Weak Galerkin Finite Element Method For Time-fractional Partial Differential Equation

Fractional differential equation is a generalization of integerorder differential equation,which is widely applied in the heat conduction of memory materials,nuclear reaction,dynamics and other problems.Many scholars at home and abroad have done a lot of research on the numerical analysis of fractional differential equations.In this paper,we mainly study the numerical calculation of time fractional equations by some numerical methods.Timefractional diffusion equations,time-fractional evolution equations and multi-term time-fractional integro-differential equations are considered,then the numerical schemes of the three equations are established respectively,and we give the theoretical analysis of each scheme,and the correctness of the algorithm and theory is verified by numerical examples.First of all,we mainly introduce the research background and significance of the problem,as well as the relevant preparatory knowledge needed in this paper.We focus on three kinds of timefractional equations and introduce the ideas and current situation of weak Galerkin finite element method in this paper.Combined with the characteristics of weak Galerkin finite element method,the research motivation of this paper is proposed.Then,we discuss the time-fractional diffusion equation and the time-fractional evolution equation in one-and two-dimensional space,respectively.The weak Galerkin finite element method is used to establish the numerical schemes of the equation,and the theoretical analysis is carried out.The stability and convergence of the schemes have been proved and we present some numerical experiments which verify our theoretical analysis.Finally,the advantages of alternating direction implicit(ADI)scheme are introduced.Combined with the finite difference method and ADI algorithm,the finite difference ADI scheme of twodimensional multi-term time-fractional integro-differential equation is designed.The truncation error,stability and convergence of the scheme are proved.Also,we give some numerical examples to verify the superiority and correctness of the algorithm,which provides ideas for the combination of ADI and weak Galerkin finite element method.At the end of the paper,we give the summary of paper and prospect the follow-up work.