On Relativistic Transformation of Coordinates and Exact Solution of Damped Wave Conduction and Relaxation Equation

Abstract

Analytical solution to the hyperbolic damped wave conduction and relaxation equation is developed by a novel method called the relativistic transformation method. The hyperbolic PDE is decomposed into a time decaying damping component and a Klein-Gardon type equation for the wave temperature. The PDE that describes the wave temperature is transformed to a Bessel differential equation by using the relativistic transformation. The relativistic transformation, η = τ2 − X2 is symmetric in space and time. The solution obtained for the transient temperature to a semi-infinite medium was compared with success to a Laplace transform solution reported by other investigators. A approximate analytical solution is obtained for the transient temperature by realizing that the integration constants from the solution of Bessel differential equation in the transformation variable is with respect to the transformation variable which is a function of space and time. So the boundary conditions can be used to solve for integration “constants” that are functions of one variable. The solution consists of three regimes. There is no discontinuity at the wave front. A inertial zero transfer regime, a second rising regime characterized by Bessel composite function of the zeroth order and first kind and a third falling regime characterized by modified Bessel composite function of the zeroth order and first kind.