Ill-posed Analysis And Regularization Methods For A Inverse Problem Of The Time-fractional Diffusion Equation

The inverse problem of time-fractional diffusion equation often used in many engineering physical problems.For example,that the inside temperature of the physical problems,only through edge temperature measurements to inversion.This paper mainly studies 0<γ<1 inverse problem for diffusion equation with a linear source F≠0:Here,we define the fractional derivative of order γ in the sense of Caputo:The inverse problem is to determine u(x,t)for x∈[0,1)from the mea-surement data of u(1,·).We show the problem is ill-posed by theorem prov-ing.Therefore,we need choose appropriate regularization method to deal with the problem.In order to guarantee the research,we study the problem in case of F=0.First of all,we assume that the prior condition:‖u(0,t)‖≤E.In addition,suppose that gδ(t)is disturbance data of the measured value g(t)and satisfying‖gδ-g‖≤δ.So we need to consider the following question:There are few results on this inverse problem.In our paper,we will propose two different regularization methods:iteration method and convolution method to deal with the problem.That are iteration scheme:and convolution scheme:Also given the way of the selection of regularization parameter and error esti-mate under the prior condition.If we select k=c[E/δ],then we can get where ε=max{e1/(?)δ,E1-xδx}Finally,we will through the corresponding numerical example to verify the feasibility and effectiveness of the convolution regularization method.