Unknown source identification problem for space-time fractional diffusion equation: optimal error bound analysis and regularization method

In this paper, the problem of unknown source identification for the space-time fractional diffusion equation is studied. In this equation, the time fractional derivative used is a new fractional derivative, namely, Caputo-Fabrizio fractional derivative. We have illustrated that this problem is an ill-posed problem. Under the assumption of a priori bound, we obtain the optimal error bound analysis of the problem under the source condition. Moreover, we use a modified quasi-boundary regularization method and Landweber iterative regularization method to solve this ill-posed problem. Based on a priori and a posteriori regularization parameter selection rules, the corresponding convergence error estimates of the two regularization methods are obtained, respectively. Compared with the modified quasi-boundary regularization method, the convergence error estimate of Landweber iterative regularization method is order-optimal. Finally, the advantages, stability and effectiveness of the two regularization methods are illustrated by examples with different properties.