A Variational Principle for the Conduction of Heat

Abstract

The study describes an expression for the approximate solution of the telegraph equation using calculus of variations. It shows that in the limit when the relaxation time is obtained, an approximate solution of the classical (Fourier) equation of heat conduction is obtained. It is known that Fourier's heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. Even if Fourier's law and the experiment accord well, this is undesirable from a physical standpoint. But when very short distances or extremely short time intervals are taken into account, as they must in some contemporary aero-thermodynamics situations, disparities are likely to arise. Cattaneo and independently Vernotte proved that such process can be described by Heaviside's telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem. In the manuscript, an approximate solution to the telegraph equation was determined using the calculus of variations. The approximate solution of the classic parabolic equation is obtained as a limiting case of the approximate solution of the telegraph equation when the relaxation time \(\tau^*\rightarrow0\). This research provides some new possibilities for applying the calculus of variations to the equation of heat conduction and applying it in practice.