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Mathematics 2016, 4(4), 67; doi:10.3390/math4040067

Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus

1
Departament de Termodinàmica Aplicada, Universitat Politècnica de Valencia, Campus de Vera s/n., 46022 Valencia, Spain
2
Unidad Multidisciplinaria de Docencia e Investigación-Juriquilla, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), Juriquilla, Querétaro CP 76230, Mexico
3
Departamento de polímeros, Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México (UNAM), Ciudad Universitaria, Apartado Postal 70-360, Coyoacán, Ciudad de México 04510, Mexico
4
Unitat de Física Estadística, Universitat Autònoma de Barcelona, Barcelona 08193, Spain
*
Author to whom correspondence should be addressed.
Academic Editor: Rui A. C. Ferreira
Received: 1 November 2016 / Revised: 29 November 2016 / Accepted: 2 December 2016 / Published: 9 December 2016
(This article belongs to the Special Issue Fractional Differential and Difference Equations)
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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. View Full-Text
Keywords: transport phenomena; anomalous transport; continued fraction; Extended Irreversible Thermodynamics transport phenomena; anomalous transport; continued fraction; Extended Irreversible Thermodynamics
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Garcia-Bernabé, A.; Hernández, S.I.; del Castillo, L.F.; Jou, D. Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus. Mathematics 2016, 4, 67.

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