Inverse Source Reconstruction in a Space–Time Fractional Diffusion Model with Variable-Order Caputo Derivative from Terminal Observations

Abstract

This paper investigates an inverse source problem for a space–time fractional diffusion equation involving a spectral fractional Laplacian in space and a variable-order Caputo derivative in time. The goal is to reconstruct an unknown spatial source term from terminal-time observations, a setting that is known to be severely ill-posed due to the strong smoothing effect of the underlying diffusion operator. From a theoretical perspective, we first establish the well-posedness of the associated forward problem in an appropriate fractional energy framework, under an explicit structural hypothesis on the variable-order Caputo derivative that is known to hold in the constant-order case and under additional regularity conditions in the genuinely variable-order case. The inverse problem is then reformulated as a linear operator equation, and its identifiability is rigorously analyzed under a companion positivity hypothesis. In particular, we provide a spectral characterization of the forward operator, which allows us to explicitly explain the mechanism of instability through the decay of modal coefficients. On the computational side, we propose an iterative reconstruction approach based on the conjugate gradient method combined with Morozov’s discrepancy principle, providing an effective regularization strategy without requiring explicit tuning of regularization parameters. The space operator is discretized using a spectral method, while the variable-order Caputo derivative is approximated by an L1-type scheme adapted to time-dependent fractional orders. The main novelty of this work lies in the unified treatment of variable-order time-fractional dynamics and inverse source reconstruction in one-dimensional settings, together with a comparison against the constant-order model; the two-dimensional experiment illustrates the reconstruction method but does not itself constitute a variable-versus-constant order comparison. Numerical experiments, including challenging nonsmooth sources and multiple noise levels, demonstrate that the proposed method is stable and accurate. The results indicate that, while variable-order effects introduce additional modeling flexibility, the reconstruction quality is predominantly governed by the noise level and the intrinsic ill-posedness of the problem.