The fractional heat equation

Abstract

Paper presented at the 9th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Malta, 16-18 July, 2012.

This paper extends the method, in which a Volterra-type integral equation that relates the local values of temperature and the corresponding heat flux within a semi-infinite domain, to a transient heat transfer process in a non-isolated system that has a memory about its previous state. To model such memory systems, the apparatus of fractional calculus is used. Based on the generalized constitutive equation with fractional order derivative, the fractional heat equation is obtained and solved. Its analytical solution is given in the form of a Volterra-type integral equation. It follows from the model, developed in this study, that the heat wave, generated in the beginning of ultrafast energy transport processes, is dissipated by thermal diffusion as the process goes on. The corresponding contributions of the wave and diffusion into the heat transfer process are quantified by a fractional parameter, H , which is a material-dependent constant.