Modeling Diffusion to Thermal Wave Heat Propagation by Using Fractional Heat Conduction Constitutive Model

The relation between the heat flux vector and temperature gradient is called heat conduction constitutive model. The most well known constitutive relation in heat transfer is Fourier model which is originally based on experimental observations. This model which is pure diffusive in nature considers the instantaneous flow of heat in the medium in the presence of even a small temperature gradient. In other words, the velocity of Thermal propagation is infinite according to Fourier model which is in conflict with physical laws. Although it works well in many physical and engineering applications, many experimental studies have shown the inadequacy of Fourier model in some situations of practical interest. Up to now, some non-Fourier constitutive models have been introduced among which Cattaneo and phaselagging models have found greater applications. On the other hand, in the past three decades fractional calculus has proved its efficiency in modeling the intermediate anomalous behaviors observed in different physical phenomena. Fractional calculus is the calculus of differentiation and integration of non-integer orders. It is as old as classical (integer-order) calculus. However, until the recent decades it had not found considerable applications in practical and engineering fields and had been studied only in pure mathematics. Today, fractional calculus has shown great promise in different fields of science and engineering because of its inherent great abilities in modeling anomalous behaviors observed in many complex processes. Some main areas of application of fractional calculus