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Symmetry 2018, 10(2), 40; doi:10.3390/sym10020040

Three Classes of Fractional Oscillators

1
Shanghai Key Laboratory of Multidimensional Information Processing, No. 500, Dong-Chuan Road, East China Normal University, Shanghai 200241, China
2
School of Information Science and Technology, East China Normal University, Shanghai 200241, China
Received: 2 November 2017 / Revised: 11 December 2017 / Accepted: 18 December 2017 / Published: 30 January 2018
(This article belongs to the Special Issue Symmetry and Complexity)

Abstract

This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice. View Full-Text
Keywords: fractional differential equations; fractional oscillations (vibrations); fractional dynamical systems; nonlinear dynamical systems fractional differential equations; fractional oscillations (vibrations); fractional dynamical systems; nonlinear dynamical systems
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Li, M. Three Classes of Fractional Oscillators. Symmetry 2018, 10, 40.

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