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Discontinuity, Nonlinearity, and Complexity 1(4) (2012) 325--335 | DOI:10.5890/DNC.2012.09.004

J.J. Rosales$^{1}$; M. Guía$^{1}$; J.F. Gómez$^{2}$; V.I. Tkach$^{3}$

$^{1}$ Departamento de Ingeniería Eléctrica. División de Ingenierías Campus Irapuato-Salamanca. Universidad de Guanajuato. Carretera Salamanca-Valle de Santiago, km. 3.5 + 1.8 km. Comunidad de Palo Blanco, Salamanca Guanajuato, México

$^{2}$ Departamento de Física. División de Ciencias e Ingenierías Campus León. Universidad de Guanajuato Lomas del Bosque s/n, Lomas del Campestre, León Guanajuato, México

$^{3}$ Department of Physics and Astronomy. Northwestern University, Evanston, IL 60208-3112, USA

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Abstract

In this article we propose an alternative procedure for constructing fractional differential equations. The order of the derivative being considered is 0 < γ ≤ 1. In particular, we will consider the propagation of electromagnetic waves in an infinitely extended homogeneous media at rest, characterized by the permittivity ε and permeability μ. Two types of the fractional differential equations will be examined separately; with fractional time derivative and the spatial fractional derivative. The parameters σt and σx are introduced which characterize the existence of the fractional time and space components, respectively. It is shown that in the first case there is a relation between σt and the period T0 of the wave given by the order γ of the fractional differential equation, and in the second case the relation is between σx and the wavelength λ . Due to these relations the solutions of the corresponding fractional differential equations are given in terms of the Mittag-Leffler function depending only on the parameter γ.

Acknowledgments

We would like to thank Juan Martínez, Edgar Alvarado, Carlos R. Montoro and Irina Lyanzuridi for interesting discussions. This work is partly supported by PROMEP: Fortalecimiento de CAs, UGTO-CA-27, with fruitful funds.

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