Comparing Numerical Methods for Solving Time ISRN Mathematical Analysis 3. Generalized Differential...

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  • International Scholarly Research NetworkISRN Mathematical AnalysisVolume 2012, Article ID 737206, 28 pagesdoi:10.5402/2012/737206

    Research ArticleComparing Numerical Methods for SolvingTime-Fractional Reaction-Diffusion Equations

    Veyis Turut1 and Nuran Guzel2

    1 Department of Mathematics, Faculty of Arts and Sciences, Batman University, 72100 Batman, Turkey2 Department of Mathematics, Faculty of Arts and Sciences, Yldz Technical University, 34220 Istanbul,Turkey

    Correspondence should be addressed to Nuran Guzel, nguzel@yildiz.edu.tr

    Received 7 March 2012; Accepted 29 April 2012

    Academic Editors: G. Schimperna and W. Shen

    Copyright q 2012 V. Turut and N. Guzel. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

    Multivariate Pade approximation MPA is applied to numerically approximate the solutionsof time-fractional reaction-diffusion equations, and the numerical results are compared withsolutions obtained by the generalized differential transform method GDTM. The fractionalderivatives are described in the Caputo sense. Two illustrative examples are given to demonstratethe effectiveness of the multivariate Pade approximation MPA. The results reveal that themultivariate Pade approximation MPA is very effective and convenient for solving time-fractional reaction-diffusion equations.

    1. Introduction

    The fractional calculus and fractional differential equations have recently become increas-ingly important topics in the literature of engineering, science, and applied mathematics.Application areas include viscoelasticity, electromagnetics, heat conduction, control theory,and diffusion 14. Reaction-diffusion equations are commonly used to model the growthand spreading of biological species. A fractional reaction-diffusion equation FRDE can bederived from a continuous-time random walk model when the transport is dispersive 5 ora continuous-time randomwalk model with temporal memory and sources 6. The topic hasreceived a great deal of attention recently, for example, in systems biology 7, chemistry, andbiochemistry applications 8.

    One of the time-fractional reaction-diffusion equations is the time-fractional Fisherequation. It was originally proposed by Fisher 9 as a model for the spatial and temporalpropagation of a virile gene in an infinite medium. It is encountered in chemical kinetics10, flame propagation 11, autocatalytic chemical reaction 12, nuclear reactor theory 13,neurophysiology 14, and branching Brownian motion process 15.

  • 2 ISRN Mathematical Analysis

    Another time-fractional reaction-diffusion equation is the time-fractional Fitzhugh-Nagumo equation. It is an important nonlinear reaction-diffusion equation and usually usedto model the transmission of nerve impulses 16, 17; it is also used in circuit theory, biology,and the area of population genetics 18 as mathematical models.

    The generalized differential transform method GDTM was presented by 1921.This method is based on differential transformmethod DTM 2225; the DTM introduces apromising approach for many applications in various domains of science. By using the DTM,a truncated series solution is obtained. This series solution does not exhibit the real behaviorsof the problem but gives a good approximation to the true solution in a very small region.Odibat et al. 26 proposed a reliable algorithm of the DTM. The new algorithm acceleratesthe convergence of the series solution over a large region and improves the accuracy of theDTM. The validity of themodified technique is varied through illustrative examples of Lotka-Volterra, Chen, and Lorenz systems. The generalized differential transform method GDTMhas been applied to differential equations of fractional order in 1921, 27.

    In the literature, the univariate Pade approximation has been used to obtainapproximate solutions of fractional order 28, 29. So the objective of the this paper is toshow the application of the multivariate Pade approximation MPA to provide approximatesolutions for time-fractional diffusion-reaction equations and to make comparison with thegeneralized differential transform method GDTM.

    2. Multivariate Pade Approximation

    The principles and theory of the multivariate Pade approximation and its applicability forvarious of differential equations are given in 3040. Consider the bivariate function fx, ywith Taylor series development

    f(x, y)

    i,j0

    cijxiyj 2.1

    around the origin. We know that a solution of univariate Pade approximation problem for

    fx

    i0

    cixi 2.2

    is given by

    px

    m

    i0cix

    i xm1

    i0cix

    i xnmn

    i0cix

    i

    cm1 cm cm1n...

    .... . .

    ...cmn cmn1 cm

    ,

    qx

    1 x xncm1 cm cm1n...

