l,q)

1+l1+l−σ . ApproximatingJ

(ξ,ζ)(b,r) (y)J (ξ,ζ)

(c,s) (z)xl−σ with three dimen-sional Jacobi polynomials as follows

J(ξ,ζ)(b,r) (y)J (ξ,ζ)

(c,s) (z)xl−σ =m∑

i=0

m∑

j=0

m∑

k=0

hqrsijkJ

(a,b,c)(i,j,k) . (3. 24)

The coefficientshqrsijk can be easily calculated by using relation (2. 14 ), that is,

hqrsijk =

1

γ(ξ,ζ)(a,i) γ

(ξ,ζ)(b,j) γ

(ξ,ζ)(c,k)

∫ c

0

∫ b

0

∫ a

0

ω(ξ,ζ)a,b,cJ

(ξ,ζ)(b,r) (y)J (ξ,ζ)

(c,s) (z)xl−σJ(a,b,c)(i,j,k) dxdydz,

which after simplification yields

hqrsijk =

δ(j,r)δ(k,s)

γ(ξ,ζ)(a,i)

∫ a

0

ω(ξ,ζ)a xl−σJ

(ξ,ζ)(a,i) (x)dx. (3. 25)

Now, we obtain

∫ a

0

ω(ξ,ζ)a xl−σJ

(ξ,ζ)(a,i) (x)dx =

i∑

l′=0

fa(l′,i)

∫ a

0

(a− x)ξxl′+ζ+l−σdx. (3. 26)

By the convolution theorem of Laplace transform, we have

£(∫ a

0

(a− x)ξxl′+ζ+l−σdx) =Γ(l′ + ζ + l − σ + 1)Γ(ξ + 1)

s(l′+ζ+l−σ+ξ+2),

and by taking the inverse Laplace transform, we obtain

∫ a

0

(a− x)ξxl′+ζ+l−σdx =Γ(l′ + ζ + l − σ + 1)Γ(ξ + 1)a(l′+ζ+l−σ+ξ+1)

Γ(l′ + ζ + l − σ + ξ + 1). (3. 27)

Using the equation (3. 27 ) and (3. 26 ) in (3. 25 ), we get

hqrsijk =

δ(j,r)δ(k,s)

γ(ξ,ζ)(a,i)

i∑

l′=0

fa(l′,i)

Γ(l′ + ζ + l − σ + 1)Γ(ξ + 1)a(l′+ζ+l−σ+ξ+1)

Γ(l′ + ζ + l − σ + ξ + 1). (3. 28)

42 H. Khalil

Also by using (3. 24 ) in (3. 23 ) we get

∂σ

∂xσJ

(a,b,c)(n) (x, y, z) =

m∑

i=0

m∑

j=0

m∑

k=0

q∑

l=dσehqrs

ijk

︷︸︸︷fa

(l,q) J(a,b,c)(i,j,k)

=m∑

i=0

m∑

j=0

m∑

k=0

hqrsijkJ

(a,b,c)(i,j,k) ,

(3. 29)

wherehqrsijk =

∑ql=dσe hqrs

ijk

︷︸︸︷fa

(l,q). Now using the notationn = M2q + Mr + s + 1,

n′ = M2i + Mj + k + 1, we have

∂σ

∂xσJ

(a,b,c)(n) (x, y, z) =

M3∑

n′=0

hnn′J

(a,b,c)(n′) (x, y, z). (3. 30)

which in view of the notationΩn,n′ = hnn′ for i, j, k, q, r, s = 0, 1, 2, 3, ..m, yields the

desired result. ¤

LEMMA 3.2. Let Λ(a,b,c)(x, y, z) be the function vector as defined in(2. 15 ), then thefractional order partial derivative of orderσ of Λ(a,b,c)(x, y, z) w.r.t y is given by

∂σ

∂yσΛ(a,b,c)(x, y, z) = yA

(σ,a,b,c)M3×M3Λ(a,b,c)(x, y, z), (3. 31)

whereyA(σ,a,b,c)M3×M3 is the operational matrix of differentiation of orderσ, and is given by

yA(σ,a,b,c)M3×M3 =

Ω1,1 Ω1,2 · · · Ω1,n′ · · · Ω1,M3

Ω2,1 Ω2,2 · · · Ω2,n′ · · · Ω2,M3

......

