Rational-harmonic balancing approach to nonlinear phenomena governed by pendulum-like differential equations

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Title: Rational-harmonic balancing approach to nonlinear phenomena governed by pendulum-like differential equations
Authors: Gimeno, Encarnación | Beléndez, Augusto
Research Group/s: Holografía y Procesado Óptico
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
Keywords: Nonlinear oscillator | Approximate solutions | Rational harmonic balance method | Nonlinear pendulum
Knowledge Area: Física Aplicada
Date Created: 6-Oct-2008
Issue Date: Dec-2009
Publisher: Verlag der Zeitschrift für Naturforschung
Citation: GIMENO NIEVES, Encarnación; BELÉNDEZ VÁZQUEZ, Augusto. "Rational-harmonic balancing approach to nonlinear phenomena governed by pendulum-like differential equations". Zeitschrift für Naturforschung A: Physical Sciences. Vol. 64a, No. 12 (2009). ISSN 0392-0784, pp. 819-826
Abstract: This paper presents a new approach for solving accurate approximate analytical solutions for nonlinear phenomena governed by pendulum-like differential equations. The new approach couples Taylor series expansion with rational harmonic balancing. An approximate rational solution depending on a small parameter is considered. After substituting the approximate solution into the governing differential equation, this equation is expanded in Taylor series of the parameter prior to harmonic balancing. The approach gives a cubic equation, which must be solved in order to obtain the value of the small parameter. A method for transforming this cubic equation into a linear equation is presented and discussed. Using this approach, accurate approximate analytical expressions for period and periodic solutions are obtained. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This allowed us to compare the coefficients for the different harmonic terms in these solutions. These analytical approximations may be of interest for those researchers working in nonlinear physical phenomena governed by pendulum-like differential equations in fields such as classical mechanics, vibrations, acoustics, electromagnetism, electronics, superconductivity, optics, gravitation, and others.
Sponsor: This work was supported by the “Generalitat Valenciana” of Spain under project ACOMP/2009/150 and by the “Vicerrectorado de Tecnologa e Innovación Educativa” of the University of Alicante, Spain (GITE-09006-UA).
URI: http://hdl.handle.net/10045/12016
ISSN: 0392-0784
Language: eng
Type: info:eu-repo/semantics/article
Peer Review: si
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