On reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval

TL;DR

This paper applies the enclosure method to an inverse parabolic problem, enabling the extraction of inclusion depth and minimum radius from boundary temperature data over finite time, advancing non-destructive evaluation techniques.

Contribution

It introduces a simple method to determine the depth of inclusions and the minimum radius of the containing ball using boundary data for a parabolic equation with discontinuous coefficients.

Findings

Method effectively extracts inclusion depth from finite boundary data.

New formula for the minimum radius of the containing ball.

Application extends the enclosure method to parabolic inverse problems.

Abstract

The enclosure method was originally introduced for inverse problems of concerning non-destructive evaluation governed by elliptic equations. It was developed as one of useful approaches in inverse problems and applied for various equations. In this paper, an application of the enclosure method to an inverse initial boundary value problem for a parabolic equation with a discontinuous coefficient is given. A simple method to extract the depth of unknown inclusions in a heat conductive body from a single set of the temperature and heat flux on the boundary observed over a finite time interval is introduced. Other related results with infinitely many data are also reported. One of them gives the minimum radius of the open ball centered at a given point that contains the inclusions. The formula for the minimum radius is newly discovered.