......

......

Ωn,1 Ωn,2 · · · Ωn,n′ · · · Ωn,M3

......

......

......

ΩM3,1 ΩM3,2 · · · ΩM3,n′ · · · ΩM3,M3

, (3. 32)

wheren′ = M2h + Mi + j + 1, n = M2q + Mr + s + 1,

i, j, k, q, r, s = 0, 1, 2, ...,m,

and


ijk =r∑

l=dσehqrs

ijk

︷︸︸︷fb

(l,r), (3. 33)

hqrsijk =

δ(i,q)δ(k,s)

γ(ξ,ζ)(b,j)

j∑

l′=0

fb(l′,j)

Γ(l′ + ζ + l − σ + 1)Γ(ξ + 1)b(l′+ζ+l−σ+ξ+1)

Γ(l′ + ζ + l − σ + ξ + 1),

︷︸︸︷fb

(l,r) = fb(l,r)

1+l1+l−σ , andfb


Proof. The proof of this lemma is similar to that of the above lemma. ¤

LEMMA 3.3. Let Λ(a,b,c)(x, y, z) be the function vector as defined in(2. 15 ), then thefractional order partial derivative of orderσ of Λ(a,b,c)(x, y, z) w.r.t z is given by

∂σ

∂zσΛ(a,b,c)(x, y, z) = zA

(σ,a,b,c)M3×M3Λ(a,b,c)(x, y, z), (3. 34)


wherezA(σ,a,b,c)M3×M3 is the operational matrix of differentiation of orderσ, and is defined by

zA(σ,a,b,c)M3×M3 =

Ω1,1 Ω1,2 · · · Ω1,n′ · · · Ω1,M3

Ω2,1 Ω2,2 · · · Ω2,n′ · · · Ω2,M3

......

......

......

Ωn,1 Ωn,2 · · · Ωn,n′ · · · Ωn,M3

......

......

......

ΩM3,1 ΩM3,2 · · · ΩM3,n′ · · · ΩM3,M3

, (3. 35)

where

n′ = M2h + Mi + j + 1, n = M2q + Mr + s + 1,

i, j, k, q, r, s = 0, 1, 2, ...,m,

and


ijk =s∑

l=dσehqrs

ijk

︷︸︸︷fc

(l,s), (3. 36)

hqrsijk =

δ(i,q)δ(j,r)

γ(ξ,ζ)(c,k)

k∑

l′=0

fc(l′,k)

Γ(l′ + ζ + l − σ + 1)Γ(ξ + 1)c(l′+ζ+l−σ+ξ+1)

Γ(l′ + ζ + l − σ + ξ + 1),

︷︸︸︷fc

(l,s) = fc(l,s)

1+l1+l−σ andfc


Proof. The proof of this lemma is similar to that of the above lemma. ¤

LEMMA 3.4. Let ΛM (t) be the function vector as defined in(2. 9 ) then theγ order inte-gration ofΛM (t) is given by

Iγ(ΛM (t)) ' Hη,γM×MΛM (t), (3. 37)

whereHη,γM×M is the operational matrix of integration of orderγ and is defined as

Hη,γM×M =

Θ0,0,k Θ0,1,k, · · · Θ0,j,k · · · Θ0,m,k

Θ1,0,k Θ1,1,k · · · Θ1,j,k · · · Θ1,m,j

......

......

......

Θi,0,k Θi,1,k · · · Θi,j,k · · · Θi,m,k

......

......

......

