Determining a Time-Varying Potential in Time-Fractional Diffusion from Observation at a Single Point

TL;DR

This paper addresses the inverse problem of identifying a time-varying potential in a time-fractional diffusion equation using single-point boundary data, establishing stability, developing an iterative reconstruction algorithm, and validating with numerical simulations.

Contribution

It introduces a new method for recovering a time-dependent potential in fractional diffusion models with proven stability and error bounds, supported by numerical demonstrations.

Findings

Conditional Lipschitz stability established

An iterative algorithm for potential recovery developed

Numerical simulations confirm theoretical results

Abstract

We discuss the identification of a time-dependent potential in a time-fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem. Numerically, we develop an easily implementable iterative algorithm to recover the unknown coefficient, and also derive rigorous error bounds for the discrete reconstruction. These results are attained by using the (discrete) solution theory of direct problems, and applying error estimates that are optimal with respect to problem data regularity. Numerical simulations are provided to demonstrate the theoretical results.