Inverse source problem of sub-diffusion of variable exponent
Zhiyuan Li, Chunlong Sun, Xiangcheng Zheng
TL;DR
The paper studies the inverse space-dependent source problem for a variable-exponent sub-diffusion equation by transforming the model to a tractable constant-exponent form and proving analytic extensibility of solutions. It establishes a weak unique continuation principle via Laplace transform and Dirichlet eigenfunction analysis, leading to uniqueness from local interior or partial boundary data. A variational identity yields a weak norm under which Lipschitz-type conditional stability is proven for interior and boundary observations. Numerically, the authors implement iterative thresholding for smooth sources and a TV-regularized primal-dual method for non-smooth sources, confirming effective reconstructions from partial observations and highlighting remaining challenges near edges and far from the observation domain.
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.