Artificial Boundary Methods And Numerical Solutions For Time Fractional Sub-diffusion Equations

This paper mainly studies the high-order difference method and spectral method of the time fractional order reaction diffusion equation under the precise artificial bound-ary condition.Firstly,the precise artificial boundary condition of the time fractional order reaction diffusion equation is obtained by using the Laplace transform.A higher order difference scheme for fractional-order reaction diffusion equation is constructed,a priori estimation is given,and convergence and stability are proved.Spectral by using Galerkin method,constructed the fractional order reaction diffusion equation of space-time spectrum format,verify the method,and proves that the spectrum format,stability and convergence of spectral accuracy is verified by numerical examples.This article provides two kinds of the format of the high order and high precision,is time fractional order reaction diffusion equation on unbounded region of a highly efficient,economic,simple,reliable and effective algorithm.The first chapter introduces the research background,research status and main contents of the structure of fractional differential equation.The second chapter introduces the definition and properties of fractional deriva-tive,Laplace transformation,L^pspace and Sobolev space.In the third chapter,the exact working boundary is constructed for the time frac-tional order reaction diffusion equation of the unbounded region,??0<?<1?the fractional derivative of order Caputo constructs a new L2 order interpolation approx-imation,improves the L1-2 format,and studies the properties of the coefficients,whose convergence order is O(?t^3-?).In this paper,the high-order L2 difference scheme of the reaction diffusion equation of fractional order under the precise artificial boundary condition is constructed,and the prior estimation is given under certain con-straints,and the stability and convergence are strictly proved,the convergence order is O(?t^3-?+h²).Finally,numerical examples are used to verify that this format is a high order,simple,economical and effective format.In the fourth chapter,Uses Galerkin spectral method,constructed in exact arti-ficial boundary conditions time fractional order reaction diffusion equation of time-space discrete format,all of weak solution is proved,and prove the convergence and stability of the spectrum format,finally the spectral accuracy is verified by a numeri-cal example,illustrates the spectrum method is processing precise artificial boundary conditions of a high precision,efficient algorithms.The fifth chapter summarizes the paper and prospects the follow-up research.