Hyperbolic heat conduction in the semi-infinite body with the heat source which capacity linearly depends on temperature

Abstract

It is commonly taken for granted that long-time solutions of Cattaneo's hyperbolic heat conduction equation tend to overlap the related parabolic solutions. Usually, for small times considerable qualitative as well as quantitative differences between the hyperbolic and parabolic solutions are observed which vanish completely for large times. However, in the case of a heat conducting body with the temperature dependent inner heat generation, the quantitative differences may grow with time. This arises from the feedback between the temperature and the source capacity. To illustrate this effect, Cattaneo's hyperbolic equation for the semi-infinite body, with the heat source which capacity is linearly dependent on temperature, is solved analytically by the Laplace transforms method.