    .... . .

    ...cmn cmn1 cm

    .

    2.3

  • ISRN Mathematical Analysis 3

    Let us now multiply jth row in px and qx by xjm1 j 2, ..., n 1 and afterwardsdivide jth column in px and qx by xj1 j 2, ..., n 1. This results in a multiplication ofnumerator and denominator by xmn. Having done so, we get

    pxqx

    mi0 cix

    i m1i0 cix

    i mni0 cixicm1x

    m1 cmxm cm1nxm1n

    ......

    . . ....

    cmnxmn cmn1xmn1 cmxm

    1 1 1cm1x

    m1 cmxm cm1nxm1n

    ......

    . . ....

    cmnxmn cmn1xmn1 cmxm

    2.4

    if D detDm,n / 0.This quotient of determinants can also immediately be written down for a bivariate

    function fx, y. The sumk

    i0 cixi will be replaced with kth partial sum of the Taylor series

    development of fx, y and the expression ckxk by an expression that contains all the termsof degree k in fx, y. Hereby, a bivariate term cijxiyj is said to be of degree i j. If we define

    p(x, y)

    m

    ij0cijx

    iyjm1

    ij0cijx

    iyj mn

    ij0cijx

    iyj

    ijm1cijx

    iyj

    ijmcijx

    iyj ijm1n

    cijxiyj

    ......

    . . ....

    ijmncijx

    iyj

    ijmn1cijx

    iyj ijm

    cijxiyj

    ,

    q(x, y)

    1 1 1

    ijm1cijx

    iyj

    ijmcijx

    iyj ijm1n

    cijxiyj

    ......

    . . ....

    ijmncijx

    iyj

    ijmn1cijx

    iyj ijm

    cijxiyj

    ,

    2.5

    then it is easy to see that px, y and qx, y are of the form

    p(x, y)

    mnm

    ijmn

    aijxiyj , q

    (x, y)

    mnn

    ijmn

    bijxiyj . 2.6

    We know that px, y and qx, y are called Pade equations 30. So the multivariate Padeapproximant of order m,n for fx, y is defined as,

    rm,n(x, y)

    p(x, y)

    q(x, y) . 2.7

  • 4 ISRN Mathematical Analysis

    3. Generalized Differential Transform Method

    The fractional derivatives are described in the Caputo sense which are defined in 41 as

    Dfx JmDmfx 1

    m x

    0x tm1fmtdt, 3.1

    for m 1 < m, m N, x > 0; for m to be the smallest integer that exceeds , the Caputotime-fractional derivative operator of order > 0 is defined as

    Dt ux, t ux, t

    t

    1m

    t

    0t m1

    mux, m

    d, for m 1 < < m,

    mux, ttm

    , for m N.

    3.2

    The basic definitions and fundamental operations of generalized differential transformmethod are defined in 1921 as follows.

    Definition 3.1. The generalized differential transform of the function ux, y is given asfollows:

    U,k, h 1

    k 1(h 1

    )[(Dx0)k(

    Dy0

    )h]

    x0,y0, 3.3

    where Dx0k Dx0 Dx0 Dx0 .

    Definition 3.2. The generalized differential inverse transform of U,k, h is defined asfollows:

    u(x, y)

    k0

    h0

    U,k, hx x0k(y y0

    )h. 3.4

    The fundamental operations of generalized differential transformmethod are listed in Table 1see 1921.

    4. Numerical Experiments

    In this section, two methods, GDTM and MPA, will be illustrated by two examples, thetime-fractional Fisher equation and the time-fractional Fitzhugh-Nagumo equation. All thenumerical results are calculated by using the software Maple12. The general model ofreaction-diffusion equations is

    u

    t D

    2u

    x2 fu, 0 < 1, t > 0, x , 4.1

    where D is the diffusion coefficient, and fu is a nonlinear function representing reactionkinetics.

  • ISRN Mathematical Analysis 5

    Table

    1:The

    operations

    oftheGDTM.

    Origina

    lfun

    ctions

    Tran

    sformed

    func

    tion

    sux,y

    x,y

    w

    x,y

    U

    ,k,hV,k,hW

    ,k,h

    ux,y

    x,y

    U

    ,k,hV,k,h

    ux,y

    D

    x0x,y

    U

    ,k,h

    k

    1

    1

    k1

    V,k

    1,h,0