Finite Difference Method For Time Fractional Integro-differential Equation With Neumann Boundary Conditions

Recently,fractional differential equations are applied in many fields.The numerical analysis of the fractional differential equations becomes an important research topic.This thesis proposes a numerical method for a kind of time fractional differential equations.The thesis is divided by three chapters.The first chapter is the introduction.It introduces the background of the fractional calculus,research significance,and its research status at home and abroad.And there will be a brief overview of the fractional integro-differential equation studied by this thesis.In the second chapter,a finite compact differential method of the fractional differential equations on one-dimension wil be proposed.We approximate the Caputo fractional derivative and Riemann-Liouville(R-L)fractional derivative,establishing the finite difference method of O(?andO(?)in the spatial direction respectively with the time stept.Because of the Neumann boundary condition,the exterior order will be one order lower than the interior order when we apply the compact operator on the equation.Therefore,we need to deal with the boundary to reach fourth order.This chapter establishes the compact difference method for the equation,then gives the proof of the stability and convergence,and proves it by numerical experiment finally.In the third chapter,we construct the finite compact difference method on two-dimension.We deal with the Caputo fractional derivative and R-L fractional derivative and the boundary like what we have done on chapter 2.The chapter establishes a compact ADI difference method with the spatial direction reaching fourth order in spatial direction and the temporal accuracy of ordermin(?).The theorical proof is given.Eventually,we prove it by the numerical experiment on MATLAB.