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Piecewise-adaptive decomposition methods

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  • Ramos, J.I.

Abstract

Piecewise-adaptive decomposition methods are developed for the solution of nonlinear ordinary differential equations. These methods are based on some theorems that show that Adomian’s decomposition method is a homotopy perturbation technique and coincides with Taylor’s series expansions for autonomous ordinary differential equations. Piecewise-decomposition methods provide series solutions in intervals which are subject to continuity conditions at the end points of each interval, and their adaption is based on the use of either a fixed number of approximants and a variable step size, a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size. It is shown that the appearance of noise terms in the decomposition method is related to both the differential equation and the manner in which the homotopy parameter is introduced, especially for the Lane–Emden equation. It is also shown that, in order to avoid the use of numerical quadrature, there is a simple way of introducing the homotopy parameter in the two first-order ordinary differential equations that correspond to the second-order Thomas–Fermi equation. It is also shown that the piecewise homotopy perturbation methods presented here provide more accurate results than a modified Adomian decomposition technique which makes use of Padé approximants and the homotopy analysis method, for the Thomas–Fermi equation.

Suggested Citation

  • Ramos, J.I., 2009. "Piecewise-adaptive decomposition methods," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1623-1636.
  • Handle: RePEc:eee:chsofr:v:40:y:2009:i:4:p:1623-1636
    DOI: 10.1016/j.chaos.2007.09.043
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    References listed on IDEAS

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    1. El-Wakil, S.A. & Abdou, M.A., 2007. "New applications of Adomian decomposition method," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 513-522.
    2. Kamdem, J. Sadefo & Qiao, Zhijun, 2007. "Decomposition method for the Camassa–Holm equation," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 437-447.
    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. Al-Khaled, Kamel, 2007. "Theory and computation in singular boundary value problems," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 678-684.
    5. He, Ji-Huan, 2005. "Application of homotopy perturbation method to nonlinear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 695-700.
    6. 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.
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    1. Ramos, J.I., 2009. "Generalized decomposition methods for nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1078-1084.

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