A Generalized Quasi-Boundary Value Method for an Inverse Source Problem in a Distributed Order Time-Fractional Diffusion Equation

Abstract

Subdiffusion equations with distributed-order fractional derivatives describe important physical phenomena. In this paper, we consider an inverse space-dependent source term problem governed by a distributed order time-fractional diffusion equation using final time data. Based on the series expression of the solution, the inverse source problem can be transformed into a first kind of Fredholm integral equation. The existence, uniqueness and a conditional stability of the considered inverse problem are established. Building upon this foundation, a generalized quasi-boundary value regularization method is proposed to solve the inverse source problem, and then we prove the well-posedness of the regularized problem. Further, we provide the convergence rates of the regularized solution by employing an a priori and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to validate the effectiveness of the proposed method.