Studies On Some Inverse Problems For Time-fractional Diffusion-wave Equations

In this thesis,we consider the following inverse problems for time fractional diffusion-wave equations,i.e.determine a space-dependent source by terminal measurement data,recover a fractional order of derivative and diffusion coefficient by boundary measurement data,determine a time-dependent zeroth-order coefficient and a time-dependent source by data from two measuring points,identify the fractional order of derivative and the time-dependent zeroth-order coefficient by an integral data.Part 1 studies an inverse space-dependent source function from noisy final time measured data.We provide a regularity of the corresponding direct problem and the existence and uniqueness of the adjoint problem.The uniqueness of the inverse problem is discussed.Using the Tikhonov regularization method,the inverse source problem is formulated into a variational problem and a conjugate gradient algorithm is proposed to solve it.The efficiency and robust of the proposed method are supported by some numerical experiments.Part 2 devotes to identifying the fractional order of derivative and a diffusion coefficient in a time fractional diffusion wave equation from boundary observation data in one dimensional case.By the Laplace transform and Gel'fand-Levitan theory,we prove the uniqueness of the inverse problem of recovering the fractional order and diffusion coefficient.In addition,we provide an iterative regularizing ensemble Kalman method to carry out a numerical implementation of the inverse problem.Four numerical examples are carried out to verify the performance of the proposed method.Part 3 aims to identify simultaneously a time-dependent zeroth-order coefficient and a time source term in a time fractional diffusion-wave equation from two points observed data.First of all,using the fixed point theorem,we prove the existence and uniqueness of the solution for the direct problem.Secondly,the stability of the inverse problem is proved and the uniqueness is a direct result of the stability estimate.In addition,we illustrate the ill-posedness of the inverse problem and use a non-stationary iterative Tikhonov regularization method to recover numerically the time dependent zeroth-order coefficient and time source term.At the same time,we give the existence of the minimizer for the minimization functional.In order to solve the minimization problem,we apply the alternating minimization method to find the minimizer and give the stability of solving sub-problem of minimization functional as well as prove the data fidelity item decreases monotonously with the iterative running.Finally,some numerical examples are provided to shed light on the validity and robustness of the numerical algorithm.Part 4 investigates a nonlinear inverse problem of identifying simultaneously fractional order and a time-dependent zeroth-order coefficient in a time-fractional diffusion wave equation from an integral data.We firstly prove that the given integral data can uniquely determine the fractional order and time-dependent zeroth-order coefficient.In addition,a Bayesian method is used to reconstruct numerically the fractional order of derivative and time-dependent zeroth-order coefficient.Further,we prove that the well-definedness and well-posedness of Bayesian posterior measure when only the time coefficient is identified.Moreover,we adopt an adaptive ensemble Kalman inversion method to solve the Bayesian inverse problem.Finally,some numerical examples are presented to confirm the effectiveness of the proposed method.