Two Kinds Of Inverse Problems Of Time Fractional Diffusion-wave Equation

We discuss two kinds of the inverse problems of time fractional diffusion wave equations:the source term identification problem and the initial value inversion problem.The ill-posed causes of two kinds of inverse problems are theoretically analyzed.Based on their ill-posedness,we propose two regularization methods to solve them.One is to construct an iterative regularization from the problem by relying on the idea of Landweber iterative regularization;the other is to transform the unit operator in Tikhonov regularization and propose a variational regularization method.The specific research content is as follows:(1)Study the source term identification problem of the time fractional diffusion wave equation,including Dirichlet boundary and Neumann boundary.In the two boundary cases,we obtain the iterative regularization solution and the variational regularization solution,as well as the error estimation between the regularization solution and the exact solution under different parameter selection rules.Finally,we take the Dirichlet boundary case as an example to give numerical examples of two regularization methods.In the numerical experiment,we first give all the definite solution conditions,solve the positive problem by the finite difference method,and then add noise to the terminal data of the solution of the positive problem.The noise-added terminal data is used as the input data with measurement error,and combined with the rest of the definite solution conditions except the source term,two regularization methods are used to identify the source term.(2)Research the initial value inversion of the time fractional diffusion wave equation.Because in the definite solution condition of the diffusion wave equation(the order a satisfies 1<?<2),there are two initial value conditions u(x,0)=?(x)and ut(x,0)=?(x).Under the Dirichlet boundary,we assume that ?(x)is known to invert the initial value ?(x);under the Neumann boundary,assume that ?(x)is known to invert the initial value ?(x).We also use iterative regularization and variational regularization to solve the two initial value inversion problems,obtain the iterative regularization solution and the variational regularization solution,and also give the error estimates under different parameter selection rules.Finally,we take the inversion ?(x)under Dirichlet boundary conditions as an example to give numerical examples of two regularization methods.The numerical process is similar to that described in(1).For the above two methods,we give Holder-type error estimates,and the order of convergence is consistent with other regularization methods.However,the iterative regularization proposed in this paper is based on the problem and is more direct.Variational regularization is to generalize the unit operator in Tikhonov regularization.Numerical experiments verify the effectiveness of the two regularization methods in solving the source term identification problem and the initial value inversion problem of the time fractional diffusion wave equation.