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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 737206, 28 pages
http://dx.doi.org/10.5402/2012/737206
Research Article

Comparing Numerical Methods for Solving Time-Fractional Reaction-Diffusion Equations

1Department of Mathematics, Faculty of Arts and Sciences, Batman University, 72100 Batman, Turkey
2Department of Mathematics, Faculty of Arts and Sciences, Yıldız Technical University, 34220 İstanbul, Turkey

Received 7 March 2012; Accepted 29 April 2012

Academic Editors: G. Schimperna and W. Shen

Copyright © 2012 Veyis Turut and Nuran Güzel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Multivariate Padé approximation (MPA) is applied to numerically approximate the solutions of time-fractional reaction-diffusion equations, and the numerical results are compared with solutions obtained by the generalized differential transform method (GDTM). The fractional derivatives are described in the Caputo sense. Two illustrative examples are given to demonstrate the effectiveness of the multivariate Padé approximation (MPA). The results reveal that the multivariate Padé 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 increasingly 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 growth and spreading of biological species. A fractional reaction-diffusion equation (FRDE) can be derived from a continuous-time random walk model when the transport is dispersive [5] or a continuous-time random walk model with temporal memory and sources [6]. The topic has received a great deal of attention recently, for example, in systems biology [7], chemistry, and biochemistry applications [8].

One of the time-fractional reaction-diffusion equations is the time-fractional Fisher equation. It was originally proposed by Fisher [9] as a model for the spatial and temporal propagation of a virile gene in an infinite medium. It is encountered in chemical kinetics [10], flame propagation [11], autocatalytic chemical reaction [12], nuclear reactor theory [13], neurophysiology [14], and branching Brownian motion process [15].

Another time-fractional reaction-diffusion equation is the time-fractional Fitzhugh-Nagumo equation. It is an important nonlinear reaction-diffusion equation and usually used to 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 transform method (DTM) [2225]; the DTM introduces a promising 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 behaviors of 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 accelerates the convergence of the series solution over a large region and improves the accuracy of the DTM. The validity of the modified technique is varied through illustrative examples of Lotka-Volterra, Chen, and Lorenz systems. The generalized differential transform method (GDTM) has been applied to differential equations of fractional order in [1921, 27].

In the literature, the univariate Padé approximation has been used to obtain approximate solutions of fractional order [28, 29]. So the objective of the this paper is to show the application of the multivariate Padé approximation (MPA) to provide approximate solutions for time-fractional diffusion-reaction equations and to make comparison with the generalized differential transform method (GDTM).

2. Multivariate Padé Approximation

The principles and theory of the multivariate Padé approximation and its applicability for various of differential equations are given in [3040]. Consider the bivariate function 𝑓(𝑥,𝑦) with Taylor series development 𝑓(𝑥,𝑦)=𝑖,𝑗=0𝑐𝑖𝑗𝑥𝑖𝑦𝑗(2.1) around the origin. We know that a solution of univariate Padé approximation problem for 𝑓(𝑥)=𝑖=0𝑐𝑖𝑥𝑖(2.2) is given by |||||||||||||𝑝(𝑥)=𝑚𝑖=0𝑐𝑖𝑥𝑖𝑥𝑚1𝑖=0𝑐𝑖𝑥𝑖𝑥𝑛𝑚𝑛𝑖=0𝑐𝑖𝑥𝑖𝑐𝑚+1𝑐𝑚𝑐𝑚+1𝑛𝑐𝑚+𝑛𝑐𝑚+𝑛1𝑐𝑚|||||||||||||,||||||||||||𝑞(𝑥)=1𝑥𝑥𝑛𝑐𝑚+1𝑐𝑚𝑐𝑚+1𝑛𝑐𝑚+𝑛𝑐𝑚+𝑛1𝑐𝑚||||||||||||.(2.3) Let us now multiply 𝑗th row in 𝑝(𝑥) and 𝑞(𝑥) by 𝑥𝑗+𝑚1(𝑗=2,...,𝑛+1) and afterwards divide 𝑗th column in 𝑝(𝑥) and 𝑞(𝑥) by 𝑥𝑗1(𝑗=2,...,𝑛+1). This results in a multiplication of numerator and denominator by 𝑥𝑚𝑛. Having done so, we get 𝑝(𝑥)=|||||𝑞(𝑥)𝑚𝑖=0𝑐𝑖𝑥𝑖𝑚1𝑖=0𝑐𝑖𝑥𝑖𝑚𝑛𝑖=0𝑐𝑖𝑥𝑖𝑐𝑚+1𝑥𝑚+1𝑐𝑚𝑥𝑚𝑐𝑚+1𝑛𝑥𝑚+1𝑛𝑐𝑚+𝑛𝑥𝑚+𝑛𝑐𝑚+𝑛1𝑥𝑚+𝑛1𝑐𝑚𝑥𝑚||||||||||𝑐111𝑚+1𝑥𝑚+1𝑐𝑚𝑥𝑚𝑐𝑚+1𝑛𝑥𝑚+1𝑛𝑐𝑚+𝑛𝑥𝑚+𝑛𝑐𝑚+𝑛1𝑥𝑚+𝑛1𝑐𝑚𝑥𝑚|||||(2.4) if (𝐷=det𝐷𝑚,𝑛0).

