Comparison of Numerical Solutions of Time-Fractional Reaction-Diffusion Equations

Abstract

Nonlinear phenomena play a crucial role in applied mathematics and physics. Although it is very easy for us now to find the solutions of nonlinear problems by means of computers, it is still rather difficult to solve nonlinear problems either numerically or theoretically. One of the most famous of the nonlinear fractional partial differential equations which called the time-fractional reaction-diffusion equation in this paper, we compare numerical solutions for time-fractional reaction-diffusion equation using variation iteration, homotopy perturbation, adomian decomposition and differential transform methods. The fractional derivatives are described in the Caputo sense. The methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the most efficient from approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. An example is given to demonstrate the effectiveness of the present method.