Third Boundary-Value Problem of the Heat Conduction Equation for a System with Plane-Parallel Boundaries

TL;DR

This paper presents a new explicit solution representation for the third boundary-value problem of heat conduction in a layered system, using superpositions of fundamental solutions with variable coefficients.

Contribution

It introduces a novel explicit solution form for the third boundary-value problem in heat conduction with boundary conditions of the third kind.

Findings

Derived a superposition-based solution for the heat equation with third-kind boundary conditions.

Expressed temperature evolution in a layered system explicitly.

Provides a new analytical tool for heat conduction problems with complex boundary conditions.

Abstract

We obtained a new representation of a solution of the heat conduction equation with boundary condition of the third kind for a layer. The result is presented as a superposition of fundamental solutions for an unbounded system with variable coefficients, the explicit form of which is given. We consider the well-known problem of the evolution of the temperature field initially uniformly distributed in a layer. The distribution of the temperature field is represented in terms of the obtained functions.