This quotient of determinants can also immediately be written down for a bivariate function 𝑓(𝑥,𝑦). The sum 𝑘𝑖=0𝑐𝑖𝑥𝑖 will be replaced with 𝑘th partial sum of the Taylor series development of 𝑓(𝑥,𝑦) and the expression 𝑐𝑘𝑥𝑘 by an expression that contains all the terms of degree 𝑘 in 𝑓(𝑥,𝑦). Hereby, a bivariate term 𝑐𝑖𝑗𝑥𝑖𝑦𝑗 is said to be of degree 𝑖+𝑗. If we define |||||||||||||||𝑝(𝑥,𝑦)=𝑚𝑖+𝑗=0𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑚1𝑖+𝑗=0𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑚𝑛𝑖+𝑗=0𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚+1𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚+1𝑛𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚+𝑛𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚+𝑛1𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚𝑐𝑖𝑗𝑥𝑖𝑦𝑗|||||||||||||||,|||||||||||||𝑞(𝑥,𝑦)=111𝑖+𝑗=𝑚+1𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚+1𝑛𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚+𝑛𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚+𝑛1𝑐𝑖𝑗𝑥𝑖𝑦𝑗𝑖+𝑗=𝑚𝑐𝑖𝑗𝑥𝑖𝑦𝑗|||||||||||||,(2.5) then it is easy to see that 𝑝(𝑥,𝑦) and 𝑞(𝑥,𝑦) are of the form 𝑝(𝑥,𝑦)=𝑚𝑛+𝑚𝑖+𝑗=𝑚𝑛𝑎𝑖𝑗𝑥𝑖𝑦𝑗,𝑞(𝑥,𝑦)=𝑚𝑛+𝑛𝑖+𝑗=𝑚𝑛𝑏𝑖𝑗𝑥𝑖𝑦𝑗.(2.6) We know that 𝑝(𝑥,𝑦) and 𝑞(𝑥,𝑦) are called Padé equations [30]. So the multivariate Padé approximant of order (𝑚,𝑛) for 𝑓(𝑥,𝑦) is defined as, 𝑟𝑚,𝑛(𝑥,𝑦)=𝑝(𝑥,𝑦).𝑞(𝑥,𝑦)(2.7)

3. Generalized Differential Transform Method

The fractional derivatives are described in the Caputo sense which are defined in [41] as 𝐷𝛼𝑓(𝑥)=𝐽𝑚𝛼𝐷𝑚1𝑓(𝑥)=Γ(𝑚𝛼)𝑥0(𝑥𝑡)𝑚𝛼1𝑓𝑚(𝑡)𝑑𝑡,(3.1) for 𝑚1<𝛼𝑚, 𝑚, 𝑥>0; for 𝑚 to be the smallest integer that exceeds 𝛼, the Caputo time-fractional derivative operator of order 𝛼>0 is defined as 𝐷𝛼𝑡𝜕𝑢(𝑥,𝑡)=𝛼𝑢(𝑥,𝑡)𝜕𝑡𝛼=1Γ(𝑚𝛼)𝑡0(𝑡𝜏)𝑚𝛼1𝜕𝑚𝑢(𝑥,𝜏)𝜕𝜏𝑚𝜕𝑑𝜏,for𝑚1<𝛼<𝑚,𝑚𝑢(𝑥,𝑡)𝜕𝑡𝑚,for𝛼=𝑚.(3.2) The basic definitions and fundamental operations of generalized differential transform method are defined in [1921] as follows.

Definition 3.1. The generalized differential transform of the function 𝑢(𝑥,𝑦) is given as follows: 𝑈𝛼,𝛽1(𝑘,)=𝐷Γ(𝛼𝑘+1)Γ(𝛽+1)𝛼𝑥0𝑘𝐷𝛽𝑦0(𝑥0,𝑦0),(3.3) where (𝐷𝛼𝑥0)𝑘=𝐷𝛼𝑥0𝐷𝛼𝑥0𝐷𝛼𝑥0.

Definition 3.2. The generalized differential inverse transform of 𝑈𝛼,𝛽(𝑘,) is defined as follows: 𝑢(𝑥,𝑦)=𝑘=0=0𝑈𝛼,𝛽(𝑘,)𝑥𝑥0𝑘𝛼𝑦𝑦0𝛽.(3.4) The fundamental operations of generalized differential transform method are listed in Table 1 (see [1921]).

tab1
Table 1: The operations of the GDTM.

