The numerical solutions of a two-dimensional space-time riesz-caputo fractional diffusion equation

Abstract

This paper is concerned with the numerical solutions of a two-dimensional space-time fractional differential equation used to model the dynamic properties of complex systems governed by anomalous diffusion. The space-time fractional anomalous diffusion equation is defined by replacing the second order space and the first order time derivatives with Riesz and Caputo operators, respectively. Using the Laplace and Fourier transforms, a general representation of analytical solution is obtained in terms of the Mittag-Leffler function. Gr¨unwald-Letnikov (GL) approximation is also used to find numerical solution of the problem. Finally, simulation results for two examples illustrate the comparison of the analytical and numerical solutions and also validity of the GL approach to this problem.