Non-classical heat conduction problem with non local source

TL;DR

This paper studies a non-classical heat conduction problem with a non-local source term depending on the boundary heat flux, providing existence, uniqueness, and explicit solutions in certain cases, relevant for temperature regulation modeling.

Contribution

It introduces a novel non-local heat conduction model involving integral boundary conditions and solves it using Volterra equations, including explicit solutions in one dimension.

Findings

Unique local solution exists and can be extended globally.

Explicit solution obtained in one-dimensional case using Adomian method.

Properties of the solution are analyzed and discussed.

Abstract

We consider the non-classical heat conduction equation, in the domain $D=\br^{n-1}\times\br^{+}$, for which the internal energy supply depends on an integral function in the time variable of % $(y , t)\mapsto \int_{0}^{t} u_{x}(0 , y , s) ds$, %where $u_{x}(0 , y , s)$ is the heat flux on the boundary $S=\partial D$, with homogeneous Dirichlet boundary condition and an initial condition. The problem is motivated by the modeling of temperature regulation in the medium. The solution to the problem is found using a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\br^{n-1}$. The solution to this Volterra equation is the heat flux $(y, s)\mapsto V(y , t)= u_{x}(0 , y , t)$ on $S$, which is an additional unknown of the considered problem. We show that a unique local solution exists, which can be extended globally in time. Finally a one-dimensional case is studied with some simplifications, we obtain the solution explicitly by using the Adomian method and we derive its properties.