Θm,0,k Θm,1,k · · · Θm,j,k · · · Θm,m,k

(3. 38)

where

Θi,j,k =i∑

k=0

Λi,k,γSj , (3. 39)

Λi,k,γ =(−1)i−kΓ(i + ζ + 1)Γ(i + k + ξ + ζ + 1)Γ(1 + k)

Γ(k + ζ + 1)Γ(i + ξ + ζ + 1)(i− k)!k!Γ(1 + k + ξ)ηk, (3. 40)

and

44 H. Khalil

Sj =j∑

l=0

(−1)j−lΓ(j + l + ξ + ζ + 1)Γ(k + ξ + l + ζ + 1)Γ(l + ζ + 1)(j − l)!l!Γ(k + γ + l + ζ + ξ + 2)

× (2j + ξ + ζ + 1)Γ(j + 1)Γ(ξ + 1)ηγ

Γ(j + ξ + 1).

Proof. For the proof of this lemma we refer to [19]. ¤

4. APPLICATION OF THE OPERATIONAL MATRICES OF INTEGRATIONS AND

DERIVATIVES TO HEAT CONDUCTION PROBLEM

Consider the following heat conduction problem posed on a cubic region


∂tσ= λx

∂2u(t, x, y, z)∂x2

+ λy∂2u(t, x, y, z)

∂y2

+ λz∂2u(t, x, y, z)

∂z2+ I(t, x, y, z), u(0, x, y, z) = f(x, y, z),

(4. 41)

whereχt is the volumetric heat capacity(J/(m3K)), λx, λy, λz are thermal conductivities(W/m.K) in x, y andz direction respectively,0 < α ≤ 1, t ∈ [0, τ ], x ∈ [0, a], y ∈ [0, b]andz ∈ [0, c]. We seek solution of this problem in terms of shifted Jacobi polynomialssuch that the following relation holds

∂σu(t, x, y, z)∂tσ

= ΛM (t)T KM×M3Λ(a,b,c)(x, y, z). (4. 42)

By the application of fractional integration of orderσ with respect to variablet on equation(4. 42 ) and making use of Lemma 3.4, we have

Iσ ∂σu(t, x, y, z)∂tσ

= ΛM (t)T (Hτ,σM×M )T KM×M3Λ(a,b,c)(x, y, z),

which implies that

u(t, x, y, z)− c1 = ΛM (t)T (Hτ,σM×M )T KM×M3Λ(a,b,c)(x, y, z). (4. 43)

The initial condition yieldsc1 = u(0, x, y, z). Hence, we have

u(t, x, y, z) = ΛM (t)T (Hτ,σM×M )T KM×M3Λ(a,b,c)(x, y, z) + f(x, y, z), (4. 44)

which can be rewritten as

u(t, x, y, z) = ΛM (t)T (Hτ,σM×M )T KM×M3 + FM×M3Λ(a,b,c)(x, y, z). (4. 45)

Using this value ofu(t, x, y, z), we obtain

∂2u(t, x, y, z)∂x2

= ΛM (t)T (Hτ,σM×M )T KM×M3 + FM×M3 xA

(σ,a,b,c)M3×M3Λ(a,b,c)(x, y, z),

(4. 46)∂2u(t, x, y, z)

∂y2= ΛM (t)T (Hτ,σ

M×M )T KM×M3 + FM×M3 yA(σ,a,b,c)M3×M3Λ(a,b,c)(x, y, z)

(4. 47)and

∂2u(t, x, y, z)∂z2

= ΛM (t)T (Hτ,σM×M )T KM×M3 + FM×M3 zA

(σ,a,b,c)M3×M3Λ(a,b,c)(x, y, z).

