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Algebraic-Numerical Method for the Slightly Perturbed Harmonic Oscillator

A. Nadeau, J. Guyard and M. R. Feix
Mathematics of Computation
Vol. 28, No. 128 (Oct., 1974), pp. 1057-1066
DOI: 10.2307/2005365
Stable URL: http://www.jstor.org/stable/2005365
Page Count: 10
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Algebraic-Numerical Method for the Slightly Perturbed Harmonic Oscillator
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Abstract

The solution of slightly perturbed harmonic oscillators can easily be obtained in the form of a series given by Poisson's method. However, this perturbation method leads to secular terms unbounded for large time (the time unit being the fundamental period of the harmonic oscillator), which prevent the use of finite series. The analytical elimination of such terms was first solved by Poincaré and, more recently, generalized by Krylov and Bogoliubov. Unfortunately, these methods are very difficult to handle and are not easily carried out for high orders. A numerical reinitialization method is combined here with the Poisson perturbation treatment to avoid the growth of secular terms and therefore to get the solution at any time. The advantages of such a method is that the analytical work can be carried to high orders keeping the step of numerical integration to a relatively large value (compared to a purely numerical method). This algorithm has been tested on the Mathieu equation. A method for the computation of the eigenvalues of this equation is given. By properly selecting the order of the perturbation and the time step of reinitialization, we can recover, at any order, all the effects of the slight perturbation (including all the unstable zones). Consequently, such a method is a useful intermediate between purely analytical and purely numerical algorithms.

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