A variational approach to recover the unknown source and initial condition for a time-space fractional diffusion equation

In this work, we investigate the inverse source and backward problems associated with the Caputo fractional derivative operator in time and the fractional Laplacian operator in space. We show that the inverse problems are ill-posed and establish uniqueness and conditional stability results. To deal with the ill-posed problems, a variational method is constructed by combining the concepts of variational regularization and mollification regularization. Next, we propose an a priori as well as an a posteriori regularization parameter selection rules and give the error estimates between the approximate and the exact solutions for both rules. We illustrate the robustness and validity of the variational approach by several numerical experiments.