Research ArticleNumerical Investigation of Chemical SchnakenbergMathematical Model

Faiz Muhammad Khan,1 Amjad Ali,1 Nawaf Hamadneh ,2 Abdullah,1

and Md Nur Alam 3

1Department of Mathematics and Statistics, University of Swat, Pakhtunkhwa, Pakistan2Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia3Department of Mathematics, Pabna University of Science and Technology, Pabna 6600, Bangladesh

Correspondence should be addressed to Nawaf Hamadneh; nhamadneh@seu.edu.saand Md Nur Alam; nuralam.pstu23@gmail.com

Received 20 September 2021; Accepted 19 October 2021; Published 9 November 2021

Academic Editor: Taza Gul

Copyright © 2021 Faiz Muhammad Khan et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Schnakenberg model is known as one of the influential model used in several biological processes. The proposed model is anautocatalytic reaction in nature that arises in various biological models. In such kind of reactions, the rate of reaction speedsup as the reaction proceeds. It is because when a product itself acts as a catalyst. In fact, model endows fractional derivativesthat got great advancement in the investigation of mathematical modeling with memory effect. Therefore, in the present paper,the authors develop a scheme for the solution of fractional order Schnakenberg model. The proposed model describes an autochemical reaction with possible oscillatory behavior which may have several applications in biological and biochemicalprocesses. In this work, the authors generalized the concept of integer order Schnakenberg model to fractional orderSchnakenberg model. We provided the approximate solution for the underlying generalized nonlinear Schnakenberg model inthe sense of Caputo differential operator via Laplace Adomian decomposition method (LADM). Furthermore, we establishedthe general scheme for the considered model in the form of infinite series by the aforementioned technique. The consequentresults obtained by the proposed technique ensure that LADM is an effective and accurate techniques to handle nonlinearpartial differential equations as compared to the other available numerical techniques. Finally, the obtained numerical solutionis visualized graphically by MATLAB to describe the dynamics of desired solution.

1. Introduction

Since the biological processes are not linear systems bynature, which happen at various time scales, therefore, sev-eral complex problems arise as a result of fast or slowresponds with following interventions or treatment. Thus,to capture an appropriate individual trajectories, the studyunderconsideration depends on a sequential sample over inappropriate time course. Although various classical methodsgive a significant insight for the better understanding of avariety of biological processes, but due to some propertieslike localizing, quantifying and pressibility of measuring rev-olutionized our thoughts and motivated the researchers and

scientists to construct some dynamical methods to tacklevarious biological phenomena. Most of dynamical and bio-logical phenomena that are involved in the study of chemicaltheory, fluid dynamics, and mathematical biology have moreimportance due to explaining the processes related to reallife. Such phenomena are usually modulated by linear ornonlinear partial differential equations (PDEs). DEs haveability to predict about the dynamical phenomena aroundthe globe and also used to describe the exponential growthand decay over the time. DEs are having a diverse range ofapplications in several field, such as physics, engineering,and biology. The researchers use the tool of differentialequation, to modulate aforesaid phenomena. Furthermore,

HindawiJournal of NanomaterialsVolume 2021, Article ID 9152972, 8 pageshttps://doi.org/10.1155/2021/9152972

some useful applications of DEs to modulate the engineeringand chemical phenomena can be found in some recent arti-cles (see [1–5]). PDEs often model multidimensionaldynamical system, i.e., it can be used to formulate naturalphenomena, such as sound, heat, electrostatics, electrody-namics, quantum mechanics, and flow of fluid (see [6, 7]).

It is important to note that reaction-diffusion systemshave been used over decades to study the deep insights ofbiological systems. More precisely, these models have beenused in several biological, physical, environmental, andchemical processes of real life. In reaction-diffusion systems,Brusselator, Lengyel-Epstein, and Schnakenberg models arethe most famous due to its applicability and reliable results.These models are used for generating patterns for both bio-logical and chemical systems and so called turning typemodels. The Schnakenberg model is one of the well-knownchemical reaction-diffusion model which was introducedby Schnakenberg in 1979. It is important to note that an autochemical reaction having oscillatory behavior is preciselydescribed by Schnakenberg model with a verity of the bio-logical and biochemical processes like pattern formationsin skin analysis and embryogenesis. Further in biology, italso models the spatial distribution of a morphogen. Sciencein several biological systems, these types of models involveauto catalytic reactions which natural arises. Therefore, insuch reactions, the rate of reaction boost with the reactionproceeds, due to the role of a product acts as catalysts. Intri-molecular reaction, the reaction under considerationplays a role of two species models. Such types of reactionbetween chemical sources A and B and products Φ and Ψare described as:

