Recovery of a time-dependent reaction coefficient in a space-time fractional diffusion model via Nyström discretization

We study an inverse problem for a space-time fractional diffusion equation posed on a bounded one-dimensional domain under an exterior Dirichlet condition. The model incorporates a fractional derivative in time of Caputo type and a nonlocal spatial diffusion operator given by the integral fractional Laplacian. The objective is to recover an unknown time-dependent reaction coefficient from a limited number of pointwise observations combined with multiple experiments with distinct initial conditions. We establish a rigorous functional framework for the direct problem and prove existence, uniqueness, and stability of solutions. A conditional identifiability result for the reaction coefficient is derived under a natural non-degeneracy assumption associated with the multi-experiment measurement setting. The inverse problem is formulated as a nonlinear least-squares optimization problem in a finite-dimensional parameter space based on piecewise linear basis functions. For the numerical implementation, the spatial nonlocal operator is discretized using a Nyström quadrature method, while the time-fractional derivative is approximated by a classical $L1$ finite difference scheme. The resulting optimization problem is solved by a Levenberg-Marquardt (LM) algorithm, in which the sensitivity information is computed using finite-difference approximations. Several numerical experiments are presented to illustrate the effectiveness and stability of the proposed approach.