Difference methods for two-dimensional space-time fractional dispersion equation

Abstract

The two-dimensional space-time fractional dispersion equation is obtained from the standard two-dimensional dispersion equation by replacing the first order time derivative by the Caputo fractional derivative,and the two second order space derivatives by the Riemann-Liouville fractional derivatives,respectively.Base on the shifted Grunwald finite difference approximation for the two space fractional derivatives,an implicit difference method and a practical alternate direction implicit difference method were proposed to approximate the fractional dispersion equation. The consistency,stability,and convergence of the two implicit difference methods were analyzed. By using mathematical induction method,it was proven that the two implicit difference methods were all unconditionally stable and convergent and the order of convergence were obtained. The convergence speed and computational complexity of the two implicit difference methods were compared. A numerical simulation for a space-time fractional dispersion equation with known exact solution was also presented,and correctness of the theoretical analysis was verified by the numerical results.