Analytical approximate solutions for the cubic-quintic Duffing oscillator in terms of elementary functions

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Title: Analytical approximate solutions for the cubic-quintic Duffing oscillator in terms of elementary functions
Authors: Beléndez, Augusto | Alvarez, Mariela L. | Francés, Jorge | Bleda, Sergio | Beléndez, Tarsicio | Nájera López, Alberto | Arribas Garde, Enrique
Research Group/s: Holografía y Procesado Óptico | GITE - Física, Óptica y Telecomunicaciones
Center, Department or Service: Universidad de Alicante. Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal | Universidad de Alicante. Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías | Universidad de Castilla-La Mancha. Departamento de Física Aplicada | Universidad de Castilla-La Mancha. Departamento de Ciencias Médicas
Keywords: Nonlinear oscillators | Approximate analytical solution | Rational harmonic balance method | Chebyshev polynomials | Cubic-quintic Duffing oscillator
Knowledge Area: Física Aplicada | Matemática Aplicada
Date Created: 28-Jun-2012
Issue Date: 30-Sep-2012
Publisher: Hindawi Publishing Corporation
Citation: BELÉNDEZ VÁZQUEZ, Augusto, et al. "Analytical approximate solutions for the cubic-quintic Duffing oscillator in terms of elementary functions". Journal of Applied Mathematics. Vol. 2012, Article ID 286290, 16 pages (2012). ISSN 1110-757X
Abstract: Accurate approximate closed-form solutions for the cubic-quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.
Sponsor: This work was supported by the “Generalitat Valenciana” of Spain, under Project PROMETEO/2011/021, and by the “Vicerrectorado de Tecnología e Innovación Educativa” of the University of Alicante, Spain, under Project GITE-09006-UA.
URI: http://hdl.handle.net/10045/24655
ISSN: 1110-757X (Print) | 1687-0042 (Online)
DOI: 10.1155/2012/286290
Language: eng
Type: info:eu-repo/semantics/article
Rights: Copyright © 2012 A. Beléndez et al. 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.
Peer Review: si
Publisher version: http://dx.doi.org/10.1155/2012/286290
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