A⇌Φ, B⟶Ψ, 2Φ +Ψ⟶ 3Φ, ð1Þ

where A and B are two chemical sources and Φ and Ψ areproducts. A system of reaction-diffusion equations isobtained by using law of mass of action, for the concentra-tions ϕðx, tÞ and ψðx, tÞ of the products Φ and Ψ describedin (1). The derived nondimensional form of the system [8, 9]is given by

∂ϕ x, tð Þ∂t

= α − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ +D1∂2ϕ x, tð Þ

∂x2,

∂ψ x, tð Þ∂t

= β − ϕ2 x, tð Þψ x, tð Þ +D2∂2ψ x, tð Þ

∂x2,

8>>><>>>:

ð2Þ

where ϕ = ϕðx, tÞ and ψ = ψðx, tÞ represent the concentrationand α and β are positive arbitrary constants and represent theconcentration of A and B. D1 and D2 are the diffusion coeffi-cients of the chemicals Φ and Ψ, for detail study (see [8–11]).

In modern era, the researchers paid keen interest toinvestigate nonlinear PDEs due to its wide range of applica-tions in physics, engineering, and modern sciences. In lasttwo decades, a considerable number of efforts have beenmade to investigate field of the fractional order partial differ-ential equations (FOPDEs) in all aspects, such as theoretical,numerical, and applications. These equations provide the

hereditary properties and description of memory effect ofdifferent phenomena. Fractional order differential operatorhas the advantage of being a nonlocal operator and possessesgreater degree of freedom as compared to conventional dif-ferential operator. Fractional calculus has got the consider-ation of researchers, due to its extensive applications in theaforesaid fields. The mathematical models involving frac-tional order derivatives are more reliable and great degreeof freedom and accuracy as compared to traditional deriva-tives. In some situation, a mathematical model involvinginteger order derivative does not describe the real situation.In such circumstances, fractional order derivatives are morereliable to describe these real word problems, (see [12–19]).In this regard, the proposed model has been studied by var-ious researchers from both analytical and numerical pointsof view (see [11, 20, 21]).

After the comprehensive literature review, it was foundthat mathematical models consist of Caputo fractional orderoperators that are more accurate and reliable instead of inte-ger order model. Keeping the aforementioned applicationsof FDEs, the researchers investigated different aspects ofvarious mathematical models. In this continuation, theresearchers well explored different aspects of mathematicalmodeling and published variety of articles (see [2, 22, 23]).Therefore, the researchers investigated different features ofaforementioned model. An important class of biochemicalmodel known as Schnakenberg model represents a chemicalprocess, where sudden fluctuation occurs during the reaction.The considered model can be well described by Caputo frac-tional differential operator instead of integer order derivatives.Therefore, the author used the idea of Caputo fractional orderderivatives to generalize the concept of model (2) intoCaputo fractional order Schnakenberg model given by

∂σϕ x, tð Þ∂tσ

= α − βϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ +D1∂2ηϕ x, tð Þ

∂x2η,

∂σψ x, tð Þ∂tσ

= γ − ϕ2 x, tð Þψ x, tð Þ +D2∂2ηψ x, tð Þ

∂x2η,

0 < σ, η ≤ 1,

8>>>>><>>>>>:

ð3Þ

subjected to the initial conditions (ICs)

ϕ x, 0ð Þ = h xð Þ,ψ x, 0ð Þ = g xð Þ:

ð4Þ

We have established the numerical scheme for aforemen-tioned model with the help of well-known numerical tech-nique called LADM. The proposed technique consists ofspecial polynomial known as Adomian polynomial. The spe-cific class of this polynomial decomposes the nonlinear terminvolving in the model in the form of series. With the help ofAdomian polynomial, the nonlinear term is decomposes as

