Finite Difference And Spectral Method For Time Fractional Diffusion Equation

The numerical approximation for fractional diffusion equations is a very importantproblem that people are concerned about. In this paper, the finite difference method andspectral method are used to solve the initial boundary problem of a class of time fractionaldiffusion equation in the sense of Caputo fractional derivative.Firstly, an implicit finite difference scheme is introduced. The Fourier analysis methodis used to prove the stability and convergence of the scheme. The numerical solution isderived by transforming the algebraic equations into a convex problem which can be solvedthrough an Matlab-based software CVX.Secondly, the semi-discrete scheme for the original problem is obtained by using finitedifference method in the time domain. Then, the stability and error estimation of the semi-discrete scheme are given. Based on the semi-discrete scheme, the full-discrete schemeis derived by exploiting Legendre collocation method in the space domain. The error es-timation between the full-discrete scheme and the original problem is proved. A specificexample is given in the end.The numerical results indicate that both methods provide good approximation to theoriginal initial boundary problem.