Coercivity-based analysis and its application to an inverse source problem for a subdiffusion equation with time-dependent principal parts

Abstract In this article, we investigate an inverse source problem for subdiffusion equations with time-dependent principal parts using distributed observations. We analyze coercivity properties of fractional derivatives. As applications, we use a coercivity property to establish a Lipschitz stability estimate for an inverse problem of determining the temporal component in the source term, which is a new result to the best knowledge of the authors. Numerically, we transform the inverse source problem into an optimization framework, incorporating Tikhonov regularization to mitigate the ill-posedness of the inverse problem. We then propose an iterative thresholding algorithm to solve this optimization problem efficiently. Finally, we present several numerical experiments to validate the accuracy and the efficiency of our proposed algorithm.