Gimeno, Encarnación, Beléndez, Augusto
Rational-harmonic balancing approach to nonlinear phenomena governed by pendulum-like differential equations
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
URI: http://hdl.handle.net/10045/12016
DOI:
ISSN: 0392-0784
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.
Keywords:Nonlinear oscillator, Approximate solutions, Rational harmonic balance method, Nonlinear pendulum
Verlag der Zeitschrift für Naturforschung
info:eu-repo/semantics/article