A Class Of Spectral Methods For Fractional Partial Differential Equations

As an important branch of mathematics,the theory of fractional calculus(including fractional differential and fractional integral)has been studied by many researchers.It has become one of the important tools to describe all kinds of complex mechanical and physical behaviors,and has been widely used in various fields,such as mathematics,physics,biology,control theory,image processing and so on.Due to the analytical solution of fractional diffusion equations is difficult to obtain,the study of its numerical methods has important theoretical and practical significance.In this thesis,we discuss a class of time spectral methods for fractional partial differential equations.The main contain of this thesis has two parts.For one-dimensional two-sided space Riemann-Liouville fractional diffusion equations,we consider the third-order WSGD finite difference method in space;In the time direction,the crank-Nicholson method and the spectral collocation method are used for discretization.In the numerical experiments,we use two kinds of orthogonal polynomials,Chebyshev polynomials and Legendre polyno-mials,and several different types of time nodes to solve the different discrete schemes.The numerical results show the effectiveness of the time spectrum collocation method proposed in this thesis.For one-dimensional Caputo time-fractional diffusion equations,we use the difference operator A to approximate the spatial derivative;In time direction,L2-1? formula and spectral collocation method are selected.The numerical solutions of the corresponding discrete schemes are obtained at the time nodes of uniform and non-uniform grids.The numerical results verify the effectiveness of the proposed method in this thesis.