Eur. Phys. J. Plus (2017) 132: 13
https://doi.org/10.1140/epjp/i2017-11293-3

Regular Article

New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications

¹ CONACyT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico
² Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, 9300, Bloemfontein, South Africa

^* e-mail: jgomez@cenidet.edu.mx

Received: 6 October 2016
Accepted: 13 December 2016
Published online: 13 January 2017

Abstract

Some physical problems found in nature can follow the power law; others can follow the Mittag-Leffler law and others the exponential decay law. On the other hand, one can observe in nature a physical problem that combines the three laws, it is therefore important to provide a new fractional operator that could possibly be used to model such physical problem. In this paper, we suggest a fractional operator power-law-exponential-Mittag-Leffler kernel with three fractional orders. Some very useful properties are obtained. Numerical solutions were obtained for three examples proposed. The results show that the new fractional operators are powerful mathematical tools to model complex problems.