The enclosure method for the heat equation

TL;DR

This paper extends the enclosure method, originally for elliptic equations, to inverse initial boundary value problems for the heat equation, providing an explicit technique to identify discontinuities within a heat-conductive body.

Contribution

It introduces a novel application of the enclosure method to parabolic equations, specifically the heat equation, with an explicit approach for detecting internal discontinuities.

Findings

Method successfully extracts support function values for unknown discontinuities.

Applicable to inverse problems with heat equations and boundary temperature data.

Provides a practical approach for identifying internal structures in heat conductive bodies.

Abstract

This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary value problems whose governing equation is the heat equation is considered. An explicit method to extract an approximation of the value of the support function at a given direction of unknown discontinuity embedded in a heat conductive body from the temperature for a suitable heat flux on the lateral boundary for a fixed observation time is given.