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Mathematical Problems in Engineering
Volume 2016, Article ID 7845874, 15 pages
http://dx.doi.org/10.1155/2016/7845874
Research Article

Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation

1CONACYT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Colonia Palmira, 62490 Cuernavaca, MOR, Mexico
2Departamento de Ingeniería Física, División de Ciencias e Ingenierías Campus León, Universidad de Guanajuato, 37150 León, GTO, Mexico
3Departamento de Electromecánica, Instituto Tecnológico Superior de Irapuato, 36821 Irapuato, GTO, Mexico
4Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Colonia Palmira, 62490 Cuernavaca, MOR, Mexico

Received 19 February 2016; Accepted 17 April 2016

Academic Editor: Juan J. Trujillo

Copyright © 2016 J. F. Gómez Aguilar et al. 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.

Abstract

The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range . In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to .