Regularization Method And Algorithm For Inverse Problem Of Time Fractional Diffusion Equation

Compared with the integer order diffusion equation,the fractional diffusion equation can describe the anomalous diffusion phenomenon more accurately.At present,in many scientific fields,researches on fractional diffusion equations have received more and more attention,especially on the inverse problem of fractional diffusion equations.This paper is divided into three parts to discuss the inverse problem of time fractional diffusion equation.The second chapter covers the inverse initial value problem of the nonhomogeneous time-fractional diffusion heat equations.This problem is ill-posed.Based on discrete random noise data,the trigonometric method in nonparametric associated with the quasi-boundary value regularization method is used to obtain the regularization solution of the problem,and the error estimate between the exact solution and the regularization solution is given.The third and fourth chapters considers the unknown source identification problem of the nonhomogeneous time-fractional diffusion-wave equation in a general bounded domain and the inverse initial value problem of the nonhomogeneous timespace fractional diffusion-wave equation.Both types of problems are ill-posed.In this paper,the Landweber iterative regularization method is used to solve these two kinds of problems.Under an a priori and an a posteriori regularization parameters choice rules,corresponding error estimation formulas are derived.Finally,numerical examples in two different dimensions are solved to show the feasibility and effectiveness of the Landweber iterative regularization method for these two kinds of problems.