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

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.