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

Sergey A. Rukolaine

TL;DR

The paper develops a comprehensive Fokas unified transform framework for initial-boundary value problems of the one-dimensional energy balance and Guyer–Krumhansl constitutive system on a finite interval. It derives local and global relations, diagonalizes the governing operator, and uses contour deformation and dispersion-symmetry to construct a final integral–residue solution that accommodates general coupled boundary conditions, including Newton-type flux relations. A numerical example illustrates the method's capability to handle laser-flash–like inputs and boundary convection, highlighting agreement with traditional Fourier-based approaches. This work extends analytical solvability of GK-type heat conduction problems with boundary couplings and provides a robust tool for studying non-Fourier heat transport in bounded domains.

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.

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

TL;DR

The paper develops a comprehensive Fokas unified transform framework for initial-boundary value problems of the one-dimensional energy balance and Guyer–Krumhansl constitutive system on a finite interval. It derives local and global relations, diagonalizes the governing operator, and uses contour deformation and dispersion-symmetry to construct a final integral–residue solution that accommodates general coupled boundary conditions, including Newton-type flux relations. A numerical example illustrates the method's capability to handle laser-flash–like inputs and boundary convection, highlighting agreement with traditional Fourier-based approaches. This work extends analytical solvability of GK-type heat conduction problems with boundary couplings and provides a robust tool for studying non-Fourier heat transport in bounded domains.

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.

Paper Structure

This paper contains 14 sections, 116 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of the initial-boundary value problem
  3. Solving the problem
  4. The local relations
  5. The global relations
  6. A preliminary expression for the solution
  7. Deformation of the integration path
  8. Using the symmetry of the dispersion relations
  9. The solution
  10. A numerical example
  11. The paths of integration in the complex plane
  12. Particular cases that allow series representation of the solutions
  13. The heat sources are absent, the initial state is zero, the heat flux is prescribed on the left boundary, and the right boundary is thermally insulating
  14. The initial state is non-zero and the boundaries are thermally insulating

Figures (2)

  • Figure 1: The solutions to the problem \ref{['eq:IBVPEqsCaseOne']}--\ref{['eq:IBVPBoundCondCaseOne']} on the right boundary, i. e., for $x = 1$ with the heat flux $g(t)$ given by Eq. \ref{['eq:g']}. The values of the parameters are $\alpha = 1$, $\tau = 0.02$, $\mu^2 = 0.02$, $\gamma_l^{} = 0$ and $\gamma_l^{} = 0.2$, $l = 1$, $\tau_\Delta^{} = 0.04$.
  • Figure 2: The solutions to the problem \ref{['eq:IBVPEqsCaseOne']}--\ref{['eq:IBVPBoundCondCaseOne']} on the right boundary, i. e., for $x = 1$ with the heat flux $g(t)$ given by Eq. \ref{['eq:g']}. The values of the parameters are $\alpha = 1$, $\tau = 0.02$, $\mu^2 = 0.2$, $\gamma_l^{} = 0$ and $\gamma_l^{} = 0.2$, $l = 1$, $\tau_\Delta^{} = 0.04$.