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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 869484, 6 pages
http://dx.doi.org/10.1155/2013/869484
Research Article

The Periodic Solution of Fractional Oscillation Equation with Periodic Input

1School of Sciences, Shanghai Institute of Technology, Shanghai 201418, China
2School of Mathematics and Information Sciences, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Received 5 July 2013; Revised 14 August 2013; Accepted 15 August 2013

Academic Editor: Ming Li

Copyright © 2013 Jun-Sheng Duan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.