Entropy 2017, 19(2), 55; https://doi.org/10.3390/e19020055
Bateman–Feshbach Tikochinsky and Caldirola–Kanai Oscillators with New Fractional Differentiation
Antonio Coronel-Escamilla 1, José Francisco Gómez-Aguilar 2,*
, Dumitru Baleanu 3,4, Teodoro Córdova-Fraga 5, Ricardo Fabricio Escobar-Jiménez 1, Victor H. Olivares-Peregrino 1 and Maysaa Mohamed Al Qurashi 6
, Dumitru Baleanu 3,4, Teodoro Córdova-Fraga 5, Ricardo Fabricio Escobar-Jiménez 1, Victor H. Olivares-Peregrino 1 and Maysaa Mohamed Al Qurashi 61
Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, 62490 Cuernavaca, Mexico
2
CONACyT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, 62490 Cuernavaca, Mexico
3
Department of Mathematics, Faculty of Art and Sciences, Cankaya University, 0630 Ankara, Turkey
4
Institute of Space Sciences, 409 Atomistilor Str., 077125 Magurele, Romania
5
Departamento de Ingeniería Física, División de Ciencias e Ingenierías Campus León, Universidad de Guanajuato, 37328 León, Mexico
6
Department of Mathematics, King Saud University, 11451 Riyadh, Saudi Arabia
*
Author to whom correspondence should be addressed.
Academic Editor: Carlo Cattani
Received: 23 November 2016 / Revised: 23 January 2017 / Accepted: 24 January 2017 / Published: 28 January 2017
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Abstract
In this work, the study of the fractional behavior of the Bateman–Feshbach–Tikochinsky and Caldirola–Kanai oscillators by using different fractional derivatives is presented. We obtained the Euler–Lagrange and the Hamiltonian formalisms in order to represent the dynamic models based on the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based on the Mittag–Leffler kernel with arbitrary order α. Simulation results are presented in order to show the fractional behavior of the oscillators, and the classical behavior is recovered when α is equal to 1. View Full-TextKeywords:
Bateman–Feshbach Tikochinsky oscillator; Caldirola–Kanai oscillator; fractional operators; Mittag–Leffler kernel
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Coronel-Escamilla, A.; Gómez-Aguilar, J.F.; Baleanu, D.; Córdova-Fraga, T.; Escobar-Jiménez, R.F.; Olivares-Peregrino, V.H.; Qurashi, M.M.A. Bateman–Feshbach Tikochinsky and Caldirola–Kanai Oscillators with New Fractional Differentiation. Entropy 2017, 19, 55.
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