The Fractional Landweber Regularization Method For Solving Inverse Source Problem Of Time-Fractional Diffusion Equation

In recent years,fractional diffusion equations have attracted more and more attention,which are used to describe a large number of anomalous diffu-sion phenomenon in nature,such as pollutant diffusion,heat transfer,etc.If the diffusion of pollutant in a river channel is simulated by fractional diffusion equation,then the problem of identifying the discharge intensity of pollution sources can be transformed into the problem of identifying source term of the equation,which belongs to a class of significant inverse problems and has im-portant applications in the field of environmental hydraulics.Since the inverse source problem is usually ill-posed,the current research on it mainly consists of two aspects,one is to find or establish an appropriate regularization method and the second is to apply the regularization method to the inverse source problem.Many experts and scholars have studied the inverse source problem of time-fractional diffusion equation,and have given some good results.But they mostly study about the case where the time derivative is single and the source term is f(x).In this paper,the fractional Landweber regularization method is improved based on Kirsch's filter function theory,and the improved method is applied to the inverse source problem of single/multi-term time-fractional diffusion equation respectively where the source term is r(t)f(x)+W(x,t).Numerical results show that the improved method is effective and stable.The paper is composed as follows:Chapter 1 is the introduction which mainly introduces the research back-ground and meaning of inverse source problem of time-fractional diffusion e-quation,and the main work of this paper is summarized.Chapt.er 2,chapter 3 and chapter 4 are the main contents of this paper.First,from the perspective of singular value decomposition of compact opera-tor,the fractional Landweber method is combined with TSVD method and an improved fractional Landweber regularization method is proposed,and the er-ror estimation under the prior choice and posterior choice of parameter is given respectively,which proves that the improved method is of optimal order.Next,the improved regularization method is applied to the inverse source problem of the single-term and multi-term time-fractional diffusion equation respectively,whose target is to obtain f(x)when the source is r(t)f(x)+W(x,t),and a more precise error estimation is given in single case.Finally,the numerical scheme for solving the inverse source problem is studied.L1 compact differ-ence scheme is used to discrete equation.Numerical examples show that the computation speed and error are better than fractional Landweber method.Chapter 5 is the conclusion of this paper.