Recovering a space-dependent source term for distributed order time-space fractional diffusion equation

This paper considers the identification of space-dependent source function in the 1D distributed order time-space fractional diffusion equation. Such a problem is obtained from the fractional equation in which the order is replaced as a function Pi(alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varpi (\alpha )$$\end{document}. We prove regularity result for the direct problem, and the ill-posedness, uniqueness and conditional stability for the inverse source problem. In addition, we use iterative regularizing ensemble Kalman method to deal with inverse source problem and provide a numerical implementation. Finally, two numerical examples are conducted to demonstrate the validity of the ensemble-based method.