Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the weakly damped harmonic oscillator

Author: Howard R. Baum
Journal: Quart. Appl. Math. 29 (1972), 573-576
DOI: https://doi.org/10.1090/qam/99746
MathSciNet review: QAM99746
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Abstract | References | Additional Information

Abstract: The initial-value problem for the differential equation

$\displaystyle u''\left( t \right) + \epsilon \left[ {a + {{\left( {u'} \right)}^2}} \right]u' + u = 0$

is studied under the assumption that $ a$ and $ \epsilon $ are positive constants and $ \epsilon $ is small. Perturbation methods are used to obtain a first approximation to the solution that is uniformly valid in $ t$. For any non-zero $ a$ the solution ultimately decays exponentially with a time scale $ {\left( { \epsilon a} \right)^{ - 1}}$. For sufficiently small $ a$, however, the damping is dominated by the cubic damping term for times of order $ { \epsilon ^{ - 1}}$ and is thus algebraic in character. The frequency of the oscillations is reduced by an amount $ \frac{1}{8}{\left( { \epsilon a} \right)^2}$ and is unaffected by the cubic damping to the order of approximation considered.

References [Enhancements On Off] (What's this?)

  • [1] G. F. Carrier and C. E. Pearson, Ordinary differential equations, Blaisdell, Waltham, Mass., 1968, p. 210
  • [2] Julian D. Cole, Perturbation methods in applied mathematics, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0246537

Additional Information

DOI: https://doi.org/10.1090/qam/99746
Article copyright: © Copyright 1972 American Mathematical Society

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