4. Numerical Experiments

In this section, two methods, GDTM and MPA, will be illustrated by two examples, the time-fractional Fisher equation and the time-fractional Fitzhugh-Nagumo equation. All the numerical results are calculated by using the software Maple12. The general model of reaction-diffusion equations is 𝜕𝛼𝑢𝜕𝑡𝛼𝜕=𝐷2𝑢𝜕𝑥2+𝑓(𝑢),0<𝛼1,𝑡>0,𝑥,(4.1) where 𝐷 is the diffusion coefficient, and 𝑓(𝑢) is a nonlinear function representing reaction kinetics.

Example 4.1. Let us consider (4.1) with 𝑓(𝑢)=6𝑢(1𝑢), then we have the time-fractional Fisher equation [27] 𝐷𝛼𝑡𝑢=𝐷2𝑥𝑢+6𝑢(1𝑢),0<𝛼1,𝑡>0,𝑥,(4.2) subject to the initial condition 1𝑢(𝑥,0)=(1+𝑒𝑥)2.(4.3) Selecting 𝛽=1 and applying the generalized differential transform of (4.2), using the related definitions in Table 1, Rida et al. [27] solved as it follows: Γ(𝛼(+1)+1)𝑈Γ(𝛼+1)𝛼,1(𝑘,+1)=(𝑘+1)(𝑘+2)𝑈𝛼,1(𝑘+2,)+6𝑈𝛼,1(𝑘,)6𝑘𝑟=0𝑠=0𝑈𝛼,1(𝑟,𝑠)𝑈𝛼,1(𝑘𝑟,𝑠),(4.4) that is, 𝑈𝛼,1=(𝑘,+1)Γ(𝛼(+1)+1)Γ(𝛼+1)(𝑘+1)(𝑘+2)𝑈𝛼,1(𝑘+2,)+6𝑈𝛼,1(𝑘,)6𝑘𝑟=0𝑠=0𝑈𝛼,1(𝑟,𝑠)𝑈𝛼,1.(𝑘𝑟,𝑠)(4.5) By equating the series form of (4.3) with (3.4), the initial transformation coefficients 𝑈𝛼,1(𝑘,0), 𝑘=0,1,2, can be obtained as follows: 𝑈𝛼,11(0,0)=4,𝑈𝛼,11(1,0)=4,𝑈𝛼,11(2,0)=,𝑈16𝛼,11(3,0)=48,𝑈𝛼,11(4,0)=.96(4.6) By applying (4.6) into (4.5), some values of 𝑈𝛼,1(𝑘,) can be obtained as given in Table 1. Consequent substitution of all 𝑈𝛼,1(𝑘,) into (3.4) and after some manipulations, the series from solutions of (4.2) and (4.3) has been obtained in [27] as 1𝑢(𝑥,𝑡)=4+5𝑡4Γ(𝛼+1)𝛼+25𝑡8Γ(2𝛼+1)2𝛼+1+45𝑡8Γ(𝛼+1)𝛼+25𝑡8Γ(2𝛼+1)2𝛼𝑥+1+516𝑡16Γ(𝛼+1)𝛼25𝑡8Γ(2𝛼+1)2𝛼𝑥+2+1548𝑡24Γ(𝛼+1)𝛼25𝑡24Γ(2𝛼+1)2𝛼𝑥+3+1+596𝑡96Γ(𝛼+1)𝛼+425𝑡384Γ(2𝛼+1)2𝛼𝑥+4.(4.7)𝑢(𝑥,𝑡) can be written in the form: 1𝑢(𝑥,𝑡)=4141𝑥+𝑥4831𝑥964+5+4585𝑥𝑥1625𝑥243+5𝑥964𝑡+𝛼Γ+(𝛼+1)258+258𝑥258𝑥225𝑥243+425𝑥3844𝑡+2𝛼.Γ(2𝛼+1)(4.8) The exact solution of (4.2), for the special case 𝛼=1.0, is given in [27] as 1𝑢(𝑥,𝑡)=1+𝑒𝑥5𝑡2.(4.9)

