Research On Numerical Solution Of Fractional Diffusion Equation

Fractional differential and integral equations are widely used in physics,biology,chemistry,and other disciplines.They are generalized from classical differential and integral equations.Its research is not only important for the development of differentiation and integration,but also for the development of physics and engineering technology.The field has broad application prospects.Therefore,exploring the theory and applications of fractional differential and integral equations has become the focus of attention.A large number of recent research results show that the value of the diffusion coefficient in the fractional differential equation is closely related to changes in time and space.However,the analytic expressions that constitute the analytic solution of such problems can only be solved in some specific cases.Therefore,to solve fractional differential equations by universal and effective numerical methods have attracted the attention and favor of scholars at home and abroad.The research content mainly includes three parts: In the first part,the process of applying the finite difference method to solving the spatial fractional-order variable coefficient diffusion equation is mainly introduced,and the steps of constructing the format of the study and the validity and convergence of the format are proved.In the second part,it mainly introduces the application of finite difference method in solving the time fractional variable coefficient diffusion equation,and studies the process and validity of the scheme.Numerical experimental results show that the newly established finite difference scheme is very reliable for solving approximate analytical solutions of fractional partial differential equations.In the third part,an implicit finite difference scheme is used to solve the numerical solution of the time fractional order and the variable coefficient convection-diffusion equation.Theoretical analysis proves the stability of the scheme and analyzes the error estimate,and the numerical results show the accuracy of the established scheme.