Fractional oscillator

The notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order. This type of equation appears in the description of chaotic dynamics, wave propagation in fractal media, and field theory. For the fractional linear oscillator the physical meaning of the derivative of order α < 2 is dissipation. In systems with many spacially coupled elements (oscillators) the fractional derivative, along the space coordinate, corresponds to a long range interaction. We discuss a method of constructing a solution using an expansion in ε = n − α with small ε and positive integer n. The method is applied to the fractional linear and nonlinear oscillators and to fractional Ginzburg-Landau or parabolic equations.

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“…LFO is an object of numerous investigations [20,21,22,23,34,35,36,37,38] because of different applications.…”

Section: ε-Expansion For Linear Oscillatormentioning

confidence: 99%

Dynamics with low-level fractionality

Tarasov

Zaslavsky

2006

Physica A: Statistical Mechanics and its Applications

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“…We denote the following oscillation equation as a fractional oscillator in Class I. The free response to (31) is in the form (Mainardi [25], Achar et al [33], Uchaikin ([38], Chapter 7)) y 1 (t) = y 10 E α,1 −(ω n t) α + y 10 tE α,2 −(ω n t) α , 1 < α ≤ 2, t ≥ 0, (32) where E a,b (z) is the generalized Mittag-Leffler function given by…”

Section: Three Classes Of Fractional Oscillatorsmentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

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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.

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“…Fractional calculus is an emerging fields and during the last decades it represents an alternative tool to solve several problems from various fields [1,2,3,4,5,6,7,8,9,10,11]. During the last years the fractional variational principles [12,13,14,15,16,17,18,19,20,21,22,23,24,25] have developed and applied to fractional optimal control problems [26,27].…”

Section: Introductionmentioning

confidence: 99%

On exact solutions of a class of fractional Euler–Lagrange equations

2007

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In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where c a D α t x(t)) and 0 < α < 1, such that the following is the corresponding Euler-LagrangeAt last, exact solutions for some Euler-Lagrange equations are presented. In particular, we consider the following equationswhere g(t) and f(t) are suitable functions.

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Fractional oscillator

Cited by 94 publications

References 18 publications

Dynamics with low-level fractionality

Dynamics with low-level fractionality

Three Classes of Fractional Oscillators

On exact solutions of a class of fractional Euler–Lagrange equations

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