Improved and more feasible numerical methods for Riesz space fractional partial differential equations

Abstract

In this article, the numerical solutions of fractional order partial differential equations with Riesz space fractional derivatives, on a finite domain, have been considered. Two kinds of equations have been considered: the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The Riesz fractional diffusion equation is obtained from the standard diffusion equation by replacing the second order space derivative with the Riesz fractional derivative of order α, where 1<α⩽2. The Riesz fractional advection–dispersion equation is obtained from the standard advection–dispersion equation by replacing the first and second order space derivatives with the Riesz fractional derivatives of order 1<β<1 and 1<α⩽2, respectively.To obtain the solutions, firstly the system of ODEs is obtained using the improved matrix transform method with respect to the space variable. After this, the (3,1) Pade approximation can be used to construct the exponential matrix in the analytic solution of the ODE, and we obtain two difference schemes. It has also been shown that the two difference schemes are unconditionally stable and feasible, using the matrix analysis method. Finally, some numerical results are given, which demonstrate the effectiveness of the two difference schemes.