A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reactiondiffusion equations. This method is a combination of the multitimestepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the timeevolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The GaussSeidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multitimestepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.
The nonlinear reactiondiffusion equations have found numerous applications in pattern formation, in many branches of biology, chemistry, and physics [
The differential transform method was used first by Zhou [
Fractional partial differential equations (FPDEs) are also an interesting and an important topic. The fractional derivatives and integrals have been occurring in many physical and engineering problems with noninteger orders. Fractional calculus is based on the definition of the fractional derivatives and integrals. They play a major role in engineering, physics, and applied mathematics. FPDEs are used to model complex phenomena since the fractional order differential equations are naturally related to the systems with memory and nonlocal relations in space and time which exist in most physical phenomena. Fractional order differential equations are as stable as their integer order counterpart. One of the fundamental equations of physics is the Schrödinger equation which describes how the quantum state of physical system changes with time. The fractional Schrödinger equation provides us with a general point of view on the relationship between statistical properties of quantum mechanical path and structure of fundamental equations of quantum mechanics [
The differential transform is well suited to combine with other numerical techniques, as shown by Yu and Chen [
In the current study, the hybrid generalized differential transform/finite difference method is used for solving time fractional nonlinear RD equations. The validity of the proposed approach has been confirmed by comparing the results derived in the literature using the GDTM method [
There are several approaches of definitions for the fractional derivative. Among them, one is called RiemannLiouville fractional derivatives and defined by
For Caputo derivative we have
We firstly introduce the main features of GDTM [
In this study, we use a hybrid method that is a combination of generalized, temporal differential transform and spatial finite difference methods to solve nonlinear fractional reaction diffusion equations.
We present a solution of a more general model of RD equation
We apply GDTM to discretize fractional order time derivative and central difference method to discretize derivatives in
The new algorithm has been developed to solve the nonlinear reaction diffusion equation and our aim of this approach is to combine the flexibility of differential transform and the efficiency of finite differences. This algorithm also provides an iterative procedure to calculate the numerical solutions; therefore, it is not necessary to carry out complicated symbolic computation. On applying the differential transform method with respect to time on the equation we are basically transforming the timeevolutionary equation to an elliptic type. In essence this means that the central finite difference approximation that is subsequently used on the transformed equation is a Poisson solver. The resulting system of linear algebraic equations is then diagonally dominant and hence the GaussSeidel iterative method used for solving the same has assured convergence as the coefficient matrix remains nonsingular throughout the computation. The algorithm used thus succeeds in segregating the time discretization from explicitly influencing the computation in the spatial domain and this presents a situation wherein the two can be handled independent of each other in the course of computation without having to bother about the stability of the solution if the differential transform part is properly handled. The latter is achieved deftly in the differential transform part of the algorithm by using the multistepping procedure as first enunciated by Yu and Chen [
To show effectiveness of the proposed numerical solution using the temporal generalized differential transform and the spatial finite difference method and to give an understandable overview of the methodology, two examples of the reaction diffusion equations will be discussed in the following section. Then our results will be compared with published work of Rida et al. [
The time fractional Fisher equation is
In this example, we have the nonlinear function
Operating the generalized differential transform on (
Now we apply the central finite difference method to the derivatives with respect to
Equation (
Some values of

 

0  1  2  
0  0.250000 




1  0.225644 




2  0.202649 


The time series solutions of (
The numerical calculation results are shown in Figures
Numerical solution for the time fractional Fisher equation with
Numerical solution for the time fractional Fisher equation with (a)
Approximate solutions are shown in Figure
The influence of
Approximate solution for the time fractional Fisher equation with different
Numerical comparison between GDTM [
Comparison of numerical results between different methods for the time fractional Fisher equation. GDTM: generalized differential transform method, Rida et al. [








Present  
0.02  2  0.0236384265  0.0241395370  0.0236296987  0.0241417516  
0.04  2  0.0325745492  0.0351552680  0.0325330330  0.0351447962  
0.06  2  0.0420810751  0.0489473130  0.0419776983  0.0488819720  
0.08  2  0.0521879221  0.0660928586  0.0519904364  0.0658898759  









Present  Exact 


0.02  2  0.0169179992  0.0169279385  0.0169177486  0.0169293607  0.0169282151 
0.04  2  0.0200377701  0.0201117284  0.0200357651  0.0201240442  0.0201217246 
0.06  2  0.0235686494  0.0238370103  0.0235618826  0.0238628770  0.0238595181 
0.08  2  0.0275106370  0.0281487518  0.0274945972  0.0282219465  0.0282178229 
It is also found that the result is in complete agreement with the result of HPM [
We investigate convergence criteria of our solutions for different values of
In Figures
Comparison of present results for
Comparison of present results for
Comparison of present results for
One important observation made from the computation is that when the number of mesh points was increased, less number of terms was required in the time series solution to have convergence for a predetermined accuracy. The hybrid method of the present study gives faster convergence than other traditional methods; for example, if we take
The time fractional FitzHughNagumo equation is
Some values of