(4. 48)


Using (4. 42 ), (4. 46 ), (4. 47 ) and (4. 48 ) in (4. 41 ), we get

χtΛM (t)T KM×M3Λ(a,b,c)(x, y, z)) = λxΛM (t)T (Hτ,σM×M )T KM×M3 + FM×M3 xA

(σ,a,b,c)M3×M3

+ λyΛM (t)T (Hτ,σM×M )T KM×M3 + FM×M3 yA

(σ,a,b,c)M3×M3Λ(a,b,c)(x, y, z)

+ λzΛM (t)T (Hτ,σM×M )T KM×M3 + FM×M3 zA

(σ,a,b,c)M3×M3Λ(a,b,c)(x, y, z)

+ ΛM (t)T F oM×M3Λ(a,b,c)(x, y, z),

(4. 49)

whereΛM (t)T F oM×M3Λ(a,b,c)(x, y, z) = I(t, x, y, z). The above equations can be sim-

plified as

ΛM (t)T [(χtKM×M3 − λx(Hτ,σM×M )T KM×M3 + FM×M3 xA

(σ,a,b,c)M3×M3−

λy(Hτ,σM×M )T KM×M3 + FM×M3 yA

(σ,a,b,c)M3×M3−

λz(Hτ,σM×M )T KM×M3 + FM×M3 zA

(σ,a,b,c)M3×M3 − F o

M×M3 ]Λ(a,b,c)(x, y, z) = 0.

which implies that

[(χtKM×M3 − λx(Hτ,σM×M )T KM×M3 + FM×M3 xA

(σ,a,b,c)M3×M3−

λy(Hτ,σM×M )T KM×M3 + FM×M3 yA

(σ,a,b,c)M3×M3−

λz(Hτ,σM×M )T KM×M3 + FM×M3 zA

(σ,a,b,c)M3×M3 − F o

M×M3 ] = 0.

In generalized form we can write it as

AM×MKM×M3BTM3×M3 −KM×M3 + CM×M3 = 0. (4. 50)

Where

AM×M = (λx

χtIM×M +

λy

χtIM×M +

λz

χtIM×M )(Hτ,σ

M×M )T ,

BTM3×M3 = xA

(2,a,b,c)M3×M3 + yA

(2,a,b,c)M3×M3 + zA

(2,a,b,c)M3×M3 ,

and

CM×M3 = FM×M3λx

χt

xA(2,a,b,c)M3×M3 +

λy

χt

yA(2,a,b,c)M3×M3 +

λz

χt

zA(2,a,b,c)M3×M3+ F o

M×M3 .

The resulting equation (4. 50 ) is an algebraic equation of Sylvester type and can beeasily solved for the unknown matrixKM×M3 by using the MatLab commanddlyap. Byusing the value of K in (4. 45 ) we can easily obtain the approximate solution of the prob-lem.

5. ILLUSTRATIVE EXAMPLES

The method mentioned above is the extension of pseudo spectral method. The basicproperty of such method is that the accuracy of the solutions depend on the smoothnessof the solutions. If the exact solution of the problem is smooth then the method will yieldmore accurate solution at small scale level. However, if the solution is not smooth, itwill be needed to simulate the algorithm at relatively higher scales. As experiment, weapproximate the solution of two different problems. In the first problem the source term iszero. However for the second problem, we select a suitable source term such that the exactsolution of the problem is known.

46 H. Khalil

EXAMPLE 1. Consider the following time fractional heat conduction problem


∂tσ= λx

∂2u(t, x, y, z)∂x2

+ λy∂2u(t, x, y, z)

∂y2

+ λz∂2u(t, x, y, z)

∂z2+ I(t, x, y, z), u(0, x, y, z) = f(x, y, z).

(5. 51)

