Numerical treatment for the fractional Fokker-Planck equation

Abstract

We consider a space-time fractional Fokker-Planck equation on a finite domain. The space-time fractional Fokker-Plank equation is obtained from the general Fokker-Planck equation by replacing the first order time derivative by the Caputo fractional derivative, the second order space derivative by the left and right Riemann-Liouville fractional derivatives. We propose a computationally effective implicit numerical method to solve this equation. Stability and convergence of the numerical method are discussed. We prove that the implicit numerical method is unconditionally stable, and convergent. The error estimate is also given. Numerical result is in good agreement with theoretical analysis.