An inverse boundary value problem for isotropic nonautonomous heat flows

TL;DR

This paper proves the unique determination of thermal properties in isotropic nonautonomous heat flows from boundary measurements, advancing inverse boundary value problem theory for heat equations.

Contribution

It establishes uniqueness results for identifying thermal conductivity and heat capacity from boundary data in all dimensions under specific diffusivity assumptions.

Findings

Uniqueness of thermal property reconstruction in all dimensions.

Extension of results to anisotropic media in two dimensions.

Use of exponential solutions related to thermal diffusivity.

Abstract

We study an inverse boundary value problem on the determination of principal order coefficients in isotropic nonautonomous heat flows stated as follows; given a medium, and in the absence of heat sources and sinks, can the time-dependent thermal conductivity and volumetric heat capacity of the medium be uniquely determined from the Cauchy data of temperature and heat flux measurements on its boundary? We prove uniqueness in all dimensions under an assumption on the thermal diffusivity of the medium, which is defined as the ratio of the thermal conductivity and volumetric heat capacity. As a corollary of our result for isotropic media, we also obtain a uniqueness result, up to a natural gauge, in two-dimensional anisotropic media. Our assumption on the thermal diffusivity is related to construction of certain families of exponential solutions to the heat equation.