Some exact and approximate solutions to a generalized Maxwell-Cattaneo equation

The simple heat conduction equation in one-space dimension does not have the property of a finite speed for information transfer. A partial resolution of this difficulty can be obtained within the context of heat conduction by the introduction of a partial differential equation (PDE) called the Maxwell-Cattaneo (M-C) equation, elsewhere called the damped wave equation, a special case of the telegraph equation. We construct a generalization to the M-C equation by allowing the relaxation time parameter to be a function of temperature. In the balance of the paper, we present a variety of special exact and approximate solutions to this nonlinear PDE.