We have the generalized differential transform method solution for the time-fractional Fisher equation (4.2) (when =1.0 ) as 𝑢(𝑥,𝑡)=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+(1.2500000000.6250000000𝑥0.3125000000𝑥20.2083333333𝑥3+0.05208333333𝑥4)𝑡+0.5000000000(3.125000000+3.125000000𝑥3.125000000𝑥21.401666667𝑥3+1.106770833𝑥4)𝑡2,(4.10)=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.250000000𝑡0.6250000000𝑥𝑡0.3125000000𝑥2𝑡0.2083333333𝑥3𝑡+0.05208333333𝑥4𝑡+1.562500000𝑡2+1.562500000𝑥𝑡21.562500000𝑥2𝑡20.5208333335𝑥3𝑡2+0.5533854165𝑥4𝑡2,(4.11) and let 𝐴=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.250000000𝑡0.6250000000𝑥𝑡0.3125000000𝑥2𝑡0.2083333333𝑥3𝑡+1.562500000𝑡2+1.562500000𝑥𝑡21.562500000𝑥2𝑡2,𝐵=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥3+1.250000000𝑡0.6250000000𝑥𝑡0.3125000000𝑥2𝑡+1.562500000𝑡2+1.562500000𝑥𝑡2,𝐶=0.25000000000.2500000000𝑥0.06250000000𝑥2+1.250000000𝑡0.6250000000𝑥𝑡+1.562500000𝑡2.(4.12) Then let us calculate the approximate solution of (4.10) for 𝑚=4 and 𝑛=2 by using multivariate Padé approximation. To obtain multivariate Padé equations of (4.10) for 𝑚=4 and 𝑛=2, we use (2.5). By using (2.5), we obtain =|||||||||𝑝(𝑥,𝑡)𝐴𝐵𝐶0.05208333333𝑥4𝑡0.5208333335𝑥3𝑡2𝒜0.5533854165𝑥4𝑡20.05208333333𝑥4𝑡0.5208333335𝑥3𝑡2𝒜|||||||||=0.5533854165𝑥40.0001225490198𝑥5𝑡0.03063725491𝑡5𝑥4+0.3082873774𝑡4𝑥3+0.02037377449𝑡3𝑥4+0.003604983663𝑡2𝑥5+0.00004901960791𝑥50.00001225490198𝑥61.470588235𝑡40.3676470590𝑡4𝑥2+0.002757352939𝑡3𝑥60.04289215684𝑡3𝑥3+0.09803921566𝑡3𝑥20.0001633986928𝑡𝑥60.2573529412𝑡3𝑥+0.1608455885𝑡4𝑥+0.0001225490196𝑥7𝑡20.01072303921𝑥5𝑡40.06318933824𝑡5𝑥50.001914828416𝑡5𝑥6+0.00004084967320𝑥8𝑡20.002323325162𝑥7𝑡3+0.02221200981𝑥6𝑡40.00002042483661𝑥9𝑡20.0006382761434𝑥8𝑡3+0.007531658495𝑥7𝑡40.4084967326×105𝑥7+0.2042483663×105𝑥89.191176472𝑡6+5.840226718𝑡6𝑥29.334788603𝑡6𝑥7.352941178𝑡5+4.049862133𝑡5𝑥32.202052696𝑡5𝑥2+1.953125001𝑡5𝑥+0.3498391544𝑡4𝑥4+0.05895118467𝑡3𝑥5+0.002024611930𝑡2𝑥6+0.0001021241832𝑡𝑥70.00004901960791𝑥40.006587009808𝑡2𝑥4+0.02205882354𝑡2𝑥3+0.001470588237𝑡𝑥40.03431372552𝑡2𝑥20.001960784315𝑡𝑥3),|||||||||𝑞(𝑥,𝑡)=1110.05208333333𝑥4𝑡0.5208333335𝑥3𝑡2𝒜0.5533854165𝑥4𝑡20.05208333333𝑥4𝑡0.5208333335𝑥3𝑡2𝒜|||||||||=0.5533854165𝑥41.029411764𝑡3𝑥+5.882352942𝑡4+0.091911764𝑡4𝑥0.001960784314𝑡2𝑥5+0.001960784314𝑡3𝑥4+0.02083333333𝑡3𝑥5+0.01531862746𝑡2𝑥4+0.1102941176𝑡3𝑥3+2.052696079𝑡4𝑥2+0.007843137258𝑡𝑥3+0.1372549020𝑡2𝑥2+0.009803921564𝑡2𝑥30.0490196080𝑡3𝑥2+0.0001960784315𝑥4+0.0009803921572𝑡𝑥4,(4.13) where 𝒜 denotes 0.01041666667𝑥40.2083333333𝑥3𝑡1.562500000𝑥2𝑡2, and denotes 0.02083333333𝑥30.3125000000𝑥2𝑡+1.562500000𝑥𝑡2. So the multivariate Padé approximation is of order (4,2) for (4.10), that