Numerical Schemes Of Time Fractional Fokker-Planck Equation

Fractional differential equations have rich theoretical connotations and far-reaching physical meanings.They can more accurately and concisely describe the complex mechanics and physical processes with historical memory in some practical problems.They have been widely used in the fields such as anomalous diffusion,fluid mechanics,signal control,image processing.In recent years,fractional differential equations have attracted the attention of more and more scholars.This paper considers the following time fractional Fokker-Planck equation(?)where ??(0,1),k? is the generalized diffusion coefficient,?? is the generalized friction coefficient,F=F(x,t)is the varying external force field,g(x,t)is the source item and(?)t1-?u is the Riemann-Liouville fractional derivative.A full-discrete numerical method,which uses the center-difference scheme and the Grunwald Letnikov approximation for spacial and time discretizations respectively,is stud-ied for the time fractional Fokker-Planck equation with space-and time-dependent forcing and its stability and convergence is theoretically proved.The error of the method is proved to be O(h2+??),where ?,h are time step size and space mesh size respectively,and A is a parameter related to the singularity of the solution.Numerical teste are conducted to support our theoretical analysis.This paper is divided into five chapters as follows:In Chapter 1,we outline the research background and current situation of the thesis and expound the research work and innovation points.In Chapter 2,we introduce some related concepts and basic properties of fractional derivatives.In Chapter 3,we design a full,diserete method for time fratctional Fokker-Planck equation.In Chapter 4,we theoretically analyze the stability and convergence of the method.In Chapter 5,we verify our theoretical analysis through three different numerical examples.