TITLE:
Variational Approach to Heat Conduction Modeling
AUTHORS:
Slavko Đurić, Ivan Aranđelović, Milan Milotić
KEYWORDS:
Telegraph Equation, Heat Equation, Heat Conduction, Calculus of Variations
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.1,
January
30,
2024
ABSTRACT: 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. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. 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.