Finite Difference Method For Time Fractional Diffusion Equations With Periodic Boundary Conditions

In recent decades,fractional differential equations(FDE)have been deeply explored and applied by many scholars.People borrow FDE to describe,interpret and speculate various natural laws.In fact,the exact solution of the fractional-order model is difficult to obtain,so it is necessary to explore the numerical solution of FDE.The current numerical solutions for FDE familiarity are:finite difference method,finite element method and spectral method.With the rapid development of science and technology,the current numerical solution of various fractional calculus can be realized by efficient algorithms,which also promotes the influence of FDE numerical solution in technology and engineering calculations.This paper focuses on the finite difference method for fractional diffusion equations with periodic boundary conditions.The main contents of the article are as follows:We briefly describe the history of FDE and the numerical solution of the fractional diffusion equation.Finally,the FDE with periodic boundary conditions studied in this paper is described.Simultaneously,we strictly discuss the FDE finite difference method with periodic boundary conditions.We use the L2-1_smethod in the time direction and the second-order difference method in the spatial direction to establish the finite difference scheme of O(t~2(10)h~2)convergence precision.Then the Fourier method is used to strictly analyze the stability and convergence of the difference scheme.Finally,we use MATLAB software for numerical verification.The numerical results verify the feasibility of theoretical analysis.In the third part of the article we mainly analyze the finite difference format of FDE with periodic boundaries.The L2-1_sformula is used in the time direction,and the fourth-order difference method is used in the spatial direction.On this basis,the convergence precision ofO(t~2(10)h~4)is obtained.Then the Fourier method is used to give a rigorous theoretical analysis,and the numerical results are verified.Finally,a brief conclusion is obtained.