Chooseχt = λx = λy = λz = 1 , 0 < σ ≤ 1, t ∈ [0, 1], x ∈ [0, 1] ,y ∈ [0, 1]andz ∈ [0, 1] and the initial conditionu(0, x, y, z) = e(x+y+z). Taking the source termI(t, x, y, z) = 0, then the exact solution of the problem for fixσ = 1 is u(t, x, y, z) =e(x+y+z+3t). To check the accuracy of the scheme we fixσ = 1(because the exact solutionat σ = 1 is known) and simulate the algorithm at different scale level. We observe thatthe accuracy of the solution increases as the scale level increase. At scale levelM = 10,we observe that the approximate solution is equal to the exact solution with difference lessthan10−3. We compare the exact solution with the approximate solution for different val-ues oft. In Fig (1), we fixt = 0.3 and compare the exact solution with the approximatesolution for values ofz, that is,z = [0.1, 0.3, 0.7, 0.9]. The surfaces in the Fig(1) rep-resents the approximate solution at some fixz, while the color dots represents the exactsolution at the corresponding values ofz. Fig (2) and Fig(3) shows the same phenomenaat some other values oft, that is, t = 0.5 and t = 0.9 respectively. The most inter-esting property of fractional differential equation is that the solution at fractional valuesapproaches to the solution at integer values as the order of derivative approaches fromfractional to integer simultaneously. We use this property to show that the method provideaccurate solution at fractional values. For this purpose, we approximate the solution atdifferent value ofσ and observe that asσ → 1 the solution approaches to the exact solu-tion. In Fig (4) and Fig(5), we show this behavior of the solution at two different pointsof yz-plane. One can easily note that the approximate solution approaches to the exactsolution( color dots). We observe that for the current problem the method provide muchmore accurate results. We approximate the absolute error at two different points of theyz-plane, as shown in the Fig(6) and Fig (7) the absolute error is less than10−5 whichis relatively accepted number for such complicated problems. The simulations of this ex-ample is carried out with selecting the parameter of the Jacobi polynomialsξ = ζ = 1.

00.5100.10.20.30.40.50.60.70.80.91

0

2

4

6

8

10

12

14

16

18

20

yx

Approximate U at z = 0.1

Exact U at z = 0.1


Exact U at z = 0.3


Exact U at z = 0.7


Exact U at z = 0.9

Fig. 1 : The comparison of the exact and approximate solution of example1 at t = 0.3ξ = ζ = 1, M = 10, a = 1, b = 1, c = 1, τ = 1 and the order of derivativeσ = 1.


00.2

0.40.6

0.81

00.2

0.40.6

0.81

0

5

10

15

20

25

30

35

yx


Exact U at z = 0.1


Exact U at z = 0.3


Exact U at z = 0.7


Exact U at z = 0.9

Fig. 2 The comparison of the exact and approximate solution of example 1 att = 0.5,ξ = ζ = 1, M = 10, a = 1, b = 1, c = 1, τ = 1 and the order of derivativeσ = 1.

0

0.2

0.4

0.6

0.8

1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

10

20

30

40

50

60

yx


Exact U at z = 0.1


Exact U at z = 0.3


Exact U at z = 0.7


Exact U at z = 0.9

Fig. 3 The comparison of the exact and approximate solution of example 1 att = 0.9ξ = ζ = 1, M = 10, a = 1, b = 1, c = 1, τ = 1 and the order of derivativeσ = 1.

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

0

20

40

60

80

100

120

140

y

x

Approximate U at σ = 0.6





Exact U at σ = 1.0

48 H. Khalil

Fig. 4 The approximate solution of example 1 at fractional value ofσ (surface). We fixz = 0.3, y = 0.3, M = 10,a = 1, b = 1, c = 1, τ = 1.

00.10.20.30.40.50.60.70.80.9100.2

0.40.6

0.810

100

200

300

400

500

600

700

y






Exact U at σ = 1.0

Fig. 5: The approximate solution of example 1 at fractional value ofσ.The blue dotsrepresents the exact solution atσ = 1.

0 0.2 0.4 0.6 0.8 1 0

0.5

1

0

0.5

1

1.5

2

x 10−5

y

x

Absolute Error atM = 10, z = 0.3, t = 0.3

Fig. 6 : The absolute error of example 1 atM = 10.Here we fixz = 0.3, t = 0.3, σ = 1,a = 1, b = 1, c = 1, τ = 1.