is, []4,2(𝑥,𝑡)=(0.0001225490198𝑥5𝑡0.03063725491𝑡5𝑥4+0.3082873774𝑡4𝑥3+0.02037377449𝑡3𝑥4+0.003604983663𝑡2𝑥5+0.00004901960791𝑥50.00001225490198𝑥61.470588235𝑡40.3676470590𝑡4𝑥2+0.002757352939𝑡3𝑥60.04289215684𝑡3𝑥3+0.09803921566𝑡3𝑥20.0001633986928𝑡𝑥60.2573529412𝑡3𝑥+0.1608455885𝑡4𝑥+0.0001225490196𝑥7𝑡20.01072303921𝑥5𝑡40.06318933824𝑡5𝑥50.001914828416𝑡5𝑥6+0.00004084967320𝑥8𝑡20.002323325162𝑥7𝑡3+0.02221200981𝑥6𝑡40.00002042483661𝑥9𝑡20.0006382761434𝑥8𝑡3+0.007531658495𝑥7𝑡40.4084967326×105𝑥7+0.2042483663×105𝑥89.191176472𝑡6+5.840226718𝑡6𝑥29.334788603𝑡6𝑥7.352941178𝑡5+4.049862133𝑡5𝑥32.202052696𝑡5𝑥2+1.953125001𝑡5𝑥+0.3498391544𝑡4𝑥4+0.05895118467𝑡3𝑥5+0.002024611930𝑡2𝑥6+0.0001021241832𝑡𝑥70.00004901960791𝑥40.006587009808𝑡2𝑥4+0.02205882354𝑡2𝑥3+0.001470588237𝑡𝑥40.03431372552𝑡2𝑥20.001960784315𝑡𝑥3)/(1.029411764𝑡3𝑥+5.882352942𝑡4+0.091911764𝑡4𝑥0.001960784314𝑡2𝑥5+0.001960784314𝑡3𝑥4+0.02083333333𝑡3𝑥5+0.01531862746𝑡2𝑥4+0.1102941176𝑡3𝑥3+2.052696079𝑡4𝑥2+0.007843137258𝑡𝑥3+0.1372549020𝑡2𝑥2+0.009803921564𝑡2𝑥30.0490196080𝑡3𝑥2+0.0001960784315𝑥4+0.0009803921572𝑡𝑥4).(4.14) The generalized differential transform method gives the solution for the time-fractional Fisher equation (4.2) (when 𝛼=0.5) which is given by 𝑢(𝑥,𝑡)=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.128379167(1.2500000000.6250000000𝑥0.3125000000𝑥20.2083333333𝑥3+0.05208333333𝑥4)𝑡0.5+(3.125000000+3.125000000𝑥3.125000000𝑥21.401666667𝑥3+1.106770833𝑥4)𝑡.(4.15) For simplicity, let 𝑡1/2=𝑎, then 𝑢(𝑥,𝑡)=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.128379167(1.2500000000.6250000000𝑥0.3125000000𝑥20.2083333333𝑥3+0.05208333333𝑥4)𝑎+(3.125000000+3.125000000𝑥3.125000000𝑥21.401666667𝑥3+1.106770833𝑥4)𝑎2,(4.16)=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.410473959𝑎0.7052369794𝑎𝑥0.3526184897𝑎𝑥20.2350789931𝑎𝑥3+0.05876974828𝑎𝑥4+3.125000000𝑎2+3.125000000𝑎2𝑥3.125000000𝑎2𝑥21.401666667𝑎2𝑥3+1.106770833𝑎2𝑥4,(4.17) and let 𝐸=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.410473959𝑎0.7052369794𝑎𝑥0.3526184897𝑎𝑥20.2350789931𝑎𝑥3+3.125000000𝑎2+3.125000000𝑎2𝑥3.125000000𝑎2𝑥2,𝐹=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥3+1.410473959𝑎0.7052369794𝑎𝑥0.3526184897𝑎𝑥2+3.125000000𝑎2+3.125000000𝑎2𝑥,𝐺=0.25000000000.2500000000𝑥0.06250000000𝑥2+1.410473959𝑎0.7052369794𝑎𝑥+3.125000000𝑎2.(4.18) Then, using (2.5) to calculate the multivariate Padé equations for (4.16), we get =|||||||||𝑝(𝑥,𝑎)𝐸𝐹𝐺0.05876974828𝑎𝑥41.401666667𝑎2𝑥3𝒞𝒟1.106770833𝑎2𝑥40.05876974828𝑎𝑥41.401666667𝑎2𝑥3𝒞|||||||||=1.106770833𝑥4(1.724963655𝑎4𝑥+0.00055331270431𝑎𝑥4+0.00002450980395𝑥50.6127450986×105𝑥60.2042483661×105𝑥7+0.1021241831×105𝑥8+0.02621254569𝑎2𝑥30.005445232711𝑎2𝑥40.03677161152𝑎2𝑥20.0008296905640𝑎𝑥30.2074226410𝑎3𝑥+0.000104497505𝑎2𝑥5+0.07657717919𝑎3𝑥4+0.0659779402𝑎4𝑥30.1247554646𝑎3𝑥31.014826625𝑎4𝑥2+0.1×1012𝑥5𝑎+0.1019438376𝑎3𝑥20.00006914088034𝑎𝑥62.941176471𝑎436.76470589𝑎6+23.36090687𝑎6𝑥237.33915441𝑎6𝑥16.59381128𝑎5+9.204379692𝑎5𝑥32.830454788𝑎5𝑥2+5.444844332𝑎5𝑥+0.3449658071𝑎4𝑥4+0.07042023202𝑎3𝑥5+0.001477816147𝑎2𝑥6+0.00004609392028𝑎𝑥70.00002450980395𝑥4=|||||||||).