Heat traveling waves in rigid thermal conductors with phase lag and stability analysis

Abstract

Recently, a model equation that describes nonlinear heat waves in a rigid thermal conductor has been derived. The system of the governing equations for temperature and heat flux is nonlinear. The objective of the present work is to find a variety of traveling wave solutions of this system of equations in the whole space. This is achieved by implementing the unified method. The obtained solutions are evaluated numerically and represented graphically. The behavior of these solutions is investigated, where it is shown that the temperature and the heat flux attain steady states in space, but increase with time. The effects of the characteristic length, time, heat flux, and reference temperature are studied via some material data. It is shown that the solutions may have the form of solitary wave, soliton, or soliton with double kinks. It is observed that the heat flux in the material is negative, this reflects the fact that heat flux is in the opposite direction of the normal vector to the material surface on which it is evaluated. The steady state solution of the considered model equation is studied. It is found that the stability of the solutions depends significantly on the wave number.