The enclosure method for the heat equation using time-reversal invariance for a wave equation

TL;DR

This paper introduces a novel method using time-reversal invariance of the wave equation to explicitly determine the minimum enclosing sphere of an unknown cavity within a heat conductive body, based on specialized heat flux data.

Contribution

It develops an enclosure method for the heat equation leveraging wave equation properties, providing a stable and explicit formula for cavity detection.

Findings

Explicit formula for cavity enclosure using heat flux data

The heat flux remains bounded as the parameter increases

The method improves stability over previous approaches

Abstract

The heat equation does not have time-reversal invariance. However, using a solution of an associated wave equation which has time-reversal invariance, one can establish an explicit extraction formula of the minimum sphere that is centered at an arbitrary given point and encloses an unknown cavity inside a heat conductive body. The data employed in the formula consist of a special heat flux depending on a large parameter prescribed on the surface of the body over an arbitrary fixed finite time interval and the corresponding temperature field. The heat flux never blows up as the parameter tends to infinity. This is different from a previous formula for the heat equation which also yields the minimum sphere. In this sense, the prescribed heat flux is moderate.