𝑞(𝑥,𝑎)1110.05876974828𝑎𝑥41.401666667𝑎2𝑥3𝒞𝒟1.106770833𝑎2𝑥40.05876974828𝑎𝑥41.401666667𝑎2𝑥3𝒞|||||||||=1.106770833𝑥4(11.76470588𝑎4+0.003318762259𝑎𝑥3+0.00009803921577𝑥4+0.1470864461𝑎2𝑥2+0.02351215238𝑎2𝑥3+0.01353735183𝑎2𝑥4+0.1244535845𝑎3𝑥3+4.105392158𝑎4𝑥20.4079311941𝑎3𝑥2+0.183823528𝑎4𝑥+0.0005531270429𝑎𝑥4+0.8296905634𝑎3𝑥).(4.19) where 𝒞 is 0.01041666667𝑥40.2350789931𝑎𝑥33.125000000𝑎2𝑥2, and 𝒟 is 0.02083333333𝑥30.3526184897𝑎𝑥20.3125000000𝑎2𝑥 recalling that 𝑡1/2=𝑎, we get multivariate Padé approximation of order (4,2) for (4.15), that is, []4,2(𝑥,𝑡)=(1.724963655𝑡2𝑥+0.00055331270431𝑡𝑥4+0.00002450980395𝑥50.6127450986×105𝑥60.2042483661×105𝑥7+0.1021241831×105𝑥8+0.02621254569𝑡𝑥30.005445232711𝑡𝑥40.03677161152𝑡𝑥20.0008296905640𝑡𝑥30.2074226410𝑡3/2𝑥+0.000104497505𝑡𝑥5+0.07657717919𝑡3/2𝑥4+0.0659779402𝑡2𝑥30.1247554646𝑡3/2𝑥31.014826625𝑡2𝑥2+0.1×1012𝑥5𝑡+0.1019438376𝑡3/2𝑥20.00006914088034𝑡𝑥62.941176471𝑡236.76470589𝑡3+23.36090687𝑡3𝑥237.33915441𝑡3𝑥16.59381128𝑡5/2+9.204379692𝑡5/2𝑥32.830454788𝑡5/2𝑥2+5.444844332𝑡5/2𝑥+0.3449658071𝑡2𝑥4+0.07042023202𝑡3/2𝑥5+0.001477816147𝑡𝑥6+0.00004609392028𝑡𝑥70.00002450980395𝑥4)/(11.76470588𝑡2+0.003318762259𝑡𝑥3+0.00009803921577𝑥4+0.1470864461𝑡𝑥2+0.02351215238𝑡𝑥3+0.01353735183𝑡𝑥4+0.1244535845𝑡3/2𝑥3+4.105392158𝑡2𝑥20.4079311941𝑡3/2𝑥2+0.183823528𝑡2𝑥+0.0005531270429𝑡𝑥4+0.8296905634𝑡3/2𝑥).(4.20) The generalized differential transform method gives the solution for the time-fractional Fisher equation (4.2) (when 𝛼=0.75) which is given by 𝑢(𝑥,𝑡)=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.088065252(1.2500000000.6250000000𝑥0.3125000000𝑥20.2083333333𝑥3+0.05208333333𝑥4)𝑡0.75+0.7522527782(3.125000000+3.125000000𝑥3.125000000𝑥21.401666667𝑥3+1.106770833𝑥4)𝑡1.50(4.21)=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.360081565𝑡0.750.6800407825𝑡0.75𝑥0.3400203912𝑡0.75𝑥20.22668002608𝑡0.75𝑥3+0.05667006520𝑡0.75𝑥4+2.350789932𝑡1.50+2.350789932𝑡1.50𝑥2.350789932𝑡1.50𝑥20.7835966442𝑡1.50𝑥3+0.8325714340𝑡1.50𝑥4.(4.22) For simplicity, let 𝑡1/4=𝑎, then 𝑢(𝑥,𝑎)=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.360081565𝑎30.6800407825𝑎3𝑥0.3400203912𝑎3𝑥20.22668002608𝑎3𝑥3+0.05667006520𝑎3𝑥4+2.350789932𝑎6+2.350789932𝑎6𝑥2.350789932𝑎6𝑥20.7835966442𝑎6𝑥3+0.8325714340𝑎6𝑥4,(4.23) and let 𝐻=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.360081565𝑎30.6800407825𝑎3𝑥0.3400203912𝑎3𝑥20.22668002608𝑎3𝑥3+0.05667006520𝑎3𝑥42.350789932𝑎6+2.350789932𝑎6𝑥,𝐾=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.360081565𝑎30.6800407825𝑎3𝑥0.3400203912𝑎3𝑥20.22668002608𝑎3𝑥3+2.350789932𝑎6,𝐿=0.25000000000.2500000000𝑥0.06250000000𝑥2+0.02083333333𝑥30.01041666667𝑥4+1.360081565𝑎30.6800407825𝑎3𝑥0.3400203912𝑎3𝑥2.