H w x, tð Þð Þ = 〠∞

m=0Am, ð5Þ

2 Journal of Nanomaterials

where Am’s are called Adomian polynomials introduced byAdomian and defined as

Am = 1Γ m + 1ð Þ

dm

dλmH〠

m

i=0λiwi x, tð Þ

" #λ=0

: ð6Þ

The technique of Laplace Adomian decomposition is thetool to obtain the approximate solution of nonlinear PDEs.LADM is the combination of two powerful techniques, i.e.,Adomian decomposition method and Laplace transform.The main advantage of LADM is that it can provide bothanalytic and numerical solution to a class of nonlinear differ-ential equations (DEs). The considered technique is moresuperior as compared to the other available techniques,because it gives us particular solutions without finding gen-eral solution for DEs. Furthermore, it does not require prede-fined size declaration like Runge-Kutta method, possess lessparameters, and requires no discretization and linearization.In comparison with other analytical techniques, the proposedtechnique is an efficient and simple tool to investigatenumerical solution of nonlinear fractional partial differentialequations. The results obtained by this method, ensure thecapability and reliability of the proposed method for nonlin-ear fractional partial differential equations (for detail, see [16,24, 25]).

In present paper, the authors have generalized the idea ofintegral order Schnakenberg model to fractional ordermodel in the terms of singular kernal operator. Moreover,we have developed the scheme for the considered fractionalmodel via LADM in the form of infinite series. We haveobtained the semianalytical solution for the considered non-linear model with the help of proposed techniques. Theresults obtained by the proposed technique ensure that theconsider technique is very effective and easy to implement.The numerical simulation is visualized graphically viaMATLAB to explain the dynamical behavior of aforemen-tioned model.

2. Preliminaries

The concerned section is devoted to the well-known defini-tions related to fractional calculus and semianalytic tech-niques, which are helpful in further corresponding in thiswork.

Definition 1 (see [25]). The LT of a function gðx, tÞ, defined∀t≥0, is denoted by Gðx, sÞ =Lfgðx, tÞg and is given as

G x, sð Þ =L g x, tð Þf g =ð∞0e−stg x, tð Þdt, ð7Þ

where “L” is called LT operator or Laplace transforma-tion and “s” is the transformed variable.

Definition 2 (see [25]). The noninteger order derivative forthe function ψ on the interval ð0,∞Þ × ð0,∞Þ in Caputo

sense is defined such as

cDαψ x, tð Þ = 1Γ m − αð Þ

ðt0t − sð Þm−α−1ψm x, sð Þds,

α ∈ m − 1,mð Þ,m ∈ℕ,ð8Þ

where m = ½α� + 1, ½α� is the integeral part of α, and αdenotes real number. Now, for α⟶m, the Caputo frac-tional derivative becomes conventional nth order derivativeof the function.

Particularly for α ∈ ð0, 1Þ,

cDαψ x, tð Þ = 1Γ m − 1ð Þ

ðt0

1t − sð Þα

∂∂s

ψ x, sð Þds: ð9Þ

Definition 3 (see [25]). The LT of Caputo derivatives is givenby

L cDαψ x, tð Þf g = sαψ x, sð Þ − 〠m−1

k=0sα−k−1ψk x, 0ð Þ,

α ∈ m − 1,mð Þ,m ∈N ,ð10Þ

where m = ½α� + 1 and ½α� denote the nonfractional partof α.

3. General Scheme for the SolutionSchnakenberg Model

This section is committed to the general scheme for the solu-tion of fractional order Schnakenberg model via LADM. Thefractional order nondimensional Schnakenberg model isgiven by

∂σϕ x, tð Þ∂tσ

= α − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ +D1∂2ηϕ x, tð Þ

∂x2η ,

∂σψ x, tð Þ∂tσ

= β − ϕ2 x, tð Þψ x, tð Þ +D2∂2ηψ x, tð Þ

∂x2η,

0 < σ, η ≤ 1,

8>>>>><>>>>>:

ð11Þ

subjected to ICs:

ϕ x, 0ð Þ = h xð Þ,ψ x, 0ð Þ = g xð Þ,

ð12Þ

where ϕ = ϕðx, tÞ and ψ = ψðx, tÞ represent the con-centrations,α and β that are positive arbitrary constantsand D1 and D2 are the diffusion coefficients of thesubstances.

