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 \(\varpi (\alpha )\) $\varpi (\alpha )$ . 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.