A Compact Finite Difference Scheme For Variable Order Time Diffusion Equation

In this paper we consider a variable order time subdiffusion equationwhere 0<a?a?(?)<1,RLD0,t?(x,t) denotes the Riemann-Liouville frac-tional derivative of order ?(x,t)define by[1,2],In this model the derivative of time is a variable that depends on time and space.In addition in the fluid mechanics,when diffusion process evolves in porous medium,the medium structure or external field changes with time,the constant order fractional diffusion equations are not capable for characterizing such phenomena.In fact,this theory of variable fractional derivatives has been proposed for some time.Please refer to[8-12]for details.The goal of this paper is to give two difference schemes for solving a time-variable fractional diffusion equation and discuss its stability and convergence.In order to obtain a high numerical accuracy,it is approximated in the time direction by using the shifted Grunwald operator,and it is approximated with two differential operators in space respectively.Then its stability and con-vergence are proved by Fourier analysis.The convergence degrees of the finite difference schemes can reach O(?3+h2)and O(?3+h4),respectively.Finally some numerical examples are provided,the results confirm the theoretical analysis and demonstrate the effectiveness of the finite difference schemes.The article gives two difference schemes and theoretical analysis of the time-variable fractional diffusion equation.The article is organized as follow five chapters:Chapter 1:Introduces the domestic and foreign literature on the devel-opment process of variable fractional calculus,and introduces the research background and significance of this paper.Chapter 2:Gives a high-order format to approximate the variable order time derivative.Chapter 3:Gives two difference scheme for the time fractional diffusion equation.Chapter 4:discusses the stability and convergence of the format,and proves that the error of the algorithm reaches O(?3 + h2).Chapter 5:Gives a numerical simulation and indicate the efficiency of the difference scheme by numerical results.