Generalized variational principles for heat conduction models based on Laplace transforms

Abstract

The classical variational principle does not exist for parabolic and hyperbolic heat conduction equations, which has led to the demand for special variational methods for heat conduction. O’Toole (1967) first used Laplace transforms for the variational principle only for Fourier’s law with the first type of boundary condition. In this paper, the Laplace transform strategy is extended to other parabolic and hyperbolic heat conduction models and other types of boundary conditions. Generalized variational principles are given for heat conduction models including Fourier’s law, the Cattaneo–Vernotte (CV) model, the Jeffrey model, the two-temperature model and the Guyer–Krumhansl (GK) model, based on Laplace transforms. The Laplace transform method transforms the heat conduction equations of these models into linear variational equations whose variational principles are already known. For the three standard types of boundary conditions, these generalized variational principles are strictly equivalent to the heat conduction equations for these models. The Laplace transform method has stronger convergence in infinite temporal domain problems. In physics, the Laplace transform method is understood as replacing the time dimension with the frequency of the temperature change and the rate of the entropy change.