Finite Difference Scheme For The Time-fractional Fokker-Planck Equation With The General Forcing

In recent years,growing attention has been focused on the fractional d-ifferential equation,and it has been successfully used in magnetic resonance imaging,biomedical porous media,molecular spectroscopy,economic and fi-nancial phenomena,signal processing and control,viscoelastic mechanics and so on.More and more results show that compared to the integer order e-quations,models about some practical problems that established through the fractional equation are more accurate,so the significance of fractional order equation is accepted by more and more experts.This paper mainly introduces the theoretical analysis of the time fractional Fokker-Planck equation.From the physical point of view,the time fractional Fokker-Planck equa-tion is the probability density function of random walk particle at time t and space x in a force field.For the time fractional Fokker-Planck equation,many experts and scholars have carried out the theoretical analysis,and gave the good results,but most of them are based on that the external force F is only a constant or only depends on the space variable x,obviously when F depends on the time variable t as well as space variable x,the time fractional Fokker-Planck equation is more general.This paper gives the theoretical analysis of the time fractional Fokker-Planck equation in this case.In this article,we con-sider the finite difference scheme of the time-fractional Fokker-Planck equation with general forcing and source term,and we give the spatial discretization by finite difference method and employ a time-stepping method and show the time error is O(τ)when the diffusion coefficient α∈(1/2,1),and the spatial is two-order accuracy in the discrete L2-norm.We establish the finite difference scheme in one-dimension and two-dimensions,give the theoretical analysis of the difference scheme,and verify our results by numerical experiments.The paper is organized as follows:The first chapter is the introduction,it mainly introduces the development history of fractional calculus,the research background and significance of the time fractional Fokker-Planck equation and give the definition of usual frac-tional derivative as well as the practical significance of this paper.The chapter 2 and chapter 3 are the main content of this article,and give respectively the finite difference scheme for the time fractional Fokker-Planck equation in one dimension and two dimensions.First,we integral about t on both sides of the equation.Then,we approximate the fractional integral operator by the right rectangle format and the spatial derivative by second order difference and central difference and obtain the difference format.For time fractional derivatives,we use the Riemann-Liouville derivative,then use energy method to analyze the convergence and stability of the scheme,finally,we give the corresponding numerical example to demonstrate our results.Chapter 4 is the conclusion of this paper.