Research On Two Kinds Of Inverse Problems Of The Multi-term Time-space Fractional Diffusion Equation

Multi-term time-space fractional diffusion equations have a wide application prospect in many fields of science and engineering,and the related inverse problems have been widely concerned by scholars,and a lot of research results have been obtained.In this paper,the backward problem and inverse source problem of the multi-term time-space fractional diffusion equation are studied.In Chapter 1,the research status,significance,and content of this paper on the direct and inverse problems of fractional diffusion equations are introduced.In Chapter 2,the definition and properties of the multinomial Mittag-Leffler functions and related fractional derivatives are introduced.In Chapter 3,the initial data for the multi-term time-space fractional diffusion equation are derived from the final data of observations.To solve the initial boundary value problem,use the method by which the matrix transfer technique is combined with the implicit finite difference technology.The Tikhonov regularization method is used to transform the inverse problem into a variational problem,and the approximate solution of the inverse problem is given by means of an optimal perturbation algorithm.Numerical examples show that the proposed algorithm is effective and stable.In Chapter 4,we use additional internal observation data to identify the time-dependent source term in the multi-term time-space fractional diffusion equation.Firstly,the existence and uniqueness of the solution to the positive problem are proved,and then the uniqueness and a stability estimate of the solution to the inverse problem are estimated.Finally,the Tikhonov regularization method is used to transform the inverse problem into a variational problem,and the approximate solution to the inverse problem is given by using the optimal perturbation algorithm.The results show that the proposed algorithm is effective and stable by one-and two-dimensional examples.In Chapter 5,we use the final data to invert the space-dependent source term in the multi-term time-space fractional diffusion equation.Firstly,the operator equation is constructed from the original equation,and some important properties of the constructed operator are proved.Then,using the above properties and the analytic Fredholm theorem,it can be obtained that the space-dependent source term is uniquely and continuously dependent on the additional final value data.Finally,the Tikhonov regularization method is used to transform the inverse problem into a variational problem and the approximate solution of the inverse problem is given by means of the optimal perturbation algorithm.Numerical examples show the effectiveness of the algorithm.The sixth chapter is a summary of the research content,and we will do follow-up work.