Inverse Problems for the Heat Equation with Memory

TL;DR

This paper investigates inverse boundary problems for a one-dimensional heat equation with memory, providing explicit solutions and uniqueness results under certain conditions, extending classical inverse problem theories to integro-differential equations.

Contribution

It offers explicit formulas for solutions and proves uniqueness for inverse problems involving heat equations with memory, a novel extension of classical inverse problem results.

Findings

Explicit solution formulas for the inverse problem

Uniqueness results similar to Borg--Marchenko theorem

Conditions on the kernel ensure solvability

Abstract

We study inverse boundary problems for a one dimensional linear integro-differential equation of the Gurtin--Pipkin type with the Dirichlet-to-Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator, we give the explicit formula for the solution of the problem with the observation on the semiaxis $t>0.$ For the observation on finite time interval, we prove the uniqueness result, which is similar to the local Borg--Marchenko theorem for the Schr\"odinger equation.