Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus

Abstract

Abstract:The main objective of this paper is to generalize the Extended Irreversible Thermodynamics in order to include the anomalous transport in systems in non-equilibrium conditions. Considering the generalized entropy, the corresponding flux and entropy production, and using the time fractional derivative, we have derived a space-time generalized telegrapher's equation with a fractional nested hierarchy which can be used in separate developments for the mass transport, for the heat conduction and for the flux of ions. We have obtained a new formalism which includes the contribution of fast of higher-order fluxes in the mesoscopic and inhomogeneous media. The results take the form of continued fraction expansions. The balance equations are used in a scheme of continued fractions, and they appear as a closure condition. In this way the transport equation and its corresponding wave number-frequency relation are obtained, both of them in the mathematical structure of the continued fraction scheme. Numerical examples are included to show the dispersive nature of the solutions, and the generalized fractional transport equation in the same mathematical form, which can be applied to the mass transport, the heat conduction and the flux of ions.

Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation

“…where τ is the so called relaxation time, and it leads to a modified version of the heat equation, In the last few years, the quest for potential applications of fractional calculus [11][12][13][14] in biology [15][16][17][18], thermodynamics [19][20][21][22], viscoelasticity [24][25][26][27] has been attracting much attention in the mathematical community. Nevertheless, despite this growing interest, not much work has been done in the study of dispersion relations for fractional models for wave propagation.…”

Advanced materials modelling via fractional calculus: challenges and perspectives

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