A non-local model of thermal energy transport: The fractional temperature equation

Non-local models of thermal energy transport have been used in recent physics and engineering applications to describe several “small-scale” and/or high frequency thermodynamic processes as shown in several engineering and physics applications. The aim of this study is to extend a recently proposed fractional-order thermodynamics ([5]), where the thermal energy transfer is due to two phenomena: A short-range heat flux ruled by a local transport equation; a long-range thermal energy transfer that represents a ballistic effects among thermal energy propagators. Long-range thermal energy transfer accounts for small-scale effects that are assumed proportional to the product of the interacting masses, to a distance-decaying function, as well as to their relative temperature. In this paper the thermodynamic consistency of the model is investigated obtaining some restrictions on the functional class of the distance decaying function that rules the strength of the long-range thermal energy transfer. As the distance-decaying function is assumed in the form of a power-law decay a novel temperature equation involving multidimensional spatial Marchaud α-order derivatives (0 ⩽ α ⩽ 1) of the temperature field in the body is obtained. Some analytical and numerical solutions of the fractional-order temperature equation have been provided in the paper to show the capabilities of the proposed model and the influence of model parameters.