An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method

TL;DR

This paper introduces a novel approach using the enclosure method to determine the minimum length of broken paths related to an unknown discontinuity inside a 3D heat conductive body from boundary temperature and heat flux data.

Contribution

It presents a new inverse problem solution for 3D heat equations, revealing the minimum path length to locate discontinuities using boundary measurements.

Findings

Minimum length of broken paths can be determined from boundary data.

Single boundary measurement suffices under certain conditions.

Enclosure method effectively identifies internal discontinuities.

Abstract

This paper studies a prototype of inverse initial boundary value problems whose governing equation is the heat equation in three dimensions. An unknown discontinuity embedded in a three-dimensional heat conductive body is considered. A {\it single} set of the temperature and heat flux on the lateral boundary for a fixed observation time is given as an observation datum. It is shown that this datum yields the minimum length of broken paths that start at a given point outside the body, go to a point on the boundary of the unknown discontinuity and return to a point on the boundary of the body under some conditions on the input heat flux, the unknown discontinuity and the body. This is new information obtained by using enclosure method.