3Journal of Nanomaterials

Applying Laplace transform on (11), we have

L∂σϕ x, tð Þ

∂tσ

� �=L α − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ +D1

∂2ηϕ x, tð Þ∂x2η

( ),

L∂σψ x, tð Þ

∂tσ

� �=L β − ϕ2 x, tð Þψ x, tð Þ +D2

∂2ηψ x, tð Þ∂x2η

( ):

8>>>>><>>>>>:

ð13Þ

By using properties of Laplace transform, (13) becomes

Φ x, sð Þ − 1sϕ x, 0ð Þ = 1

sσL α − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ +D1

∂2ηϕ x, tð Þ∂x2η

( ),

Ψ x, sð Þ − 1sψ x, 0ð Þ = 1

sσL β − ϕ2 x, tð Þψ x, tð Þ +D2

∂2ηψ x, tð Þ∂x2η

( ):

8>>>>><>>>>>:

ð14Þ

Now applying inverse Laplace transform on (14) andusing ICs, we get

ϕ x, tð Þ = h xð Þ +L−1 1sσL α − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ +D1

∂2ηϕ x, tð Þ∂x2η

( )" #,

ψ x, tð Þ = g xð Þ +L−1 1sσL β − ϕ2 x, tð Þψ x, tð Þ +D2

∂2ηψ x, tð Þ∂x2η

( )" #:

8>>>>><>>>>>:

ð15Þ

The nonlinear term ϕ2ðx, tÞψðx, tÞ presenting in (15) isdecomposed as

ϕ2 x, tð Þψ x, tð Þ = 〠∞

n=0Am, ð16Þ

where Am is called Adomian polynomial and defined as

Am = 1Γ m + 1ð Þ

dm

dλm〠m

i=0λiϕi x, tð Þ

!2

〠m

i=0λiψi x, tð Þ

!" #λ=0

:

ð17Þ

For m = 0

A0 = ϕ20 x, tð Þψ0 x, tð Þ: ð18Þ

For m = 1

A1 = ϕ20 x, tð Þψ1 x, tð Þ + 2ϕ0 x, tð Þψ0 x, tð Þϕ1 x, tð Þ: ð19Þ

The assumed solutions ϕðx, tÞ and ψiðx, tÞ are in theform of

ϕ x, tð Þ = 〠∞

i=0ϕi x, tð Þ,

ψ x, tð Þ = 〠∞

i=0ψi x, tð Þ:

ð20Þ

Plugging these values in system (15), we have

〠∞

i=0ϕi x, tð Þ = h xð Þ +L−1 1

sσL α − 〠

∞

i=0ϕi x, tð Þ + 〠

∞

n=0An +D1

∂2η

∂x2η〠∞

i=0ϕi x, tð Þ

( )" #,

〠∞

i=0ψi x, tð Þ = g xð Þ +L−1 1

sσL β − 〠

∞

n=0An +D2

∂2η

∂x2η〠∞

i=0ψi x, tð Þ

( )" #:

8>>>>><>>>>>:

ð21Þ

Comparing both sides of system (21), we have

ϕ0 x, tð Þ = h xð Þ,ψ0 x, tð Þ = g xð Þ,

(

ϕ1 x, tð Þ =L−1 1sσL α − ϕ0 x, tð Þ + A0 +D1

∂2η

∂x2ηϕ0 x, tð Þ

( )" #,

ψ1 x, tð Þ =L−1 1sσL β − A0 +D2

∂2η

∂x2ηψ0 x, tð Þ

( )" #,

8>>>>><>>>>>:ϕn x, tð Þ =L−1 1

sσL α − ϕn−1 x, tð Þ + An−1 +D1

∂2η

∂x2ηϕn−1 x, tð Þ

( )" #,

ψn x, tð Þ =L−1 1sσL β − An−1 +D2

∂2η

∂x2ηψn−1 x, tð Þ

( )" #:

8>>>>><>>>>>:

ð22Þ

In this manner, we obtain the desired solution given by

ϕ x, tð Þ = 〠∞

i=0ϕi x, tð Þ,

ψ x, tð Þ = 〠∞

i=0ψi x, tð Þ:

