Numerical solution of a two-dimensional anomalous diffusion problem

Abstract

In this chapter, we present the numerical solution of a space-time fractional anomalous diffusion problem in two-dimensional space. Space derivatives with respect to x and y variables are defined in terms of Riesz-Feller derivatives of order 0< α < 1 and 1 < µ = 2, respectively; θ 1 θ 1 = minα, 1-α and θ 2 θ 2 ≤ minµ, 2-µ are skewness parameters; and the time derivative is defined in sense of Caputo of order ß 0 < ß = 1. It is assumed that the solution and the initial condition functions can be expanded in a complex Fourier series. Grünwald-Letnikov approximation of Caputo derivative is used to take numerical solutions. Furthermore, the comparison of analytical and numerical solutions is proposed by an example and variation of problem parameters are analyzed. Finally, the convergence of analytical and numerical solutions to each other shows the effectiveness of the numerical methods to the present problem.