Two regularization methods for identifying the source term of Caputo-Hadamard time-fractional diffusion equation

In this paper, the inverse problem for identifying the unknown source on time-fractional diffusion equation with Caputo-Hadamard derivative is considered. This problem is ill-posed, and two regularization methods are used to solve it. Firstly, we prove that the problem is ill-posed and give the conditional stability and the optimal error bound. Secondly, the error estimates of the fractional Landweber iterative regularization method and the fractional Tikhonov regularization method under a priori and a posteriori parameter selection rules are given, respectively. Finally, numerical examples are given to illustrate the effectiveness of two regularization methods.

Automated summary

This paper considers the inverse problem of identifying the unknown source in the time-fractional diffusion equation with Caputo-Hadamard derivative. The problem is proved to be ill-posed, and two regularization methods are used to solve it. The error estimates of the fractional Landweber iterative regularization method and the fractional Tikhonov regularization method under different parameter selection rules are given. Numerical examples are provided to demonstrate the effectiveness of both regularization methods.