Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems

Abstract

This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the sum of an infinite series. Which converges rapidly to exact solution. The Homotopy Perturbation Method is no need to use Adomian's polynomials to calculate the nonlinear terms; we test the proposed method to solve nonlinear fractional systems of redaction diffusion equations in one dimension, two dimensions and three dimensions. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with ordinary derivative order index <i>α</i>₁=<i>α</i>₂=1 for nonlinear fractional reaction diffusion systems. Approximate solutions for different values of fractional derivatives index <i>α</i>₁=0.5 and <i>α</i>₂=0.5 together with non-fractional derivative index α₁=1 and <i>α</i>₂=1 and absolute errors are represented graphically in two and three dimensions. In addition, the graphical represented the solutions, which had been given by MATLAB program. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods.