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.
Author(s) Details
Slavko Duric
Faculty of Traffic Engineering, University of East Sarajevo, Doboj, Bosnia
and Herzegovina.
Ivan Arandelovic
Faculty of Mechanical Engineering, University of Belgrade, Belgrade,
Republic of Serbia.
Milan Milotic
Faculty of Traffic Engineering, University of East Sarajevo, Doboj, Bosnia
and Herzegovina.
Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v2/921
