Spectral Methods For Some Types Of Fractional Partial Differential Equations

Fractional order models are more accurate in describing many physical phenomena with memory and heredity characteristics.Up to now,they have been widely applied in many areas,such as quantum mechanics,system control,economics and biomedical sciences.Due to the nonlocal properties of the fractional order operator,it is very hard to find the analytical solu-tions of fractional order models.So there has been substantial interest in designing efficient numerical methods.It is well known that spectral method is global and has high precision,therefore it is very suitable for solving the fractional order differential equations with nonlocal operators.In this paper,we focus on constructing spectral methods for solving several types of fractional partial differential equations,including the two-dimensional distributed-order time fractional subdiffusion equation,nonlinear space fractional Schr(?)dinger equation and nonlin-ear space fractional Ginzburg-Landau equation.The work of this thesis mainly includes the following four aspects:First of all,we construct a Legendre spectral method for solving the two-dimensional distributed-order time fractional subdiffusion equation.By utilizing the composite Simpson formula to discretize the distributed-order integral,we transform the considered equation into a multi-term time fractional sub-diffusion equation.Then the L2-1_?formula is used to approx-imate the multi-term Caputo fractional derivatives.By virtue of the properties of coefficients in the L2-1_?formula and the energy method,the proposed spectral scheme is shown to be unconditionally stable and convergent.Secondly,we give a rigorous error analysis of the multi-symplectic method for the fractional Schr(?)dinger equation.By rewriting the multi-symplectic Fourier pseudospectral method into a matrix form,we can establish the error estimate in the discrete L~2norm via the discrete energy method and cut-off technique.The convergence result for the symplectic method can be established by a similar way.Next,we propose a linearized Legendre spectral scheme,which preserves both the mass and energy conservations,for numerically solving the nonlinear coupled fractional Schr(?)dinger equations.The conservation and convergence properties of the proposed scheme are discussed in detail.Numerical experiment results show that the spectral method has good conservation properties,and also has second order accuracy in time as well as spectral accura-cy in space.Finally,we first construct a linearized Legendre spectral method for the one-dimensional nonlinear fractional Ginzburg-Landau equation,where a three-level linearized Crank-Nicolson scheme is used for time discretization.The unique solvability and boundedness properties of the fully discrete scheme are analyzed.It is shown that the method is unconditional convergent in the maximum norm.Then,two-dimensional problems are considered and a split-step ADI Galerkin-Legendre spectral method is introduced without theoretical analysis.Finally,some numerical examples are presented to illustrate the effectiveness of the two proposed schemes.In summary,in this paper,we not only further study the structure-preserving Fourier pseudospectral methods for the fractional Schr(?)dinger equations,but also construct some efficient Galerkin-Legendre spectral schemes to solve several types of fractional partial differential equations,and give rigorous theoretical analysis for the fully discrete spectral schemes.These numerical schemes take the advantage of high precision and less amount of calculation,thus our work provides some effective ways to solve the fractional partial differential equations.