In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. Based on an a priori assumption, we give the optimal error bound analysis and a conditional stability result. Moreover, we use the Fourier regularization method to deal with this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Meanwhile, a new a posteriori parameter choice rule is also proposed. For the a priori and the a posteriori regularization parameters choice rules, we all obtain the convergence error estimates which are all order optimal. Numerical examples are presented to illustrate the validity and effectiveness of this method.

In this paper, the Cauchy problem of inhomogeneous Helmholtz equation is investigated. This problem is ill-posed and the truncation method is used to solve this inverse problem. Under the a priori and a posteriori parameter choice rules, the error estimates between the regularization solution and the exact solution is obtained, respectively. Numerical examples show that the truncation method works effectively.

In this paper, an inverse problem to identify the initial value for high dimension time fractional diffusion equation on spherically symmetric domain is considered. This problem is ill-posed in the sense of Hadamard, so the quasi-boundary regularization method is proposed to solve the problem. The convergence estimates between the regularization solution and the exact solution are presented under the a priori and a posteriori regularization parameter choice rules. Numerical examples are provided to show the effectiveness and stability of the proposed method.