Integral representation of solutions to initial-boundary value problems in the framework of the Guyer-Krumhansl heat equation

TL;DR

This paper develops an integral solution framework for initial-boundary value problems involving the Guyer-Krumhansl heat equation, using the Fokas method, and demonstrates it with a numerical example involving Newton's law boundary conditions.

Contribution

It introduces an integral representation for IBVPs with the Guyer-Krumhansl equation using the Fokas method, extending solution techniques to complex boundary conditions.

Findings

Derived explicit integral formulas for solutions

Applied the method to a boundary condition with Newton's law

Provided a numerical example illustrating the approach

Abstract

We consider initial-boundary value problems (IBVPs) on a finite interval for the system of the energy balance equation and Guyer-Krumhansl constitutive equation. Boundary conditions comprise various models of behavior of a physical system at the boundaries, including boundary conditions describing Newton's law, which states that the heat flux at the boundary is directly proportional to the difference in the temperature of the physical system and ambient temperature. In this case the boundary conditions express the relationship of unknown functions (temperature or internal energy and heat flux) with each other. To solve the problems, we apply the Fokas unified transform method. To illustrate the final formulae, we consider a numerical example for a special case in which heat exchange at one of the boundaries obeys Newton's law.