IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v41y2009i1p380-393.html
   My bibliography  Save this article

An artificial parameter–Linstedt–Poincaré method for oscillators with smooth odd nonlinearities

Author

Listed:
  • Ramos, J.I.

Abstract

An artificial parameter method for obtaining the periodic solutions of oscillators with smooth odd nonlinearities is presented. The method is based on the introduction of a linear stiffness term and a new dependent variable both of which are proportional to the unknown frequency of oscillation, the introduction of an artificial parameter and the expansion of both the solution and the unknown frequency of oscillation in series of the artificial parameter. The method results in linear ordinary differential equations at each order in the parameter. By imposing the nonsecularity condition at each order in the expansion, the method provides different approximations to both the solution and the frequency of oscillation. The method does not require any minimization procedure; neither does it require the expansion of constants in terms of the artificial parameter. It is shown that the method presented here is also a decomposition technique and a homotopy perturbation method provided that in these techniques the unknown frequency of oscillation is expanded in terms of an artificial parameter and the nonsecularity condition is imposed at each order in the expansion procedure. It is also shown by means of six examples that the first approximation to the frequency of oscillation coincides with that obtained by means of harmonic balance methods, two- and three-level iterative techniques, and modified Linstedt–Poincaré procedures based on the expansion of the solution and constants that appear in the differential equation in terms of an artificial parameter.

Suggested Citation

  • Ramos, J.I., 2009. "An artificial parameter–Linstedt–Poincaré method for oscillators with smooth odd nonlinearities," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 380-393.
  • Handle: RePEc:eee:chsofr:v:41:y:2009:i:1:p:380-393
    DOI: 10.1016/j.chaos.2008.01.009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077908000167
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2008.01.009?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. Wang, Shu-Qiang & He, Ji-Huan, 2008. "Nonlinear oscillator with discontinuity by parameter-expansion method," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 688-691.
    2. Cveticanin, L., 2006. "Homotopy–perturbation method for pure nonlinear differential equation," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1221-1230.
    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. Zeng, De-Qiang, 2009. "Nonlinear oscillator with discontinuity by the max–min approach," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2885-2889.
    2. Wang, Shu-Qiang & He, Ji-Huan, 2008. "Nonlinear oscillator with discontinuity by parameter-expansion method," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 688-691.
    3. H. X. Mamatova & Z. K. Eshkuvatov & Sh. Ismail, 2023. "A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind," Mathematics, MDPI, vol. 11(20), pages 1-30, October.
    4. Biazar, J. & Eslami, M. & Aminikhah, H., 2009. "Application of homotopy perturbation method for systems of Volterra integral equations of the first kind," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3020-3026.
    5. Tao, Zhao-Ling, 2009. "Frequency–amplitude relationship of nonlinear oscillators by He’s parameter-expanding method," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 642-645.
    6. Fathizadeh, M. & Rashidi, F., 2009. "Boundary layer convective heat transfer with pressure gradient using Homotopy Perturbation Method (HPM) over a flat plate," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2413-2419.
    7. Biazar, J. & Ghazvini, H., 2009. "He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 770-777.
    8. Yusufoğlu (Agadjanov), Elcin, 2009. "Improved homotopy perturbation method for solving Fredholm type integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 28-37.
    9. Golbabai, A. & Keramati, B., 2009. "Solution of non-linear Fredholm integral equations of the first kind using modified homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2316-2321.
    10. Nicolae Herisanu & Bogdan Marinca & Vasile Marinca, 2022. "Dynamics of the Vibro-Impact Nonlinear Damped and Forced Oscillator under the Influence of the Electromagnetic Actuation," Mathematics, MDPI, vol. 10(18), pages 1-16, September.
    11. Biazar, J. & Ghazvini, H. & Eslami, M., 2009. "He’s homotopy perturbation method for systems of integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1253-1258.
    12. Kaya, M.O. & Altay Demirbağ, S., 2009. "Application of parameter expansion method to the generalized nonlinear discontinuity equation," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 1967-1973.
    13. Golbabai, A. & Keramati, B., 2008. "Modified homotopy perturbation method for solving Fredholm integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1528-1537.
    14. Ghorbani, Asghar, 2009. "Beyond Adomian polynomials: He polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1486-1492.
    15. Cai, Xu-Chu & Wu, Wen-Ying, 2009. "Homotopy perturbation method for nonlinear oscillator equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2581-2583.
    16. Ravi Kanth, A.S.V. & Aruna, K., 2009. "He’s homotopy-perturbation method for solving higher-order boundary value problems," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1905-1909.
    17. Cveticanin, L., 2009. "Application of homotopy-perturbation to non-linear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 221-228.
    18. 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.
    19. 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.

    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:1:p:380-393. 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.