- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
International Journal of Differential Equations
Volume 2011 (2011), Article ID 852919, 9 pages
Solutions of the Force-Free Duffing-van der Pol Oscillator Equation
1Department of Mathematics, University of Karachi, Karachi 75270, Pakistan
2Abdul Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
3Department of Mathematics, NED University of Engineering and Technology, Karachi 75270, Pakistan
Received 31 May 2011; Revised 5 August 2011; Accepted 22 August 2011
Academic Editor: Yuriy Rogovchenko
Copyright © 2011 Najeeb Alam Khan 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.
- M. Rafei, D. D. Ganji, H. Daniali, and H. Pashaei, “The variational iteration method for nonlinear oscillators with discontinuities,” Journal of Sound and Vibration, vol. 305, no. 4-5, pp. 614–620, 2007.
- Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 27–34, 2006.
- T. Öziş and A. Yildirim, “A study of nonlinear oscillators with force by He's variational iteration method,” Journal of Sound and Vibration, vol. 306, no. 1-2, pp. 372–376, 2007.
- A. Beléndez, C. Pascual, M. Ortuño, T. Beléndez, and S. Gallego, “Application of a modified He's homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities,” Nonlinear Analysis, vol. 10, no. 2, pp. 601–610, 2009.
- J.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004.
- A. Beléndez, A. Hernández, T. Beléndez, E. Fernández, M. L. Álvarez, and C. Neipp, “Application of he's homotopy perturbation method to the duffin-harmonic oscillator,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, pp. 79–88, 2007.
- N. A. Khan, M. Jamil, and A. Ara, “Multiple-parameter Hamiltonian approach for higher accurate approximations of a nonlinear oscillator with discontinuity,” International Journal of Differential Equations, vol. 2011, Article ID 649748, 7 pages, 2011.
- H. M. Liu, “Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 577–579, 2005.
- J. H. He, “Variational approach for nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1430–1439, 2007.
- D. H. Shou, “Variational approach to the nonlinear oscillator of a mass attached to a stretched wire,” Physica Scripta, vol. 77, no. 4, Article ID 045006, 2008.
- F. Ö. Zengin, M. O. Kaya, and S. A. Demirbaǧ, “Application of parameter-expansion method to nonlinear oscillators with discontinuities,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 3, pp. 267–270, 2008.
- J. H. He, “Max-min approach to nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 207–210, 2008.
- Z. Guo and A. Y. T. Leung, “The iterative homotopy harmonic balance method for conservative Helmholtz-Duffing oscillators,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3163–3169, 2010.
- A. E. Ebaid, “A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 528–536, 2011.
- V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, “New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems,” Journal of Physics: A, vol. 37, no. 16, pp. 4527–4534, 2004.
- S. Mukherjee, B. Roy, and S. Dutta, “Solution of the Duffing-van der pol oscillator equation by a differential transform method,” Physica Scripta, vol. 83, Article ID 015006, 2011.
- N. A. Khan, A. Ara, S. A. Ali, and A. Mahmood, “Analytical study of Navier-Stokes equation with fractional orders using He's homotopy perturbation and variational iteration methods,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 9, pp. 1127–1134, 2009.
- N. A. Khan, A. Ara, and A. Mahmood, “Approximate solution of time-fractional chemical engineering equations: a comparative study,” International Journal of Chemical Reactor Engineering, vol. 8, article A19, 2010.
- N. A. Khan, M. Jamil, and A. Ara, “An efficient approach for solving the Riccati equation with fractional orders,” Computers and Mathematics with Applications, vol. 61, pp. 2683–2689, 2011.
- M. Madani and M. Fathizadeh, “Homotopy perturbation algorithm using Laplace transformation,” Nonlinear Science Letters: A, vol. 1, pp. 263–267, 2010.
- N. A. Khan, M. Jamil, A. Ara, and N.-U. Khan, “On efficient method for system of fractional differential equations,” Advances in Difference Equations, Article ID 303472, 15 pages, 2011.