Application of Parameter Expanding Methods to Nonlinear Oscillators with Discontinuities

In this paper He's modified Lindstedt-Poincare method and bookkeeping parameter method known as Parameter Expanding Methods are applied to nonlinear oscillators with discontinuities. We obtained sufficiently accurate solutions with first-order approximations which are valid for whole domain. Comparison of the obtained solutions with the exact ones show the proposed methods are very effective and convenient.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  Physics Letters , 1962, Nature.

[3]  Ahmet Yildirim,et al.  A Comparative Study of He's Homotopy Perturbation Method for Determining Frequency-amplitude Relation of a Nonlinear Oscillator with Discontinuities , 2007 .

[4]  Livija Cveticanin,et al.  The homotopy-perturbation method applied for solving complex-valued differential equations with strong cubic nonlinearity , 2005 .

[5]  Ahmet Yildirim,et al.  A study of nonlinear oscillators with u1/3 force by He's variational iteration method , 2007 .

[6]  Ji-Huan He,et al.  Variational principles for some nonlinear partial differential equations with variable coefficients , 2004 .

[7]  Ji-Huan He,et al.  Nonlinear oscillator with discontinuity by parameter-expansion method , 2008 .

[8]  Augusto Beléndez,et al.  Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method , 2009 .

[9]  Ji-Huan He,et al.  The homotopy perturbation method for nonlinear oscillators with discontinuities , 2004, Appl. Math. Comput..

[10]  A. A. Hemeda.  Variational iteration method for solving non-linear partial differential equations , 2009 .

[11]  Ji-Huan He,et al.  Bookkeeping Parameter in Perturbation Methods , 2001 .

[12]  Augusto Beléndez,et al.  Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators , 2008 .

[13]  Nicolae Herisanu,et al.  Periodic solutions for some strongly nonlinear oscillations by He's variational iteration method , 2007, Comput. Math. Appl..

[14]  Ahmet Yildirim,et al.  Application of He's semi-inverse method to the nonlinear Schrödinger equation , 2007, Comput. Math. Appl..

[15]  Hong-Mei Liu,et al.  Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–Poincare method , 2005 .

[16]  H. M. Daniali,et al.  The variational iteration method for nonlinear oscillators with discontinuities , 2007 .

[17]  Mary L. Boas,et al.  Solving Equations with Physical Understanding , 1985 .

[18]  Ahmet Yildirim,et al.  A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity , 2007 .

[19]  S. Gallego,et al.  Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities , 2009 .