On the Strict Heat Conduction Model with Stochastic Inputs

Abstract

This paper is concerned with the construction of the strict model of the heat conduction phenomena with random inputs. It is well known that the parabolic heat equation has an infinite thermal propagation speed. This fact is a drawback of the parabolic heat conduction model. Since this drawback comes from Fourier's law, by revising Fourier's law from the physical view point, the stochastic hyperbolic heat conduction model with stochastic inputs is proposed. In the stochastic hyperbolic heat conduction model, thermal propagation speed becomes finite. It should be noted that the influence of the input to the hyperbolic heat conduction model is not simply additive, but the term related to the time delay of the thermal propagation appeared. In this paper, the existence of the unique solution of the stochastic hyperbolic heat equation is proved. Finally, the comparison between the solution processes of the parabolic equation and the hyperbolic one is shown through simulation experiments.