The Numerical Solutions of a Two-Dimensional Space-Time Riesz-Caputo Fractional Diffusion Equation

This paper is concerned with the numerical solutions of a two-dimensional space-timefractional differential equation used to model the dynamic properties of complex systems governedby anomalous diffusion. The space-time fractional anomalous diffusion equation is definedby replacing the second order space and the first order time derivatives with Riesz and Caputooperators, respectively. Using the Laplace and Fourier transforms, a general representation ofanalytical 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 resultsfor two examples illustrate the comparison of the analytical and numerical solutions and alsovalidity of the GL approach to this problem.