Finite difference/predictor–corrector approximations for the space and time fractional Fokker–Planck equation

Abstract

In this work, by using the properties of the Riemann–Liouville derivative and the Caputo derivative, we firstly transform the space and time fractional, in the sense of the Riemann–Liouville derivative, Fokker–Planck equation to a new fractional PDE with a Caputo time derivative. After discretizing the spatial (classical and fractional) derivatives of the new fractional PDE using a finite difference method, we use the predictor–corrector approach to approximate the FODEs obtained. Conditional stability and convergence of the numerical scheme are rigorously established. We prove that the numerical scheme is stable and that the numerical solution converges to the exact solution with order O ( h + k min { 1 + 2 α , 2 } ) if k α / h μ < C . Numerical experiments are performed to demonstrate the effectiveness of the algorithm and confirm the theoretical claims.