Studies On Determination Of Source And Coefficient For The Time-fractional Diffusion Equations

In recent years,the time fractional diffusion equations(TFDEs)have attracted wide attentions motivated by practical problems.The studies on the direct problems for the time-fractional diffusion equations have made great progress.However,the studies on the inverse problems of TFDEs,including theoretical analysis and numerical computation is still in its infancy.In view of the above situation,we discuss the following classes of inverse problems for the time-fractional diffusion equation in the present paper.In Part 1,we discuss a space-dependent source identification problem in a timefractional diffusion equation,and mainly study the uniqueness of inverse source problem in a bounded domain of multidimensional case.The existence and uniqueness of the weak solution for the direct problem is proved by the separation variable method.Then we prove the uniqueness for the inverse source problem by the Laplace transformation.In Part 2,we study an identification of the space-dependent zeroth-order coefficient in a time-fractional diffusion equation,and mainly investigate the uniqueness of determining the zeroth-order coefficient and its numerical reconstruction from two boundary measurement data in one-dimensional case,which is a nonlinear inverse problem.The existence and uniqueness of two kinds of weak solutions for the direct problem are proved.Then we provide the uniqueness for recovering the zeroth-order coefficient and fractional order simultaneously by the Laplace transformation and the Gel'fand-Levitan theory.The identification of the zeroth-order coefficient is formulated into a variational problem by the Tikhonov regularization.The existence,stability and convergence of the solution for the variational problem are provided.Finally,the conjugate gradient method is used to solve it.Numerical examples are provided to show the effectiveness of our method.In Part 3,we devote our efforts to an inverse space-dependent source problem in a multi-term time-fractional diffusion equation from boundary measured data,which is an extension of Part 1,and obtain the uniqueness of inverse source problem in a bounded domain of multidimensional case.First,we give a weak formulation by the variational method for the direct problem,and obtain the existence and uniqueness of a weak solution for the direct problem by using the well-known Lax-Milgram theorem.Then by employing the Laplace transformation and related properties of multinomial Mittag-Leffler function,the uniqueness for the inverse source problem is derived.In Part 4,we investigate a nonlinear inverse problem for identifying the spacedependent zeroth-order coefficient and fractional orders in a multi-term time-fractional diffusion equation,which is an extension of Part 2,and we mainly discuss the uniqueness for recovering the zeroth-order coefficient and fractional orders simultaneously and its numerical reconstruction from boundary measurement data in one-dimensional case.First,we give a weak formulation for the direct problem,and obtain the existence and uniqueness of a weak solution for the direct problem.Then we provide the uniqueness of the inverse problems by the Laplace transformation,the Gel'fand-Levitan theory and analytic continuation skills.Finally,the identification of the zeroth-order coefficient and the fractional orders is formulated into a variational problem by the Tikhonov regularization,and the variational problem is solved by an alterative iteration method.With the help of sensitivity problem and adjoint problem we use a conjugate gradient method to find the approximate zeroth-order coefficient and use the optimal perturbation algorithm to recover the fractional orders.Numerical examples are provided to show the effectiveness of the proposed method.In Part 5,we devote our effort to a nonlinear inverse problem for identifying a time-dependent convection coefficient in a time-fractional convection diffusion equation,and we mainly obtained the stability for recovering the convection coefficient and its numerical reconstruction from the measured data at an interior point for onedimensional case.We prove the existence,uniqueness and regularity of solution for the direct problem by using the fixed point theorem.The stability of inverse convection coefficient problem is obtained based on the regularity of solution of direct problem and some generalized Gronwall's inequalities.We use the optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function to solve the inverse convection coefficient problem.Two numerical examples are provided to show the effectiveness of the proposed method.