Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions
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 d...
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| Main Authors: | , , , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/286290 |
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| author | A. Beléndez M. L. Alvarez J. Francés S. Bleda T. Beléndez A. Nájera E. Arribas |
| author_facet | A. Beléndez M. L. Alvarez J. Francés S. Bleda T. Beléndez A. Nájera E. Arribas |
| author_sort | A. Beléndez |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b33d76baffd74f07bfa181da2eca70c8 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-b33d76baffd74f07bfa181da2eca70c82025-02-03T01:33:10ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/286290286290Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary FunctionsA. Beléndez0M. L. Alvarez1J. Francés2S. Bleda3T. Beléndez4A. Nájera5E. Arribas6Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, 03080 Alicante, SpainDepartamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, 03080 Alicante, SpainDepartamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, 03080 Alicante, SpainDepartamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, 03080 Alicante, SpainDepartamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, 03080 Alicante, SpainDepartamento de Ciencias Médicas, Facultad de Medicina, Universidad de Castilla-La Mancha, C/Almansa No. 14, 02006 Albacete, SpainDepartamento de Física Aplicada, Escuela Superior de Ingeniería Informática, Universidad de Castilla-La Mancha, Avenida de España s/n, 02071 Albacete, SpainAccurate 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.http://dx.doi.org/10.1155/2012/286290 |
| spellingShingle | A. Beléndez M. L. Alvarez J. Francés S. Bleda T. Beléndez A. Nájera E. Arribas Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions Journal of Applied Mathematics |
| title | Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions |
| title_full | Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions |
| title_fullStr | Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions |
| title_full_unstemmed | Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions |
| title_short | Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions |
| title_sort | analytical approximate solutions for the cubic quintic duffing oscillator in terms of elementary functions |
| url | http://dx.doi.org/10.1155/2012/286290 |
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