8>>>><>>>>:

ð23Þ

By simple computational work, we get

ϕ0 x, tð Þ = h xð Þ,ψ0 x, tð Þ = g xð Þ,

(

ϕ1 x, tð Þ = α − ϕ0 x, tð Þ + ϕ20 x, tð Þψ0 x, tð Þ +D1∂2η

∂x2ηϕ0 x, tð Þ

!tσ

Γ σ + 1ð Þ ,

ψ1 x, tð Þ = β − ϕ20 x, tð Þψ0 x, tð Þ +D2∂2η

∂x2ηψ0 x, tð Þ

!tσ

Γ σ + 1ð Þ ,

8>>>>><>>>>>:

ϕ2 x, tð Þ = αtσ

Γ σ + 1ð Þ − ξ − ϕ20 x, tð Þζ − 2ϕ0 x, tð Þψ0 x, tð Þξ −D1∂2η

∂x2ηξ

!t2σ

Γ 2σ + 1ð Þ ,

ψ2 x, tð Þ = βtσ

Γ σ + 1ð Þ − ϕ20 x, tð Þζ + 2ϕ0 x, tð Þψ0 x, tð Þξ −D2∂2η

∂x2ηζ

!t2σ

Γ 2σ + 1ð Þ ,

8>>>>><>>>>>:

ð24Þ

where

ξ = α − ϕ0 x, tð Þ + ϕ20 x, tð Þψ0 x, tð Þ +D1∂2η

∂x2ηϕ0 x, tð Þ,

ζ = β − ϕ20 x, tð Þψ0 x, tð Þ +D2∂2η

∂x2ηψ0 x, tð Þ:

ð25ÞThus, the three term solutions are given by

4 Journal of Nanomaterials

4. Numerical Discussion

In this section of research work, we provide some numericalexample to illustrate the main work.

Example 1. Let α = β = η = 1 and D1 =D2 = 2, so the pro-posed model becomes

∂σϕ x, tð Þ∂tσ

= 1 − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ + 2 ∂2

∂x2ϕ x, tð Þ,

∂σψ x, tð Þ∂tσ

= 1 − ϕ2 x, tð Þψ x, tð Þ + 2 ∂2

∂x2ψ x, tð Þ,

0 < σ,≤1,

8>>>>><>>>>>:

ð27Þ

subjected to ICs:

ϕ x, 0ð Þ = ex sin x,ψ x, 0ð Þ = xex:

ð28Þ

Applying Laplace transform to system (27), we get

L∂σϕ x, tð Þ

∂tσ

� �=L 1 − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ + 2 ∂

2ϕ x, tð Þ∂x2

( ),

L∂σψ x, tð Þ

∂tσ

� �=L 1 − ϕ2 x, tð Þψ x, tð Þ + 2 ∂

2ψ x, tð Þ∂x2

( ):

8>>>>><>>>>>:

ð29Þ

By using properties of Laplace transform, system (29)becomes

Φ x, sð Þ − 1su x, 0ð Þ = 1

sσL 1 − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ + 2 ∂

2ϕ x, tð Þ∂x2

( ),

Ψ x, sð Þ − 1sv x, 0ð Þ = 1

sσL 1 − ϕ2 x, tð Þψ x, tð Þ + 2 ∂

2ψ x, tð Þ∂x2

( ):

8>>>>><>>>>>:

ð30Þ

Now applying inverse Laplace transform on (30) andusing ICs, we obtain

ϕ x, tð Þ = ex sin x +L−1 1sσL 1 − ϕ x, tð Þ + ϕ2 x, tð Þψ x, tð Þ + 2 ∂

2ϕ x, tð Þ∂x2

( )" #,

ψ x, tð Þ = xex +L−1 1sσL 1 − ϕ2 x, tð Þψ x, tð Þ + 2 ∂

2ψ x, tð Þ∂x2

( )" #:

8>>>>><>>>>>:

ð31Þ

The nonlinear term ϕ2ðx, tÞψðx, tÞ involved in (31) isdecomposed by Adomian polynomial

