Regularization Methods For Two Inverse Problems Of Two Kinds Of Fractional Time Diffusion Equation

In this paper,we mainly consider the ill-posed problem of the wave equation in the special region,specifically,we consider the inverse initial value problem of the time fractional wave equation in the spherically symmetric region.We also consider the ill-posed problem of the diffusion equation in the general domain.Specifically,we consider the inverse initial value problem of the time fractional diffusion equation with Caputo-like hyper-Bessel operator and the problem of identifying the source term.In chapter 2,we consider the inverse initial value problem of the time-fractional wave equation in the spherically symmetric region.Secondly,three regularization solutions of the problem are given by using three Landweber regularization methods,and the convergence error estimators between the corresponding regularization solutions and the exact solutions are given.Finally,numerical examples are used to demonstrate that one of the fractional Landweber regularization methods is the most effective method to restore the stability of the problem.In chapter 3,we consider the inverse initial value problem of the fractional diffusion equation with Caputo-like hyper-Bessel operator.Firstly,the expression of the exact solution of the problem is given,and we find that the problem is ill-posed from the form of the solution.Secondly,the regular solution of the problem is given by the fractional Landweber regularization method,and the H¨older-type error estimator between the exact solution and the regular solution is given.Finally,numerical results are given to illustrate the effectiveness of the fractional Landweber regularization method to recover the ill-nature of the problem.In chapter 4,we consider the problem of identifying source terms for the time fractional diffusion equations with Caputo-like hyper-Bessel operators.Firstly,we know that the problem is ill-posed by its solution form.Secondly,the fractional Landweber regularization method and the fractional Tikhonov regularization method are used to obtain the regular solution of the problem,respectively.The prior and posterior H¨older-type error estimates of the corresponding regular solution and the exact solution are given under different regularization parameters.Finally,a numerical example is given to illustrate the effectiveness of the regularization method.So far,Landweber iterative regularization methods can be divided into three types: the classic Landweber iterative regularization method,the fractional Landweber iterative regularization method,and the improved Landweber iterative regularization method.However,there is no research result indicating which regularization method is The best,so in the second chapter of this article,different methods are used to solve the same problem,and it is concluded that the fractional Landweber iterative regularization method is the most effective.Regarding the time fractional diffusion with Caputo-like hyper-Bessel operator,there are few researches on the inverse problem of equations,so the third and fourth chapters of this article have studied its related problems.