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

TL;DR

This paper introduces an exactly solvable model for second sound heat propagation in heated dielectrics, transforming a nonlinear PDE into a linear telegrapher's equation to find explicit solutions for heat transfer problems.

Contribution

It derives a $C$-integrable nonlinear heat conduction equation and provides exact solutions using integral transforms, advancing analytical methods in second sound heat transfer modeling.

Findings

The nonlinear PDE maps exactly to the linear telegrapher's equation.

Explicit solutions for initial-value and boundary-value problems are constructed.

Restrictions on parameters and second law considerations are discussed.

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.