ϕ2 x, tð Þψ x, tð Þ = 〠∞

n=0Am: ð32Þ

Plugging these valves in system (31), we have

〠∞

i=0ϕi x, tð Þ = ex sin x +L−1 1

sσL 1 − 〠

∞

i=0ϕi x, tð Þ + 〠

∞

m=0Am + 2 ∂2

∂x2〠∞

i=0ϕi x, tð Þ

( )" #,

〠∞

i=0ψi x, tð Þ = xex +L−1 1

sσL 1 − 〠

∞

m=0Am + 2 ∂2

∂x2〠∞

i=0ψi x, tð Þ

( )" #:

8>>>>><>>>>>:

ð33Þ

Comparing both sides of system (33), we have

ϕ0 x, tð Þ = ex sin x,ψ0 x, tð Þ = xex,

(

ϕ1 x, tð Þ =L−1 1sσL 1 − ϕ0 x, tð Þ + A0 + 2 ∂2

∂x2ϕ0 x, tð Þ

( )" #,

ψ1 x, tð Þ =L−1 1sσL 1 − A0 + 2 ∂2

∂x2ψ0 x, tð Þ

( )" #,

8>>>>><>>>>>:ϕn x, tð Þ =L−1 1

sσL 1 − un−1 x, tð Þ + Rn−1 + 2 ∂2

∂x2ϕn−1 x, tð Þ

( )" #,

ψn x, tð Þ =L−1 1sσL 1 − Rn−1 + 2 ∂2

∂x2ψn−1 x, tð Þ

( )" #:

8>>>>><>>>>>:

ð34Þ

Assume the solution in the form of

ϕ x, tð Þ = 〠∞

i=0ϕi x, tð Þ,

ψ x, tð Þ = 〠∞

i=0ψi x, tð Þ:

8>>>><>>>>:

ð35Þ

By simple computational work, we obtain

ϕ x, tð Þ = h xð Þ + 2α − ϕ0 x, tð Þ + ϕ20 x, tð Þψ0 x, tð Þ +D1∂2η

∂x2ηϕ0 x, tð Þ

!tσ

Γ σ + 1ð Þ − ξ − ϕ20 x, tð Þζ − 2ϕ0 x, tð Þψ0 x, tð Þξ −D1∂2η

∂x2ηξ

!t2σ

Γ 2σ + 1ð Þ+⋯

ψ x, tð Þ = g xð Þ + 2β − ϕ20 x, tð Þψ0 x, tð Þ +D2∂2η

∂x2ηψ0 x, tð Þ

!tσ

Γ σ + 1ð Þ− ϕ20 x, tð Þζ + 2ϕ0 x, tð Þψ0 x, tð Þξ −D2∂2η

∂x2ηζ

!t2σ

Γ 2σ + 1ð Þ+⋯

8>>>>><>>>>>:

ð26Þ

5Journal of Nanomaterials

–2001

0

200

1

400

600

t0.5

800

x0.5

0 00

100

200

300

400

500

600

700

800

Plot of 𝜙(x,t) for 𝜎=1

(a)

–1 –0.5 0 0.5 1t

–400

–200

0

200

400

600

800

𝜙(x

,t) a

t x=1

Figure 2: plot of 𝜙(x,t) for different value of 𝜎.

𝜎=0.7𝜎=0.8

𝜎=0.9𝜎=1

(b)

Figure 1: (a, b) Spatial numerical solution of ψðx, tÞ in 3D and 2D, respectively.

0–8000

–600

x0.5

–400

t0.5

–2000

200

11–900

–800

–700

–600

–500

–400

–300

–200

–100

0Plot of 𝜓(x,t) for 𝜎=1

(a)

𝜓(x

,t) a

t x=1

–1 –0.5 0 0.5 1t

–800

–600

–400

–200

0

200

400Figure 2: plot of 𝜓(x,t) for different value of 𝜎.

𝜎=0.7𝜎=0.8

𝜎=0.9𝜎=1

(b)

Figure 2: (a, b) Spatial numerical solution of ϕðx, tÞ in 3D and 2D, respectively.

