Three Classes of Fractional Oscillators

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.

Lyapunov Characterization of the Fractional Nonlinear Systems with Exogenous Input

“…For more information on Mittag-Leffler functions see [35]. Now let us obtain the solution of the following fractional differential equation, whose solution is fundamental for our main results of this paper.…”

Solution of Fractional Differential Equation Systems and Computation of Matrix Mittag–Leffler Functions

“…Fractional calculus has been made use of to model memory phenomena and hereditary properties [2][3][4][5][6]8]. Its application areas have been widely exploited, such as non-Newtonian flow, damping material [4,[8][9][10], viscoelasticity theory, anomalous diffusion [11][12][13][14], and control theory [15].…”

A generalization of the Mittag–Leffler function and solution of system of fractional differential equations

“…It is found that fractional calculus can describe memory phenomena and hereditary properties of various materials and processes [2][3][4][5][6][7]10]. In recent decades, fractional calculus has been applied to different fields of science and engineering, covering viscoelasticity theory, non-Newtonian flow, damping materials [4,7,11,12], anomalous diffusion [13][14][15][16], control and optimization theory [17][18][19], financial modeling [20,21], and so on.…”