0 0.2 0.4 0.6 0.8 1

0

0.5

10

0.5

1

1.5

2

2.5x 10

−3

x

y

Absolute error inU atM = 10, t = 0.7, z = 0.7

Fig. 7 : The absolute error of example 1 atM = 10.Here we fixz = 0.7, t = 0.7, σ = 1,a = 1, b = 1, c = 1, τ = 1.

EXAMPLE 2. Consider the time fractional heat conduction problem


∂tσ= λx

∂2u(t, x, y, z)∂x2

+ λy∂2u(t, x, y, z)

∂y2+ λz

∂2u(t, x, y, z)∂z2

+ I(t, x, y, z),

u(0, x, y, z) = f(x, y, z).(5. 52)

Also chooseχt = λx = λy = λz = 1 , 0 < σ ≤ 1, t ∈ [0, 1], x ∈ [0, 1], y ∈ [0, 1] andz ∈ [0, 1].If we let the initial condition

u(0, x, y, z) = x2 y2 z2 − 2 x + y + z

and take the source term

I(t, x, y, z) = 2 x y (t x y + x y z)−2 (t x + x z)2−2 (t y + y z)2−2 x2 y2−6 t3 x z3−6 t3 x3 z + 3 t2 x3 z3 − 12 t4 x2 y4 z4 − 12 t4 x4 y2 z4 − 12 t4 x4 y4 z2 + 4 t3 x4 y4 z4

then the exact solution of the problem forσ = 1 is

u(t, x, y, z) = y − 2 x + z + (t x y + x y z)2 + t3 x3 z3 + t4 x4 y4 z4.

We approximate the solution of this problem with the new technique and as expected weget a high accuracy of the approximate solution. We observe that at scale levelM = 6( which is much more small scale for such problem) the approximate solution is equal tothe exact solution with maximum difference less than10−15 which is highly negligible. Wecompare the approximate solution with the exact solution at three different value oft. Thecomparison att = 0.3 is displayed in Fig.(8), while att = 0.5 and t = 0.9 is shown inFig. (9) and Fig(10).One can easily note that the exact solution matches very well with theapproximate solution.Note that the dots in these figures represents the exact solution andthe surface in these figures represents the approximate solution.We also approximate thesolution for fractional value ofσ, and as expected, the approximate solution approachesthe exact solution as the order of derivativeσ → 1.Fig (11) and Fig (12) shows thisphenomena at some fixed points of the yz-plane. The absolute amount of error is displayedin Fig (13) and Fig(14) at some point of the yz-plane.One can see that the absolute amount

50 H. Khalil

of error is less than10−16. This example is analyzed with choosing the parameters of theJacobi polynomials asξ = ζ = 0.

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

yx

Exact U at z = 0.2


Exact U at z = 0.4


Exact U at z = 0.6


Exact U at z = 0.8


Fig. 8 The comparison of the exact and approximate solution example 2 att = 0.3,ξ = ζ = 0, M = 6, a = 1, b = 1, c = 1, τ = 1 and the order of derivativeσ = 1.

0 0.2 0.4 0.6 0.8 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

Exact U at z = 0.2


Exact U at z = 0.4


Exact U at z = 0.6


Exact U at z = 0.8




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 0.2 0.4 0.6 0.8 1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

yx

Exact U at z = 0.2


Exact U at z = 0.4


Exact U at z = 0.6


Exact U at z = 0.8



0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

−50

0

50

100

150

z

y






Exact U at σ = 1.0

Fig. 11 The approximate solution of example 2 at fractional value ofσ (surface). Here wefix x = 0.3, t = 0.3, M = 6,a = 1, b = 1, c = 1, τ = 1.

00.10.20.30.40.50.60.70.80.91

0

0.2

0.4

0.6

0.8

1

−2

0

2

4

6

8

10

12

14

16

z

y






Exact U at σ = 1.0

52 H. Khalil

Fig. 12 The approximate solution of example 2 at fractional value ofσ (surface). Here wefix x = 0.7, t = 0.7, M = 6,a = 1, b = 1, c = 1, τ = 1.

