Inverse reconstruction of a spatial source term in a space-fractional diffusion equation via Nyström discretization

We study an inverse source problem for a diffusion equation governed by the integral fractional Laplacian on a bounded domain. The objective is to identify the unknown spatial component of a separable source term using final-time measurement data. Due to the ill-posed nature of the problem, small perturbations in the terminal observations may lead to significant reconstruction errors. To overcome this difficulty, the inverse problem is recast within a Tikhonov-regularization framework. An explicit expression of the Fréchet derivative and the associated gradient is derived through an adjoint-based analysis. The resulting optimization problem is solved using a conjugate gradient algorithm combined with Morozov’s discrepancy principle as a stopping rule. The nonlocal fractional Laplacian is discretized by means of a Nyström scheme based on its integral representation. Numerical experiments in both one- and two-dimensional settings are presented to demonstrate the accuracy and robustness of the proposed approach.