Exponential Order Regularization Method For Source Term And Initial Value Inversion Of Time Fractional Diffusion Equations

Fractional diffusion equations and their inverse problems have a wide range of applications in many fields of science and engineering.Therefore,the study of inverse problems of fractional diffusion equations is not only of mathematical theoretical significance,but also of great scientific significance.This paper mainly considers time fractional.There are two types of ill-posed problems in the inverse problem of diffusion equations,that is,the source term inversion problem and the initial value inversion problem of the time fractional diffusion equation are studied separately.The first chapter introduces the significance and current situation of the research on the inverse problem of fractional diffusion equation,the main contents of this paper,the innovations,and the related definitions and basic lemmas of fractional calculus.The second chapter introduces the related definition of fractional calculus and a finite difference method for the numerical solution of the problem.The third chapter considers the source term inversion problem of time fractional diffusion equations in the bounded domain.First,the ill-posedness and conditional stability of the source term inversion problem are analyzed.Then,an index is constructed for the measurement data with noise.The order regularization method is used to reconstruct the source term,and the convergence rate of the regularization solution under the strategy of a priori selecting regularization parameters and a posteriori selecting regularization parameters is proved.Finally,several numerical examples are given.The calculation results show that the proposed algorithm is Effective.In Chapter 4,the exponential order regularization method is used to consider the initial value inversion problem of the time fractional order diffusion equation.Similar to Chapter 3,the ill-posedness and conditional stability of the Initial value inversion problem are analyzed.The initial distribution is reconstructed,and the convergence rate of the regularization solution under the strategy of selecting the regularization parameters a priori and choosing the regularization parameters a posteriori is proved.Finally,the numerical algorithm is used to verify the effectiveness of the proposed algorithm.The fifth chapter discusses a part of the research carried out before determining the research theme of the graduation thesis,that is,the problem of finding roots of nonlinear equations is considered.Starting from a special type of integral,a method of rooting of nonlinear equations is obtained.Combining the obtained method with the deformed Newton iterative method,a practical prediction-correction scheme for finding the roots of nonlinear equations is obtained,and it is proved that the scheme has at least local square convergence when ? = 1/2.Chapter 6 summarizes the full text and looks forward to the future.