(4.24) Then, using (2.5) to calculate the multivariate Padé equations for (4.23), we get |||||||||𝑝(𝑥,𝑎)=𝐻𝐾𝐿2.350789932𝑎6𝑥20.7835966442𝑎6𝑥32.350789932𝑎6𝑥2|||||||||=1.842071102𝑥2(0.55703900549𝑎6𝑥4+14.10473959𝑎120.9467234329𝑎6𝑥30.00001816058365𝑥10+0.00003632116729𝑥9+0.0004358540080𝑥60.0004358540080𝑥7+0.0001089635019𝑥8+0.07834715868𝑎3𝑥40.03616022708𝑎3𝑥3+1.500000000𝑎6+0.8750000002𝑎6𝑥20.9999999998𝑎6𝑥0.04520028386𝑎3𝑥5+2.040122348𝑎9𝑥2+18.80631945𝑎12𝑥+0.0000987995000𝑎3𝑥105.100305867𝑎9𝑥3+0.00387786414𝑎3𝑥60.001848360442𝑎3𝑥80.0006463106887𝑎3𝑥9+0.005343335889𝑎3𝑥70.05054699319𝑎6𝑥7+0.0297127611𝑎6𝑥50.07855231035𝑎6𝑥61.416751630𝑎9𝑥41.360081565𝑎9𝑥+8.160489388𝑎9)𝑎6|||||||||𝑞(𝑥,𝑎)=1112.350789932𝑎6𝑥20.7835966442𝑎6𝑥32.350789932𝑎6𝑥2|||||||||=1.842071102𝑥2(0.1446409084𝑎3𝑥3+0.1687477264𝑎3𝑥4+5.999999998𝑎6+1.999999999𝑎6𝑥+3.999999999𝑎6𝑥2+0.001743416031𝑥6+0.02410681806𝑎3𝑥5)𝑎6,(4.25) where denotes 0.05667006520𝑎3𝑥4+2.350789932𝑎6𝑥, and denotes 2.350789932𝑎60.22668002608𝑎3𝑥3; recalling that 𝑡1/4=𝑎, we get multivariate Padé approximation of order (7,2) for (4.21), that is, []7,2(𝑥,𝑡)=(0.55703900549𝑡3/2𝑥4+14.10473959𝑡30.9467234329𝑡3/2𝑥30.00001816058365𝑥10+0.00003632116729𝑥9+0.0004358540080𝑥60.0004358540080𝑥7+0.0001089635019𝑥8+0.07834715868𝑡3/4𝑥40.03616022708𝑡3/4𝑥3+1.500000000𝑡3/2+0.8750000002𝑡3/2𝑥20.9999999998𝑡3/2𝑥0.04520028386𝑡3/4𝑥5+2.040122348𝑡9/4𝑥2+18.80631945𝑡3𝑥+0.0000987995000𝑡3/4𝑥105.100305867𝑡9/4𝑥3+0.00387786414𝑡3/4𝑥60.001848360442𝑡3/4𝑥80.0006463106887𝑡3/4𝑥9+0.005343335889𝑡3/4𝑥70.05054699319𝑡3/2𝑥7+0.0297127611𝑡3/2𝑥50.07855231035𝑡3/2𝑥61.416751630𝑡9/4𝑥41.360081565𝑡9/4𝑥+8.160489388𝑡9/4)/(0.1446409084𝑡3/4𝑥3+0.1687477264𝑡3/4𝑥4+5.999999998𝑡3/2+1.999999999𝑡3/2𝑥+3.999999999𝑡3/2𝑥2+0.001743416031𝑥6+0.02410681806𝑡3/4𝑥5).(4.26) As it is presented above, we obtained multivariate Padé approximations of the generalized differential transform method solution of the time-fractional Fisher equation (4.2) for values of 𝛼=1.0, 𝛼=0.50, and 𝛼=0.75. Table 2 shows the approximate solutions for (4.2) obtained for different values of 𝛼 using the generalized differential transform method (GDTM) and the multivariate padé approximation (MPA). The values of 𝛼=1.0 are the only case for which we know the exact solution 𝑢(𝑥,𝑡)=1/(1+𝑒𝑥5𝑡)2, and the results of multivariate padé approximation (MPA) are in excellent agreement with the exact solution and those obtained by the generalized differential transform method (GDTM).

