AN INVERSE TIME-DEPENDENT SOURCE PROBLEM FOR DISTRIBUTED-ORDER TIME-SPACE FRACTIONAL DIFFUSION EQUATION

This paper focuses on the inverse time-dependent source term problem in a distributed-order time-space fractional diffusion equation (DTSFDE) using initial and boundary conditions and boundary Cauchy data. Firstly, we prove the existence and uniqueness of the solution to the direct problem under homogeneous Neumann boundary conditions. Additionally, based on regularity of the solution to the direct problem, uniqueness and stability estimates for the inverse problem are established. Subsequently, we convert the inverse problem into a variational problem using the Tikhonov regularization method, and used the conjugate gradient algorithm to solve the variational problem, obtaining an approximate solution to the inverse source problem. Finally, we validate the effectiveness and stability of the proposed algorithm through numerical examples.