Research On Uniqueness And Regularization Algorithms Of Three Kinds Of Inverse Problems For Time-fractional Diffusion-wave Equations

In this thesis,we consider three kinds of inverse problems for time-fractional diffusion-wave equations,i.e.the inverse initial value problem,the multiple parameters identification problem,and the zeroth-order coefficient identification problem.In Part 1,we investigate an inverse initial value problem for a time-fractional diffusion-wave equation to determine the initial value from an additional final time data.By using the separation variable method,we provide a regularity of the weak solution for the direct problem and we also prove the existence and uniqueness of a weak solution for the adjoint problem.The considered inverse problem is formulated into a variational problem by the Tikhonov regularization method.Based on a new formula of fractional order integration by parts and the strict deduction of the adjoint problem which can be discretized numerically,we derive the gradient of the Tikhonov regularization functional.Then we propose a conjugate gradient algorithm combined with the Morozov's discrepancy principle to solve the inverse initial value problem numerically,and four numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness and stability of the proposed algorithm.In Part 2,we discuss a multiple parameters identification problem in a onedimensional time-fractional diffusion-wave equation for determining the fractional order,the initial flux speed and the boundary Neumann data simultaneously from the Cauchy observation data at the end of the interval.We prove the uniqueness result for this inverse problem by using a new estimate for the Mittag-Leffler function and the mean value theorem,the Laplace transform combining with analytic continuation.Based on the application of Bayesian framework in inverse problems,we use an iterative regularizing ensemble Kalman method to solve the considered inverse problem numerically.And four numerical examples are provided to show the effectiveness and stability of the proposed algorithm.In Part 3,we consider a zeroth-order coefficient identification problem in a timefractional diffusion-wave equation for recovering a time-dependent zeroth-order coefficient from an additional integral condition.We prove the uniqueness and a conditional stability for such an inverse problem by using some regularity estimates for the direct problem and the Gronwall inequality.Then the two-point gradient method is used to solve the inverse zeroth-order coefficient problem numerically.Some properties of the forward operator are obtained,such as the Frechet differentiability,the Lipschitz continuity and the tangential cone condition to guarantee the convergence of the twopoint gradient method while applying on the inverse zeroth-order coefficient problem.Four numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness and stability of the suggested algorithm.