# A Heat Conduction Problem with Sources Depending on the Average of the Heat Flux on the Boundary

## TL;DR

This paper studies a non-classical heat conduction problem where the internal energy source depends on the average heat flux at the boundary, deriving integral equations and explicit solutions in specific cases.

## Contribution

It introduces a novel heat conduction model with boundary-dependent sources and provides existence, uniqueness, and explicit solutions for the problem.

## Key findings

- The heat flux satisfies a Volterra integral equation of the second kind.
- A unique local solution exists and can be extended globally.
- Explicit solutions are obtained in the one-dimensional case using Laplace transform and Adomian decomposition.

## Abstract

Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0 , y , s)$ on the boundary $S=\partial D$. The solution to the problem is found for an integral representation depending on the heat flux on $S$ which is an additional unknown of the considered problem. We obtain that the heat flux $V$ must satisfy a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method.