Two analytical formulae of the temperature inside a body by using partial lateral and initial data

TL;DR

This paper presents two analytical formulas for determining the temperature at a point inside a domain using partial initial and boundary data, advancing inverse heat conduction problem solutions.

Contribution

It introduces novel analytical formulas leveraging a special fundamental solution for the backward heat equation to solve inverse temperature problems.

Findings

Two explicit formulas derived for temperature estimation inside a domain.

Formulas utilize partial boundary and initial data over finite time.

Method enhances inverse heat problem solving with analytical solutions.

Abstract

This paper considerers the problem of computing the value of a solution of the heat equation at a given point inside a bounded domain after the initial time. It is assumed that the initial value of the solution inside the domain (possibly in a part of the domain) is known; the boundary value and the normal derivative on a part of the boundary of the domain over a finite time interval are known. Two analytical formulae for the problem are given. Both formulae make use of a special fundamental solution having a large parameter of the backward heat equation.