Generalized Hermite Spectral Method For Nonlinear Fokker-Planck Equations On The Whole Line

In statistical mechanics,the Fokker–Planck equation is a partial differential equation defined on unbounded domain that describes the time evolution of the probability density function of the velocity and position of a particle under the influence of drag forces and random forces,as in Brownian motion.While,it is difficult to find the exact solution because of the finite knowledge for the equation which has nonlinear characteristics,so it is very necessary to design efficient numerical algorithm for solving nonlinear Fokker-Planck equations.Pioneers propose some numerical approaches,such as composite generalized Laguerre spectral method with domain decomposition.These will bring about some difficulties in the theoretical analysis and numerical calculation by using domain decomposition of the existing literature,such as how to match the numerical solutions on the common boundary of adjacent subdomains.Thus it seems better to solve such problems directly using the Hermite function with the weight function X(v)? 1as the base function,which would be much stabler for a long time and simplifies theoretical analysis and brings more precise error estimates.The purpose of this work is to develop the spectral and pseudospectral methods for simplified and nonlinear Fokker-Planck equations defined on the whole line.In chapter 2,some results of generalized Hermite orthogonal approximations,interpolation approximations and some difference approximations in time- direction are introduced.These results are the mathematical foundation of our paper,with which we will build up fully discrete spectral and pseudospectral methods for Fokker-Planck equations on the whole.In chapter 3,generalized Hermite functions with the scaling factor are used as basic functions to expand numerical solution to approximate the solution of simplified Fokker-Planck equation on the whole line.Algorithms scheme is constructed.The convergence of the scheme is proved.Numerical results demonstrate the efficiency and high accuracy of this approach.In chapter 4,firstly,a fully discrete generalized Hermite spectral method for nonlinear Fokker-Planck equations is proposed.A fully discrete scheme is given using Crank-Nicolson discretization in time- direction,with the mesh size ?.The convergence and stability of the proposed scheme are analyzed.Numerical experiments substantiate the theoretical analysis and show the efficiency of our approach.Secondly,we consider generalized Hermite pseudospectral method for nonlinear Fokker-Planck equations,analyze the convergence,and so on.The chapter 5 is for some concluding remarks and the problems which can be considered in the future.