tab2
Table 2: Numerical values when 𝛼=0.50, 𝛼=0.75, and 𝛼=1.0 for Example 4.1.

Example 4.2. Let us consider (4.1) with 𝑓(𝑢)=𝑢(1𝑢)(𝑢𝜇), then we have the time-fractional Fitzhugh-Nagumo equation [27] 𝐷𝛼𝑡𝑢=𝐷2𝑥𝑢+𝑢(1𝑢)(𝑢𝜇),𝜇>0,0<𝛼1,𝑡>0,𝑥,(4.27) subject to the initial condition 1𝑢(𝑥,0)=1+𝑒𝑥/2.(4.28) Taking the generalized differential transform of (4.27), using the related definitions in Table 1, Rida et al. [27] solved it as follows: Γ(𝛼(+1)+1)𝑈Γ(𝛼+1)𝛼,1(𝑘,+1)=(𝑘+1)(𝑘+2)𝑈𝛼,1(𝑘+2,)𝜇𝑈𝛼,1(𝑘,)+(1+𝜇)𝑘𝑟=0𝑠=0𝑈𝛼,1(𝑟,𝑠)𝑈𝛼,1(𝑘𝑟,𝑠)𝑘𝑟=0𝑘𝑟𝑡=0𝑠=0𝑠𝑝=0𝑈𝛼,1(𝑟,𝑠𝑝)𝑈𝛼,1(𝑡,𝑠)𝑈𝛼,1(𝑘𝑟𝑡,𝑝),(4.29) that is, 𝑈𝛼,1=(𝑘,+1)Γ(𝛼(+1)+1)Γ(𝛼+1)(𝑘+1)(𝑘+2)𝑈𝛼,1(𝑘+2,)𝜇𝑈𝛼,1(𝑘,)+(1+𝜇)𝑘𝑟=0𝑠=0𝑈𝛼,1(𝑟,𝑠)𝑈𝛼,1(𝑘𝑟,𝑠)𝑘𝑟=0𝑘𝑟𝑡=0𝑠=0𝑠𝑝=0𝑈𝛼,1(𝑟,𝑠𝑝)𝑈𝛼,1(𝑡,𝑠)𝑈𝛼,1.(𝑘𝑟𝑡,𝑝)(4.30) By equating the series form of (4.28) with (3.4), the initial transformation coefficients 𝑈𝛼,1(𝑘,0), 𝑘=0,1,2, can be obtained as follows: 𝑈𝛼,11(0,0)=2,𝑈𝛼,11(1,0)=42,𝑈𝛼,1𝑈(2,0)=0,𝛼,11(3,0)=96,𝑈𝛼,1(4,0)=0.(4.31) By applying (4.31) into (4.30), some values of 𝑈𝛼,1(𝑘,) can be obtained as given in Table 1. Consequent substitution of all 𝑈𝛼,1(𝑘,) into (3.4) and after some manipulations, the series from solutions of (4.27) and (4.28) has been obtained in [27] as: 1𝑢(𝑥,𝑡)=2+12𝜇𝑡8Γ(𝛼+1)𝛼+(12𝜇)2𝑡8Γ(2𝛼+1)2𝛼+1+42(12𝜇)232𝑡2Γ(2𝛼+1)2𝛼𝑥++12𝜇𝑡64Γ(𝛼+1)𝛼+(12𝜇)2𝑡64Γ(2𝛼+1)2𝛼𝑥+2+1962+(12𝜇)2192𝑡2Γ(2𝛼+1)2𝛼𝑥+3+12𝜇𝑡768Γ(𝛼+1)𝛼(12𝜇)2𝑡768Γ(2𝛼+1)2𝛼𝑥+4.(4.32)𝑢(𝑥,𝑡) can be written in the form: 1𝑢(𝑥,𝑡)=21421𝑥962𝑥3+119202𝑥5++12𝜇2141𝑥322+1𝑥384417𝑥921606𝑡+𝛼(Γ(𝛼+1)12𝜇)2212+1821𝑥𝑥1621482𝑥3+1𝑥1924𝑡+2𝛼.Γ(2𝛼+1)(4.33) The exact solution of (4.27), for the special case 𝛼=1.0, is given in [27] 1𝑢(𝑥,𝑡)=1+𝑒(1/2)(𝑥+((12𝜇)/2)𝑡).(4.34)

We have the generalized differential transform method solution for the time-fractional Fitzhugh-Nagumo equation (4.27) (when 𝛼=1.0 and 𝜇=0.45) as 𝑢(𝑥,𝑡)=0.50000000000.1767766952𝑥0.007365695635𝑥3+0.0003682847818𝑥5+0.05000000000(0.25000000000.03125000000𝑥2+0.002604166667𝑥40.0001844618056𝑥6)𝑡0.001250000000(0.50000000000.08838834762𝑥0.06250000000𝑥20.01473139128𝑥3+0.005208333333𝑥4)𝑡2(4.35)=0.50000000000.1767766952𝑥0.007365695635𝑥3+0.0003682847818𝑥5+0.01250000000𝑡0.001562500000𝑡𝑥2+0.0001302088884𝑡𝑥40.9223090280×105𝑡𝑥60.0006250000000𝑡2+0.0001104854345𝑡2𝑥+0.00007812500000𝑡2𝑥2+0.00001841423910𝑡