Numerical Recovery of a Time-Dependent Potential in Subdiffusion

TL;DR

This paper addresses the inverse problem of recovering a time-dependent potential in a subdiffusion model with fractional derivatives, providing theoretical stability results and a practical fixed point method validated by numerical experiments.

Contribution

It introduces a novel conditional Lipschitz stability theorem and a simple fixed point iteration method for coefficient recovery in subdiffusion models.

Findings

Proves a new stability estimate for the inverse problem.

Develops an efficient fixed point algorithm for potential recovery.

Demonstrates accuracy and efficiency through numerical experiments.

Abstract

In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian--Caputo fractional derivative in time. Theoretically, we prove a novel conditional Lipschitz stability result, and numerically, we develop an easy-to-implement fixed point iteration for recovering the unknown coefficient. In addition, we establish rigorous error bounds on the discrete approximation. These results are obtained by crucially using smoothing properties of the solution operators and suitable choice of a weighted $L^p(0,T)$ norm. The efficiency and accuracy of the scheme are showcased on several numerical experiments in one- and two-dimensions.