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International Journal of Differential Equations
Volume 2010 (2010), Article ID 197020, 12 pages
http://dx.doi.org/10.1155/2010/197020
Research Article

Linear Fractionally Damped Oscillator

Department of Mathematics, Monroe County Community College, Monroe, MI 48161-9746, USA

Received 8 July 2009; Accepted 11 August 2009

Academic Editor: Mark M. Meerschaert

Copyright © 2010 Mark Naber. 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 linearly damped oscillator equation is considered with the damping term generalized to a Caputo fractional derivative. The order of the derivative being considered is 0 𝑣 1 . At the lower end ( 𝑣 = 0 ) the equation represents an undamped oscillator and at the upper end ( 𝑣 = 1 ) the ordinary linearly damped oscillator equation is recovered. A solution is found analytically, and a comparison with the ordinary linearly damped oscillator is made. It is found that there are nine distinct cases as opposed to the usual three for the ordinary equation (damped, over-damped, and critically damped). For three of these cases it is shown that the frequency of oscillation actually increases with increasing damping order before eventually falling to the limiting value given by the ordinary damped oscillator equation. For the other six cases the behavior is as expected, the frequency of oscillation decreases with increasing order of the derivative (damping term).