On a $C$-integrable equation for second sound propagation in heated dielectrics

Abstract

An exactly solvable model in heat conduction is considered. The $C$-integrable (i.e., change-of-variables-integrable) equation for second sound (i.e., heat wave) propagation in a thin, rigid dielectric heat conductor uniformly heated on its lateral side by a surrounding medium under the Stefan--Boltzmann law is derived. A simple change-of-variables transformation is shown to exactly map the nonlinear governing partial differential equation to the classical linear telegrapher's equation. In a one-dimensional context, known integral-transform solutions of the latter are adapted to construct exact solutions relevant to heat transfer applications: (i) the initial-value problem on an infinite domain (the real line), and (ii) the initial-boundary-value problem on a semi-infinite domain (the half-line). Possible "second law violations" and restrictions on the $C$-transformation are noted for some sets of parameters.