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Manuscript Title: A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach
Authors: Dongmei Wu, Zhongcheng Wang
Program title: AnalyDuffing.nb
Catalogue identifier: ADWR_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 174(2006)447
Programming language: Mathematica 4.2, 5.0 and 5.1.
Computer: IBM PC;.
Operating system: Windows Xp.
RAM: 51712 Bytes
Keywords: Undamped Duffing equation, Approximate analytical solution, Nonlinear second order ordinary differential equation, High-accurate methods, Nonlinear oscillator, harmonic balance method.
PACS: 02.30.Hq, 02.60.Cb, 02.70.-c.
Classification: 5.

Nature of problem:
To find an approximate solution with analytical expressions for the undamped nonlinear Duffing equation with periodic driving force when the fundamental frequency is identical to the driving force.

Solution method:
In the frame of the general HB method, by using a new iteration algorithm to calculate the coefficients of the Fourier series, we can obtain an approximate analytical solution with high-accuracy efficiently.

Restrictions:
For problems, which have a large driving frequency, the convergence may be a little slow, because more iterative times are needed.

Unusual features:
For an undamped Duffing equation, it can provide all the solutions or the oscillation modes with real displacement for any interesting parameters, for the required accuracy, efficiently. The program can be used to study the dynamically periodic behavior of a nonlinear oscillator, and can provide a high-accurate approximate analytical solution for developing high-accurate numerical method.

Running time:
several seconds

References:
[1] R. E. Mickens, J. Sound Vib., `Comments on " a Generalized Galerkin's method for non-linear oscillators"', 118, 563 (1987)
[2] M. Urabe and A. Reiter, `Numerical computation of nonlinear forced oscillations by Galerkin's procedure', J. Math. Anal. Appl. 14, 107-140 (1966).
[3] R. van Dooren, `Stabilization of Cowell's classic finite difference method for numerical integration', J. Comput. Phys. 16, 186-192 (1974).