Spectral Methods For Fractional Ginzburg-Landau Equation And Fractional Fokker-Planck Equation

In recent years,the numerical methods of the fractional equations have become a hot research field.There are two main difficulties for solving the fractional equations.First,fractional derivatives are nonlocal operators.Second,the solutions of the frac-tional equations are usually singular near the initial value and boundaries.For the first difficulty,it would be suggested that global methods such as spectral methods are appropriate tools for discretizing the fractional equations.However,the tradition-al local-methods lead to full matrices in discretization of the fractional derivatives.For the second difficulty,the spectral methods based on classic polynomials bases would fail to achieve high accuracy when the solutions of the fractional equations are nonsmooth.As is known,the usual spectral methods can provide spectral accura-cy for the problems with smooth solutions.However,the solutions of the fractional equations are usually of low regularity both in time or space due to time or space fractional derivatives.In other words,the accuracy of the usual spectral methods is limited by the regularity of solutions in the usual Sobolev spaces.Our work in this dissertation is focused on the high accuracy spectral methods for the nonlinear fractional Ginzburg-Landau equation and time,time-space fractional Fokker-Planck equation.Firstly,we propose the Fourier pseudo-spectral method for the multi-dimensional nonlinear fractional Ginzburg-Landau equation.The continuous mass and energy in-equalities as well as their discrete versions are presented.Based on these inequalities,we give the error estimate of the semi-discrete Fourier pseudo-spectral scheme.In addition,both the Crank-Nicolson Fourier spectral and split-step Fourier spectral schemes have been proven to perserve a good approximation to the dispersion rela-tion.Several numerical examples are given to study the dynamic behavior of the solutions and to verify our theoretical results.Secondly,we develop two types of space-time spectral methods for the time fractional Fokker-Planck equation:The first type is the space-time Petrov-Galerkin spectral method based on the generalised Jacobi functions.The basis functions used in time are generalised Jacobi functions,which can match the leading singularity(The solution is like u(t)=t~?g(t),where g(t)is sufficiently smooth).The standard Legen-dre polynomials are used as basis functions in space.Moreover,we give the stability and convergence analysis,and verify our theoretical results by some numerical exam-ples.Another space-time spectral method based on the Müntz Jacobi polynomials is developed for the equation.The method is also based on the Legendre polynomials in space,and the Müntz Jacobi polynomials are used as the basis functions in time.The well-posedness and stability of the discrete scheme as well as its variational formula of the continuous problem are established.The error estimate of the proposed method is given.Moreover,the Müntz spectral method and the spectral method based on the generalised Jacobi functions are compared by the numerical experiments.Finally,a space-time Galerkin/Petrov-Galerkin spectral method is established for the time-space fractional Fokker-Planck equation.This approach is based on t-wo types of the generalized Jacobi functions in space and time,respectively.The well-posedness and stability of the variational formulation of the equation are estab-lished.Moreover,a complete error estimate for the proposed method is carried out.Numerical experiments substantiate the theoretical results.