Fractional electromagnetic waves in conducting media

Abstract

We present the fractional wave equation in a conducting material. We used a Maxwell’s equations with the assumptions that the charge density and current density J were zero, and that the permeability and permittivity were constants. The fractional wave equation will be examined separately; with fractional spatial derivative and fractional temporal derivative, finally, consider a Dirichlet conditions, the Fourier method was used to find the full solution of the fractional equation in analytic way. Two auxiliary parameters and are introduced; these parameters characterize consistently the existence of the fractional space-time derivatives into the fractional wave equation. A physical relation between these parameters is reported. The fractional derivative of Caputo type is considered and the corresponding solutions are given in terms of the Mittag-Leffler function show fractal space-time geometry different from the classical integer-order model.

Keywords:

• fractional wave equation
• Caputo derivative
• Mittag-Leffler function
• anomalous behavior
• fractional space-time components

Acknowledgements

The authors appreciate the constructive remarks and suggestions of the anonymous referees that helped to improve the paper. We would like to thank Mayra Martínez and Prof. Dumitru Baleanu for the interesting discussions. This work was supported by CONACYT. José Francisco Gómez Aguilar acknowledges the support provided by CONACYT: catedras CONACYT para jovenes investigadores 2014.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by CONACYT; CONACYT: catedras CONACYT para jovenes investigadores 2014.