Simultaneous uniqueness for a coefficient inverse problem in one-dimensional fractional diffusion equation from an interior point measurement

TL;DR

This paper proves the uniqueness of simultaneously determining a spatially varying coefficient and a Robin coefficient in a one-dimensional fractional diffusion equation using interior point measurements over time.

Contribution

It establishes a new uniqueness result for an inverse problem involving fractional diffusion equations with interior data, utilizing spectral analysis and the Weyl m-function.

Findings

Uniqueness of the inverse problem is proven.

The method applies spectral theory to fractional diffusion.

Interior point measurements suffice for coefficient recovery.

Abstract

This article is concerned with an inverse problem of simultaneously determining a spatially varying coefficient and a Robin coefficient for a one-dimensional fractional diffusion equation with a time-fractional derivative of order $\alpha\in(0,1)$. We prove the uniqueness for the inverse problem by observation data at one interior point over a finite time interval, provided that a coefficient is known on a subinterval. Our proof is based on the uniqueness in the inverse spectracl problem for a Sturm-Liouville problem by means of the Weyl $m$-function and the spectral representation of the solution to an initial-boundary value problem for the fractional diffusion equation.