The Spectral Method Of Tempered Fractional Differential Equations Using The Hermite Function

In the past several decades,spectral method as one of the important tools for scientific computing developed rapidly.In this thesis,we focus on designing efficient spectral methods for solving space tempered fractional differential equations on the whole line.Firstly,we propose Galerkin and collocation schemes for the steady space tempered fractional differential equation.Next,we propose Galerkin and collocation methods for evolutionary space tempered fractional differential equation.We prove the stability and convergence of the proposed methods.In order to overcome the difficulty in computation of the tempered derivatives ?_+,x^?,?and ?_-,x^?,?.We first transform the underlying equations to algebra one by Fourier transformation.Then,we convert the proposed spectral Galerkin and spectral collocation schemes to equivalent ones using the property that Hermite function is eigenfunciton of Fourier transfrom operator.So,we can implement the concrete calculation efficiently.Ample numerical evidences are provided to highlight the efficiency,confirm the theoretical analysis and validate the expected behaviours of numerical solutions.