The Convergence Studies Of Finite Difference Approximations For The Fractional Fokker - Planck Equation

This paper introduces a new analysis of stability and convergence for finite differ-ence methods used to solve the time fractional Fokker-Planck equation. It is a kind of typical fractional order differential equation and can be used to model the sub-diffusion processes in an external force fields. It has very important practical application back-ground.Firstly, we reviewed the history and current situation of the development of frac-tional order differential equation, and gives the definition of three kinds of fractional order derivative. A brief introduction to this article research content and research sig-nificance. Secondly, we introduced the five-time fractional differential approach of the Fokker-Planck equation and the proof of the stability and convergence of difference format.Based on the stability and convergence are obtained under somewhat strong con-ditions of the function f(x), we present a new analysis of stability and convergence for the above five schemes under the more broad conditions.We choose the scheme L1-CDIA as example to conduct our analysis, firstly, we show the monotone property for the considered scheme; secondly, we prove the stability and convergence of the scheme under discrete L1 norm; finally,we carry out the numerical tests,and calculate the convergence rate of time and space,Space and time rate of convergence is consistent with theoretical analysis.At the end of the article, our numerical method of this article made a simple summary, at the same time, we can put forward further research problems, and look forward to have a better application prospect.