Regularization Methods And Algorithms For Three Kinds Ill-posed Problems On Spherically Symmetric Domain

This paper mainly considers three kinds of ill-posed problems on spherically symmetric regions,which are the unknown source identification problem and the initial value inversion problem for time-fractional diffusion equation on the spherically symmetric regions,the unknown source identification problem for time fractional wave equations on the spherically symmetric regions.In the second chapter,we discuss the unknown source identification problem of time fractional diffusion equations on the spherically symmetric domain.Firstly,we analyse the ill-posedness of the problem.Next,we use the truncation regularization method to restore the stability of the solution and gives the Holder error estimation formula under an a prior and an a posterior regularization parameter selection rules.Finally,numerical simulations verify the effectiveness of the truncated regularization method for solving such problems.In the third chapter,the inversion initial value problem of the time fractional diffusion equations is discussed on the spherically symmetric domain.Firstly,the illposedness of the problem is described.We apply the quasi-boundary regularization method to obtain the regularization solution and give the Holder error estimation formula under two regularization parameter choice rules.Finally,the numerical results show that the quasi-boundary regularization method is very stable for solving such problems.In the fourth chapter,the unknown source identification problem of time fractional wave equations is considered on spherically symmetric domain.This chapter first finds the solution of the problem and proves the ill-posedness of the problem.Secondly,the Landweber iterative method is used to obtain the regularization solution of the problem.And the Holder error estimation formula is given under an a priori and an a posterior regularization parameter selection rules.Finally,the numerical results show the stability of the Landweber iterative regularization method for solving such problems.