Open Access
2010 Linear Fractionally Damped Oscillator
Mark Naber
Int. J. Differ. Equ. 2010(SI1): 1-12 (2010). DOI: 10.1155/2010/197020


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 0v1. 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).


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Mark Naber. "Linear Fractionally Damped Oscillator." Int. J. Differ. Equ. 2010 (SI1) 1 - 12, 2010.


Received: 8 July 2009; Accepted: 11 August 2009; Published: 2010
First available in Project Euclid: 26 January 2017

zbMATH: 1207.34010
MathSciNet: MR2557328
Digital Object Identifier: 10.1155/2010/197020

Rights: Copyright © 2010 Hindawi

Vol.2010 • No. SI1 • 2010
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