IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v41y2009i3p1078-1084.html
   My bibliography  Save this article

Generalized decomposition methods for nonlinear oscillators

Author

Listed:
  • Ramos, J.I.

Abstract

Two generalized decomposition methods for nonlinear oscillators are presented and the second one is applied to a nonlinear ordinary differential equation whose velocity is bounded between −1 and +1. The first generalized decomposition method proposed here is based on the introduction of a linear stiffness term, a change of independent variable, the introduction of an artificial parameter and the expansion of the solution in power series of this parameter, and yields, at first-order, a periodic solution in the new independent variable. By requiring that the forcing be an analytic function of its arguments, higher-order approximations may be obtained by either solving an integral or a linear ordinary differential equation. For oscillators with periodic solutions, the frequency of oscillation is also expanded in power series of the artificial parameter in the second generalized decomposition method which is set to unity at the end of the calculations, and further approximations to the frequency can be obtained by requiring that, at each order in the expansion, the solution be free from secular terms. It is shown that the second generalized decomposition method is identical to an artificial parameter-Linstedt–Poincaré method recently introduced by the author and it predicts a frequency of oscillation, at first-order, which is more accurate than that provided by a first-order harmonic balance procedure.

Suggested Citation

  • Ramos, J.I., 2009. "Generalized decomposition methods for nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1078-1084.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:3:p:1078-1084
    DOI: 10.1016/j.chaos.2008.04.037
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077908002142
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2008.04.037?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Elgazery, Nasser S., 2008. "Numerical solution for the Falkner–Skan equation," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 738-746.
    2. Helal, M.A. & Mehanna, M.S., 2006. "A comparison between two different methods for solving KdV–Burgers equation," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 320-326.
    3. Bildik, Necdet & Inc, Mustafa, 2007. "Modified decomposition method for nonlinear Volterra–Fredholm integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 308-313.
    4. Ramos, J.I., 2008. "Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 400-408.
    5. Memarbashi, Reza, 2008. "Numerical solution of the Laplace equation in annulus by Adomian decomposition method," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 138-143.
    6. Ramos, J.I., 2009. "Piecewise-adaptive decomposition methods," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1623-1636.
    7. Abbasbandy, S., 2007. "A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 257-260.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ramos, J.I., 2009. "Piecewise-adaptive decomposition methods," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1623-1636.
    2. Abdel-Halim Hassan, I.H., 2008. "Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 53-65.
    3. Bengochea, Gabriel, 2014. "Algebraic approach to the Lane–Emden equation," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 424-430.
    4. Tajvidi, T. & Razzaghi, M. & Dehghan, M., 2008. "Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 59-66.
    5. Jules Sadefo-Kamdem, 2011. "Integral Transforms With The Homotopy Perturbation Method And Some Applications," Working Papers hal-00580023, HAL.
    6. Memarbashi, Reza, 2008. "Numerical solution of the Laplace equation in annulus by Adomian decomposition method," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 138-143.
    7. Izadi, Mohammad, 2021. "A discontinuous finite element approximation to singular Lane-Emden type equations," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    8. Jafarimoghaddam, A. & Roşca, N.C. & Roşca, A.V. & Pop, I., 2021. "The universal Blasius problem: New results by Duan–Rach Adomian Decomposition Method with Jafarimoghaddam contraction mapping theorem and numerical solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 60-76.
    9. Beiglo, H. & Gachpazan, M., 2020. "Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    10. Gupta, A.K. & Ray, S. Saha, 2018. "On the solution of time-fractional KdV–Burgers equation using Petrov–Galerkin method for propagation of long wave in shallow water," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 376-380.
    11. Chen, Zhong & Jiang, Wei, 2015. "An efficient algorithm for solving nonlinear Volterra–Fredholm integral equations," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 614-619.
    12. Dubey, Ved Prakash & Kumar, Rajnesh & Kumar, Devendra, 2020. "A hybrid analytical scheme for the numerical computation of time fractional computer virus propagation model and its stability analysis," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    13. Dehghan, Mehdi & Shakourifar, Mohammad & Hamidi, Asgar, 2009. "The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2509-2521.
    14. Dubey, Ved Prakash & Kumar, Rajnesh & Kumar, Devendra, 2019. "Analytical study of fractional Bratu-type equation arising in electro-spun organic nanofibers elaboration," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 762-772.
    15. Ahmad Sami Bataineh & Osman Rasit Isik & Abedel-Karrem Alomari & Mohammad Shatnawi & Ishak Hashim, 2020. "An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials," Mathematics, MDPI, vol. 8(9), pages 1-17, September.
    16. Beléndez, A. & Beléndez, T. & Neipp, C. & Hernández, A. & Álvarez, M.L., 2009. "Approximate solutions of a nonlinear oscillator typified as a mass attached to a stretched elastic wire by the homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 746-764.
    17. Chai, Zhenhua & Shi, Baochang & Zheng, Lin, 2008. "A unified lattice Boltzmann model for some nonlinear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 36(4), pages 874-882.
    18. Golbabai, A. & Keramati, B., 2008. "Easy computational approach to solution of system of linear Fredholm integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 568-574.
    19. Lai, Huilin & Ma, Changfeng, 2009. "Lattice Boltzmann method for the generalized Kuramoto–Sivashinsky equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1405-1412.
    20. Alan Jalal Abdulqader, 2018. "Numerical Solution for Solving System of Fuzzy Nonlinear Integral Equation by Using Modified Decomposition Method," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 10(1), pages 32-43, February.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:41:y:2009:i:3:p:1078-1084. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.