The Study Of Numerical Algorithm For The Time Fractional Diffusion Equation

Fractional differential equations can be obtained from the classic integer order differential equations by substituting the derivatives with fractional order derivatives.Compared with the integer order differential equations,fractional partial differential equations are capable of describing many phenomena in the natural world,such as chaotic mechanics,fluid mechanics,porous material dynamics and mathematical biology.However,it is difficult to derive the analytic solutions of these equations,so the study on numerical methods to these equations has important theoretical and practical significances.This paper investigates the numerical methods for solving time fractional diffusion equations.The main work includes: 1.To design a "time-direction" higher-order numerical method solving the time fractional diffusion equations;2.On the non-uniform time grid,to use the finite difference methods both in time and space to discrete the variable-order time fractional diffusion equations,and analyze the stability and convergence of the numerical schemes at infinity norm.This paper is organized as follows:In chapter one,we present the background,research status,significance,introduce several definitions and properties of the fractional derivative and outline the main content of this paper.In chapter two,we design a "time-direction" higher-order method for solving the time fractional diffusion equation;then analyze the truncation error of corresponding numerical schemes,and finally verify the effectiveness of the method through numerical tests.In chapter three,based on non-uniform time grid,we investigates the finite difference schemes to solve the variable-order time fractional diffusion equation;then discuss the stability and convergence of the numerical schemes at infinity norm;finally,a numerical example is presented.