0

0.5

1

00.20.40.60.810

0.2

0.4

0.6

0.8

1

x 10−15

x

y

Absolute Error atM = 6

Fig. 13 The absolute error of example 2 atM = 6, we fixz = 0.4, t = 0.4.

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

0

2

4

6

8

x 10−16

xy

Absolute Error atM = 6

Fig. 14 The absolute error of example 2 atM = 6, we fixz = 0.8, t = 0.8

EXAMPLE 3. Consider the following integer order heat conduction problem

∂U(t, x, y, z)∂t

=∂2U(t, x, y, z)

∂x2+

∂2U(t, x, y, z)∂y2

+∂2U(t, x, y, z)

∂z2

u(0, x, y, z) = f(x, y, z).(5. 53)

where t ∈ [0, 1], x ∈ [0, 1] ,y ∈ [0, 1] and z ∈ [0, 1].If we let the initial conditionU(0, x, y, z) = (1− y)e(x+z), then the exact solution of the problem is

U(t, x, y, z) = (1− y)e(x+z+2t).

This problem is also solved in[2] using homotopy analysis method and variational it-eration method. We approximate the solution of this problem with our new technique.Andas expected we found that the approximate solution matches very well with the exact so-lution. The comparison of exact and approximate solution at some fixed value oft,ie t =0.3, 0.6, 0.9 and at each value oft the solution is displayed at fix value ofz.Fig(15),Fig(16)andFig(17) shows the comparison of exact and approximate solution att = 0.3, 0.6 and 0.9


respectively. Note that here we fix the scale levelM = 9. We observe that the method yieldsa very high accurate estimate of the solution. And the error of approximation (absolute er-ror) decreases significantly by the increase of the scale levelM . In order to compare theaccuracy of the method we compare our results with the analytic solution obtained in[2].We calculate the absolute difference of the exact and approximate solution using the cur-rent method. We also calculate the difference of exact solution and n-th order analyticsolution reported in[2]. We observe that the error obtained with this new method in muchmore less than that reported in[2]. Fig(18) andFig(19) shows comparison of absoluteerror at two different points of the space.

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

0

1

2

3

4

5

6

7

8

9

xy

Exact U at z = 0.2

Approximate U atz = 0.2

Exact U at z = 0.4

Apprximate U at z = 0.4

Exact U at z = 0.6

Approximte U at z = 0.6

Exact U atz = 0.8


Fig. 15 The approximate solution of example 3 at different value ofz whereσ = 1,M = 9,t = 0.3.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

2

4

6

8

10

12

14

x

y

Exact U at z = 0.2

ApproximateU at z = 0.2

Exact U at z = 0.4


Exact U at z = 0.6


Exact U at z = 0.8



54 H. Khalil

00.2

0.40.6

0.81

00.2

0.40.6

0.81

0

2

4

6

8

10

12

14

16

18

20

xy

Exact U at z = 0.2


Exact U at z = 0.4


Exact U at z = 0.6


Exact U at z = 0.8



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Error reported in [21]at n = 7

Error using thismethodatM = 7.

Fig. 18 Comparison of absolute error of example at(x, y, z) = (0.5, 0.5, 0.5) at M = 7.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Error reported in [21]at n = 10

Error using newmethodatM = 10


Fig. 19 Comparison of absolute error at(x, y, z) = (0.9, 0.9, 0.9) at M = 10.

6. CONCLUSION

From the above analysis and calculation we concluded that the method provide a verygood approximation to the problems under consideration.This method can efficiently solvepartial differential equation in four variables.The main advantage of the method is its highaccuracy.The method can be easily extended to solve more complicated problems.We be-lieve that one may obtain a more accurate solution by using some other kinds of orthogonalpolynomials like Bernstein or Laguerre.Our future work is related to the extension of themethod to solve such problems under different kinds of boundary conditions.We expectthat the reader may find the work interesting and useful.

7. ACKNOWLEDGMENTS

The authors are extremely thankful to the reviewers for their useful comments that im-proves the quality of the manuscript.

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