Some Numerical Methods For The Space Fractional Diffusion Equations

Fractional order differential equation is a class of differential equations which is from the practical problems.As an important part of the theory of differential equation, the study of it can enrich mathematical theory of differential equation,on the other hand,it is also provide many mathematical models for the study of chemistry, biology, physics, engineering and economic phenomenon and the process of many aspects,which promote the research and development of these disciplines. As we know, compared with integer order differential equation, fractional order differential equation is less perfect.It is short for scientific solution formula.At present the study of it is still in its infancy.In order to thoroughly promote the research and development of fractional order differential equations,especially the research of numerical solution,it is necessary to make a lot of work.In this paper, we mainly study the numerical methods for solving the spatial fractional diffusion equation.The fractional order derivative is Grunwald-Letnikov definition in terms of the fractional order derivative.Firstly,the source of the fractional order differential equations,significance,research status at home and abroad,preliminary knowledge and the structure of this article are given.Secondly, the fractional diffusion equation of one dimensional space is studied by constructing a weighted explicit finite difference scheme.And making analysis of the stability and convergence of this format.Finally a numerical example is given to verify the accuracy and reliability of format.Again, this paper discusses the one-dimensional with fractional order variable coefficient of the source term space convection-diffusion equation.A kind of weighted implicit finite difference scheme is constructed by taking the weighted parameter ε=α/2.It is proved that this kind of difference scheme is unconditional convergence.Finally a numerical example is given to verify the reliability and accuracy of this format.At last, a kind of two dimensional spacial fractional order convection-diffusion which is derived from the porous media seepage problems. An ADI-CN difference method is constructed for solving this kind of equation.In order to improve the accuracy,Richardson extrapolation method is used to make the equation to solve further.At the end, two numerical examples are provided to verify the effectiveness, the accuracy and reliability of this format.