 

0  1  2  
0  0.5 




1  0.517670 




2  0.535296 


The time series solution for the above IBVP at different times is
Numerical solution for the time fractional FitzHughNagumo equation with
Numerical solution for the time fractional FitzHughNagumo equation with
For
Approximate solution for the time fractional FitzHughNagumo equation with different
As shown in the Table
Coefficients of
Rida et al. [ 
Exact  Present  



Coef. of 
0.5  0.5  0.5 
Coef. of 
−0.05  −0.05  −0.049999 
Coef. of 
0.02  0  −0.000011 




Coef. of 
0.517670  0.517670  0.517670 
Coef. of 
−0.049937  −0.0499937  −0.049933 
Coef. of 
0.020327  −0.000176  −0.000181 




Coef. of 
0.535296  0.535296  0.535296 
Coef. of 
−0.049937  −0.049937  −0.049743 
Coef. of 
0.020602  −0.000351  −0.000352 




Coef. of 
0.552835  0.552835  0.552835 
Coef. of 
−0.049441  −0.049441  −0.049430 
Coef. of 
0.020821  −0.000522  −0.000542 
Numerical comparison between GDTM, FVIM, and hybrid method is shown in Table
Comparison of numerical results between different methods for the time fractional FitzHughNagumo equation. GDTM: generalized differential transform method, Rida et al. [







Present  
0.2  0  0.49150484  0.48896821  0.48191202  
—  0.25  0.53566881  0.53312591  0.53520376  
—  0.5  0.57927974  0.57677042  0.57880163  
—  0.75  0.62168644  0.61924846  0.62118883  
—  1  0.66230919  0.65997604  0.74220689  








Present  Exact 


0.2  0  0.49510000  0.49450005  0.49473894  0.49500016 
—  0.25  0.53922189  0.53862130  0.53911415  0.53911409 
—  0.5  0.58273747  0.58214558  0.58262443  0.58262371 
—  0.75  0.62500118  0.62442684  0.62488468  0.62488385 
—  1  0.66544142  0.66489240  0.66761495  0.66532300 








Present  


0  0.2  0.53529653  0.53529653  0.53529653  
0.05  —  0.53199661  0.53163167  0.53193245  
0.1  —  0.52999926  0.52903570  0.52983082  
0.15  —  0.52833265  0.52663211  0.52803767  
0.2  —  0.52685856  0.52431371  0.52642106  








Present  Exact 


0  0.2  0.53529653  0.53529653  0.53529653  0.53529653 
0.05  —  0.53405919  0.53402165  0.53405262  0.53405254 
0.1  —  0.53283474  0.53268452  0.53280831  0.53280813 
0.15  —  0.53162316  0.53128507  0.53156363  0.53156331 
0.2  —  0.53042446  0.52982323  0.53031869  0.53031809 
Many real physical problems can be best modelled with fractional differential equations but the fact is when the equation is nonlinear there are very few reliable methods. The numerical methods that can be used to solve fractional differential equations are known to have problems of convergence and stability. These aspects are well addressed in the paper by suggesting a new procedure that uses a combination of the generalized differential transform and central difference methods. The Appendix clearly spells out the fact that the error as a result of discretization and computation is bounded and hence implies stability of the method. Lax equivalence theorem further implies convergence of the scheme. Two time fractional nonlinear reactiondiffusion equations considered for illustration of the hybrid method highlight the usefulness of the method in obtaining the solution of IBVPs involving time fractional derivatives. The control of convergence through a judicious choice of time and spatial step sizes and also the number of terms in the time series solution spells assured convergence. The segregation of the time domain from the spatial domain in the solution method ensures the fact that problem of stability does not arise. Diagonal dominance of the coefficient matrix in the system of linear algebraic equations resulting from the use of the central difference approximation in the Poisson equation ensures the fact that the matrix remains nonsingular during iterations and hence has assured convergence. An appropriate computational decision on the number of terms to be taken in the time series solution results in a convergent solution with fast convergence. Excellent comparison of the present results with the previous works on generalized differential transform method [
Consider the fractional differential equations (
Let
The paper uses an adaptive step size in computing the results. This is because such a procedure succeeds in keeping the error bounded and ensures convergence as a consequence of Lax equivalence theorem. To see what the adaptive step size produces and to show how such a procedure keeps the error bounded we start with the premise that
Thus
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are thankful to Ondokuz Mayıs University, Samsun, Turkey, for providing financial support to carry out this work under a major research project (Grant no. pyo.fen.1901.13.003).