Nonlinear oscillator with power-form elastic-term: Fourier series expansion of the exact solution
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Title: | Nonlinear oscillator with power-form elastic-term: Fourier series expansion of the exact solution |
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Authors: | Beléndez, Augusto | Francés, Jorge | Beléndez, Tarsicio | Bleda, Sergio | Pascual, Carolina | Arribas Garde, Enrique |
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 | Universidad de Castilla-La Mancha. Departamento de Física Aplicada |
Keywords: | Dynamical systems | Nonlinear oscillators | Conservative systems | Truly nonlinear oscillators | Fourier series expansion | Approximate solutions | Symbolic computation |
Knowledge Area: | Física Aplicada | Matemática Aplicada |
Date Created: | Aug-2014 |
Issue Date: | 1-May-2015 |
Publisher: | Elsevier |
Citation: | Communications in Nonlinear Science and Numerical Simulation. 2015, 22(1-3): 134-148. doi:10.1016/j.cnsns.2014.10.012 |
Abstract: | A family of conservative, truly nonlinear, oscillators with integer or non-integer order nonlinearity is considered. These oscillators have only one odd power-form elastic-term and exact expressions for their period and solution were found in terms of Gamma functions and a cosine-Ateb function, respectively. Only for a few values of the order of nonlinearity, is it possible to obtain the periodic solution in terms of more common functions. However, for this family of conservative truly nonlinear oscillators we show in this paper that it is possible to obtain the Fourier series expansion of the exact solution, even though this exact solution is unknown. The coefficients of the Fourier series expansion of the exact solution are obtained as an integral expression in which a regularized incomplete Beta function appears. These coefficients are a function of the order of nonlinearity only and are computed numerically. One application of this technique is to compare the amplitudes for the different harmonics of the solution obtained using approximate methods with the exact ones computed numerically as shown in this paper. As an example, the approximate amplitudes obtained via a modified Ritz method are compared with the exact ones computed numerically. |
Sponsor: | This work was supported by the “Generalitat Valenciana” of Spain, under projects PROMETEO/2011/021 and ISIC/2012/013, and by the “Vicerrectorado de Tecnologías de la Información” of the University of Alicante, Spain, under project GITE-09006-UA. |
URI: | http://hdl.handle.net/10045/43240 |
ISSN: | 1007-5704 (Print) | 1878-7274 (Online) |
DOI: | 10.1016/j.cnsns.2014.10.012 |
Language: | eng |
Type: | info:eu-repo/semantics/article |
Peer Review: | si |
Publisher version: | http://dx.doi.org/10.1016/j.cnsns.2014.10.012 |
Appears in Collections: | INV - GHPO - Artículos de Revistas INV - GMECA - Artículos de Revistas |
Files in This Item:
File | Description | Size | Format | |
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CNSNS_v22_p134_2015.pdf | Versión final (acceso restringido) | 380,91 kB | Adobe PDF | Open Request a copy |
CNSNS-D-14-00408R1-RUA.pdf | Versión revisada (acceso abierto) | 3,27 MB | Adobe PDF | Open Preview |
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