ϕ1 x, tð Þ = 1 − ex sin x + xe3xsin2x + 4ex cos x� � tσ

Γ σ + 1ð Þ ,

ψ1 x, tð Þ = 1 − xe3xsin2x + 2xex + 4ex� � tσ

Γ σ + 1ð Þ ,

8>>><>>>:

ϕ2 x, tð Þ = tσ

Γ σ + 1ð Þ + 17xe3xsin2x − 1 − 15ex sin x − 8ex cos x + e2xsin2x − xe5xsin4x + 16e3xsin2x + 2xe2x sin x + 2x2e5xsin3x + 16xe3x sin 2x + 4xe3x cos 2x + 4e3x sin 2x� � t2σ

Γ 2σ + 1ð Þ ,

ψ2 x, tð Þ = tσ

Γ σ + 1ð Þ − e2xsin2x − xe5xsin4x + 16e3xsin2x + 2xe2x sin x + 2x2e5xsin3x + 4xe3x cos 2x + 16xe3x sin 2x + 4e3x sin 2x + 18xe3x sin x − 4xex − 16ex� � t2σ

Γ 2σ + 1ð Þ :

8>>><>>>:

ð36Þ

6 Journal of Nanomaterials

Continuing the similar fashion, the solution terminatedafter three terms is given by

For classical order σ = 1, the solution become

The Schnakenberg mathematical model actually repre-sents the three steps biochemical processes. In such process,the final product is in the form of ϕðx, tÞ, whose details aregiven in the introduction of this work. As evident form,the graphical solution of Figures 1(a) and 1(b) shows thatthe concentration of ψðx, tÞ increases and then decreases.While from Figures 2(a) and 2(b), the graph shows thatthe concentration of ϕðx, tÞ decreases and then increases.This is a clear evidence that the final product will beobtained in terms of ϕðx, tÞ with the passage of times whichjustify the aforementioned three-step process.

5. Conclusion

The authors have successfully established the numericalscheme for the generalized fractional order Schnakenbergbiochemical model. In order to obtain the desired results,we utilized the tools of well-known numerical techniquecalled Laplace Adomian decomposition method. We haveobtained the semianalytic solution for the nonlinear Schna-kenberg model in the sense of Caputo differential operatorwith the help of proposed method. To elaborate our mainresults, we have provided a numerical example to illustrateour main work. The numerical simulation has been visual-ized graphically via MATLAB to explain the model’s dynam-ical behavior.

Data Availability

The data will be available for public after publication.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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ϕ x, tð Þ = ex sin x + 2 − ex sin x + xe3xsin2x + 4ex cos xÞ� � tσ

Γ σ + 1ð Þ + 17xe3xsin2x − 1 − 15ex sin x − 8ex cos x + e2xsin2x − xe5xsin4x + 16e3xsin2x + 2xe2x sin x + 2x2e5xsin3x + 16xe3x sin 2x + 4xe3x cos 2x + 4e3x sin 2x� � t2σ

Γ 2σ + 1ð Þ ,

ψ x, tð Þ = xex + 2 − xe3xsin2x + 2xex + 4ex� � tσ

Γ σ + 1ð Þ − e2xsin2x − xe5xsin4x + 16e3xsin2x + 2xe2x sin x + 2x2e5xsin3x + 4xe3x cos 2x + 16xe3x sin 2x + 4e3x sin 2x + 18xe3x sin x − 4xex − 16ex� � t2σ

Γ 2σ + 1ð Þ :

8>>><>>>:

ð37Þ

ϕ x, tð Þ = ex sin x + 2 − ex sin x + xe3xsin2x + 4ex cos xÞ� �t + 17xe3xsin2x − 1 − 15ex sin x − 8ex cos x + e2xsin2x − xe5xsin4x + 16e3xsin2x + 2xe2x sin x + 2x2e5xsin3x + 16xe3x sin 2x + 4xe3x cos 2x + 4e3x sin 2x� �

t2,

ψ x, tð Þ = xex + 2 − xe3xsin2x + 2xex + 4ex� �

t − e2xsin2x − xe5xsin4x + 16e3xsin2x + 2xe2x sin x + 2x2e5xsin3x + 4xe3x cos 2x + 16xe3x sin 2x + 4e3x sin 2x + 18xe3x sin x − 4xex − 16ex� �

t2:

(

ð38Þ

7Journal of Nanomaterials

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8 Journal of Nanomaterials