Relaxation equation of heat conduction and generation - an analytical solution by Laplace transforms method

The relaxation equation of heat conduction and generation permits the relaxation of heat flux (a finite speed of heat propagation) as well as the relaxation of heat source capacity. The parabolic and hyperbolic heat conduction equations can be treated as special cases of the relaxation equation. A one-dimensional case of the relaxation equation, in which the relaxation of heat flux is neglected, is solved analytically by the Laplace transforms method to investigate the effect of the inertia of the heat source on the temperature field. The results of sample calculations show that as the relaxation time of heat source capacity increases from zero to infinity the temperature profile for a given time moves from the parabolic solution with heat generation towards the parabolic solution without heat generation. It is also demonstrated that differences between relaxation solutions and the related parabolic solutions do not vanish with time.