Inverse source problem of sub-diffusion of variable exponent

TL;DR

This paper studies the inverse source problem in variable-exponent sub-diffusion models, proving uniqueness and stability, and proposing numerical algorithms for source reconstruction with numerical validation.

Contribution

It introduces a new approach combining perturbation methods, weak norms, and iterative algorithms for inverse problems in variable-exponent sub-diffusion models, with theoretical and numerical results.

Findings

Proved uniqueness of the inverse source problem from local data.

Established conditional stability in a weak norm.

Demonstrated effectiveness of numerical reconstruction algorithms.

Abstract

This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which transfers the original model to an equivalent but more tractable form, the analytical extensibility of the solutions and the weak unique continuation principle are proved, which results in the uniqueness of the inverse space-dependent source problem from local internal observation. Then, based on the variational identity connecting the inversion input data with the unknown source function, we propose a weak norm and prove the conditional stability for the inverse problem in this norm. The iterative thresholding algorithm and Nesterov iteration scheme are employed to numerically reconstruct the smooth and non-smooth sources, respectively. Numerical experiments are performed to investigate their effectiveness.