A heat conduction problem with sources depending on the average of the heat flux on the boundary

Abstract

Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain D = R n−1 × R + for which the internal energy supply depends on an average in the time variable of the heat flux (y, s) → V (y, s) = ux(0, y, s) on the boundary S = ∂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 the second kind in the time variable t with a parameter in 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.

Explicit solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms

“…Problems of this type are related to the thermostat problem [21,29,30,31,32,34,35]. For example, we will use mathematical ideas developed for the one-dimensional case in [2,25,46,49,50] and for the n-dimensional case in [8,9,10]. The first paper connecting the nonclassical heat equation with a phase-change process (i.e.…”