Two Kinds Of Ill-posed Problems For Time-fractional Diffusion Equation

In this thesis, we consider two kinds of ill-posed problems for the time-fractional diffusion equation, i.e.. the backward problem and the inverse source problem.The backward problem is a classical ill-posed problem. For the backward prob-lem for a time-fractional diffusion equation with variable coefficients in a general bounded domain, we first analyse the ill-posedness of the problem, then the opti-mal error bound for the problem under a source condition is obtained. We apply the Tikhonov regularization method, a simplified Tikhonov regularization method, the quasi-boundary value regularization method, a modified quasi-boundary value regularization method and an iteration method to deal with this problem. The corresponding convergence rates for these methods are analyzed under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule, respectively. For the numerical experiment, we use the finite difference method for1-d case and the finite element method combined with finite difference method for2-d case.In the second part of this thesis, the inverse problem of identifying a space-dependent source for the time-fractional diffusion equation is investigated. We apply the Tikhonov regularization method, a simplified Tikhonov regularization method and the quasi-reversibility method to solve it. The corresponding convergence es-timates for these methods are obtained under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule, respectively. Many numerical examples are given to show that the regularization methods are effective and stable.Among all these works, the modified quasi-boundary value regularization method and an iteration method are two new and interesting methods. The a posteriori regu-larization parameter choice rules for the quasi-boundary value regularization method and the quasi-reversibility method are proposed for first time. These methods and results can be applied to other problems and be extensively studied further.The numerical results are consistent with the theoretical results. These results show that our regularization methods for these ill-posed problems work effectively.