The role of the initial states for non-Fourier heat equations

TL;DR

This paper analyzes the impact of nonhomogeneous initial conditions on non-Fourier heat equations, specifically the Maxwell-Cattaneo-Vernotte and Guyer-Krumhansl models, ensuring solutions remain physically consistent.

Contribution

It introduces a method to determine initial time derivatives consistent with thermodynamics for non-Fourier heat models with nonhomogeneous initial states.

Findings

Derived conditions for physically admissible initial states.

Developed a method to compute initial derivatives consistent with thermodynamics.

Highlighted the importance of initial conditions in non-Fourier heat conduction models.

Abstract

There are several models for heat conduction - non-Fourier equations - in the literature that are important for various practical problems. These models manifest themselves in partial differential equations, and the application of which requires developing efficient and reliable solution methods. In the present paper, we focus on the analytical solutions of two non-Fourier models, specifically on the Maxwell-Cattaneo-Vernotte and Guyer-Krumhansl equations, as they share an established thermodynamic background, and find numerous applications. Although initial conditions are usually homogeneous in space in many situations, real applications can easily point beyond such a simple initial state. Therefore, we aim to investigate the consequences of nonhomogeneous initial conditions, emphasising the physical requirements to keep the solution physically admissible. We conclude the calculations with a method for determining the initial time derivatives in consistence with thermodynamics, avoiding contradictions.