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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Feng, Jonathan

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Volume 13, Issue 1 (Feb 2015)

Issues

Fractional thermal diffusion and the heat equation

Francisco Gómez
  • Centro Nacional de Investigación y Desarrollo Tecnológico. Tecnológico Nacional de México. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México
/ Luis Morales
  • Facultad de Ingeniería en Electrónica y Comunicaciones. Campus: Poza Rica - Tuxpan. Universidad Veracruzana. Av. Venustiano Carranza s/n, Col. Revolución, C.P. 93390, Poza Rica Veracruz, México
/ Mario González
  • Facultad de Ingeniería en Electrónica y Comunicaciones. Campus: Poza Rica - Tuxpan. Universidad Veracruzana. Av. Venustiano Carranza s/n, Col. Revolución, C.P. 93390, Poza Rica Veracruz, México
/ Victor Alvarado
  • Centro Nacional de Investigación y Desarrollo Tecnológico. Tecnológico Nacional de México. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México
/ Guadalupe López
  • Centro Nacional de Investigación y Desarrollo Tecnológico. Tecnológico Nacional de México. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México
Published Online: 2015-02-17 | DOI: https://doi.org/10.1515/phys-2015-0023

Abstract

Fractional calculus is the branch of mathematical analysis that deals with operators interpreted as derivatives and integrals of non-integer order. This mathematical representation is used in the description of non-local behaviors and anomalous complex processes. Fourier’s lawfor the conduction of heat exhibit anomalous behaviors when the order of the derivative is considered as 0 < β,ϒ ≤ 1 for the space-time domain respectively. In this paper we proposed an alternative representation of the fractional Fourier’s law equation, three cases are presented; with fractional spatial derivative, fractional temporal derivative and fractional space-time derivative (both derivatives in simultaneous form). In this analysis we introduce fractional dimensional parameters σx and σt with dimensions of meters and seconds respectively. The fractional derivative of Caputo type is considered and the analytical solutions are given in terms of the Mittag-Leffler function. The generalization of the equations in spacetime exhibit different cases of anomalous behavior and Non-Fourier heat conduction processes. An illustrative example is presented.

Keywords: anomalous diffusion; Caputo derivative; fractional differential equations; Mittag-Leffler function

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About the article

Received: 2014-09-09

Accepted: 2014-10-02

Published Online: 2015-02-17



Citation Information: Open Physics, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2015-0023. Export Citation

©2015 F. Gómez et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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