%0 Journal Article
%T The Periodic Solution of Fractional Oscillation Equation with Periodic Input
%A Jun-Sheng Duan
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/869484
%X The periodic solution of fractional oscillation equation with periodic input is considered in this work. The fractional derivative operator is taken as , where the initial time is ; hence, initial conditions are not needed in the model of the present fractional oscillation equation. With the input of the harmonic oscillation, the solution is derived to be a periodic function of time t with the same circular frequency as the input, and the frequency of the solution is not affected by the system frequency c as is affected in the integer-order case. These results are similar to the case of a damped oscillation with a periodic input in the integer-order case. Properties of the periodic solution are discussed, and the fractional resonance frequency is introduced. 1. Introduction Fractional calculus has been used in the mathematical description of real problems arising in different fields of science. It covers the fields of viscoelasticity, anomalous diffusion, analysis of feedback amplifiers, capacitor theory, fractances, generalized voltage dividers, electrode-electrolyte interface models, fractional multipoles, fitting of experimental data, and so on [1¨C5]. Scientists and engineers became aware of the fact that the description of some phenomena is more accurate when the fractional derivative is used. In recent years, even fractional-order models of happiness [6] and love [7] have been developed, and they are claimed to give a better representation than the integer-order dynamical systems approach. The fractional differential and integral operators have been extensively applied to the field of viscoelasticity [8]. The use of fractional calculus for the mathematical modelling of viscoelastic materials is quite natural. The main reasons for the theoretical development are the wide use of polymers in various fields of engineering. The theorem of existence and uniqueness of solutions for fractional differential equations has been presented in [1, 2, 9, 10]. The theory and applications of fractional differential equations are much involved [1¨C5, 11¨C17]. Fractional oscillation equations were introduced and discussed by Caputo [18], Bagley and Torvik [19], Beyer and Kempfle [20], Mainardi [21], Gorenflo and Mainardi [22], and others. Fractional oscillators and fractional dynamical systems were investigated in [23¨C28]. Achar et al. [23] and Al-rabtah et al. [24] studied the response characteristics of the fractional oscillator. Li et al. [25] considered the impulse response and the stability behavior of a class of fractional oscillators. Lim et al. [26] established
%U http://www.hindawi.com/journals/amp/2013/869484/