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

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 timefractional reaction-diffusion equations.


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 1-4 . 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 .

Multivariate Padé Approximation
The principles and theory of the multivariate Padé approximation and its applicability for various of differential equations are given in 30-40 . Consider the bivariate function f x, y with Taylor series development   if D det D m,n / 0 . This quotient of determinants can also immediately be written down for a bivariate function f x, y . The sum k i 0 c i x i will be replaced with kth partial sum of the Taylor series development of f x, y and the expression c k x k by an expression that contains all the terms of degree k in f x, y . Hereby, a bivariate term c ij x i y j is said to be of degree i j. If we define

3.2
The basic definitions and fundamental operations of generalized differential transform method are defined in 19-21 as follows.
Definition 3.1. The generalized differential transform of the function u x, y is given as follows: Definition 3.2. The generalized differential inverse transform of U α,β k, h is defined as follows: The fundamental operations of generalized differential transform method are listed in Table 1 see 19-21 .

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 where D is the diffusion coefficient, and f u is a nonlinear function representing reaction kinetics.
subject to the initial condition 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:

4.7
ISRN Mathematical Analysis 7 u x, t can be written in the form:

4.8
The exact solution of 4.2 , for the special case α 1.0, is given in 27 as We have the generalized differential transform method solution for the time-fractional Fisher equation 4.2 when 1.0 as and let

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 So the multivariate Padé approximation is of order 4, 2 for 4.10 , that is,

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 For simplicity, let t 1/2 a, then and let

4.18
Then, using 2.5 to calculate the multivariate Padé equations for 4.16 , we get 3125000000a 2 x recalling that t 1/2 a, we get multivariate Padé approximation of order 4, 2 for 4.15 , that is, The generalized differential transform method gives the solution for the time-fractional Fisher equation 4.2 when α 0.75 which is given by For simplicity, let t 1/4 a, then and let

4.24
Then, using 2.5 to calculate the multivariate Padé equations for 4.23 , we get

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 e x−5t 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 .

4.28
Taking the generalized differential transform of 4.27 , using the related definitions in Table 1   that is,

4.31
By applying 4.31 into 4.30 , some values of U α,1 k, h can be obtained as given in Table 1. Consequent substitution of all U α,1 k, h into 3.4 and after some manipulations, the series from solutions of 4.27 and 4.28 has been obtained in 27 as:

ISRN Mathematical Analysis
The exact solution of 4.27 , for the special case α 1.0, is given in 27

4.34
We have the generalized differential transform method solution for the time-   where I denotes 0.00002770432505a 6 x 3 − 0.00001003532405a 3 x 6 , and J denotes 0.0001662259502a 6 x 0.0001416751630a 3 x 4 ; recalling that t 1/4 a we get multivariate Padé approximation of order 8, 2 for 4.46 , that is,

4.51
As it is presented above, we obtained multivariate Padé approximations of the generalized differential transform method solution of the time-fractional Fitzhugh-Nagumo equation 4.27 for values of α 1.0, α 0.50, and α 0.75. Table 3 shows the approximate solutions for 4.27 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 e 1/ √ 2 x 1−2μ / √ 2 t , 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 .

Conclusion
By comparison with the generalized differential transform method GDTM , the fundamental goal of this work has been to construct an approximate solution for time-fractional reactiondiffusion equations by using multivariate Padé approximation. The goal has been achieved by using the multivariate Padé approximation MPA and the generalized differential transform method GDTM . The present work shows the validity and great potential of the multivariate Padé approximation for solving time-fractional reaction-diffusion equations from the numerical results. For the values of α 1.0 in Example 4.1 and for the values of α 1.0 in Example 4.2, numerical results obtained using the multivariate Padé approximation MPA and the generalized differential transform method GDTM are in excellent agreement with exact solutions and each other. For the values of α 0.50, α 0.75, in Example 4.1 and for the values of α 0.50, α 0.75 in Example 4.2, numerical results show that the results of multivariate Padé approximation are in excellent agreement with those results obtained by the generalized differential transform method GDTM . The basic idea described in this paper is expected to be further employed to solve other similar problems in fractional calculus.