Inverse source problem for the time-space fractional diffusion equation involving the fractional Sturm–Liouville operator

In this work, we consider an inverse source problem for the time-space fractional diffusion equation with homogeneous Dirichlet boundary conditions, in which the spatial operator under consideration is the fractional Sturm–Liouville operator. We demonstrate that this inverse source problem is ill-posed in the sense of Hadamard and exhibit the uniqueness and conditional stability of its solution. To recover a stable solution, we employ an iterative generalized quasi-boundary value regularization method. We derive error estimates between the exact and regularized solutions based on both a-priori and a-posteriori choice rules for the regularization parameters. Notably, our method reduces to the generalized quasi-boundary value regularization method after one iteration and exhibits a better error convergence rate. Finally, we present three numerical examples to validate the effectiveness of the used regularization method.