Finite Difference Method For The Variable-order Time Fractional Fourth Order Sub-diffusion Equations

The theoretical research of fractional partial differential equation is one of the hot issues at present,which has aroused wide attention of scholars in different fields.Due to the non-local nature of its fractional differential operator,fractional partial differential equations can accurately describe many complex physical phenomena.Among them,the time-fractional sub-diffusion equation can describe the mechanical behavior of some specific materials,and scholars have done a lot of theoretical research and numerical calculation work on it.In recent years,however,it has been found that many complex diffusion processes represent fractional properties that change with external factors,such as time,space and other factors.It shows that the fractional derivative of constant order can be extended to the variable fractional order,and the study of the variable fractional sub-diffusion equation is of great practical significance for complex dynamics problems.At present,the variable-order fractional diffusion model has been used to describe many complex physical problems,such as the evolution of the diffusion process in porous media as the external field changes with time and temperature changes when the particles in a particular material diffusion process.However,the uncertainty of fractional derivative of variable order makes it difficult for us to obtain the numerical solution of its equation.Therefore,solving the variable-order fractional sub-diffusion equation is our key research problem.Finite difference method is a common method to solve partial differential equations.In this paper,the finite difference methods for solving variable-order time fractional fourth order sub-diffusion equations with Dirichlet boundary conditions are studied.The main work is divided into the following two parts.In the first part,for the variable-order time fractional fourth order subdiffusion equation under the second Dirichlet boundary condition,the fourth derivative of space is reduced by taking into account the reduced order of the equations at a special point,and then the second order difference scheme is obtained.On this basis,the space order of the difference scheme is improved by acting compact operator,and the difference scheme with second-order in time and fourth-order in space is obtained.The stability and convergence of the two schemes are proved by energy analysis.Numerical examples are given to verify the theoretical results of the two schemes.In the second part,for the variable-order time fractional sub-diffusion equation with variable coefficients in space under the first Dirichlet boundary condition,we also consider the equation at a special point by reducing the order,and use an operator to deal with the boundary condition,and obtain the difference scheme with second order in time and space.The lemma of variable coefficients is given,and the stability and convergence of the difference scheme are proved by energy analysis.Finally,numerical examples are given to verify the theoretical results.