Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method

This paper deals with the nonlinear oscillations of a particle which moves on a rotating parabola. An analytic approximate technique, namely optimal homotopy asymptotic method (OHAM) is employed to propose an analytic approach to solve nonlinear oscillations. The validity of the OHAM is independent on whether or not there exist small or large parameters in the considered nonlinear equations. Our procedure provides us with a convenient way to optimally control the convergence of the approximate solutions. An example is given and the results reveal that this procedure is very effective, simple and accurate. This paper demonstrates the general validity and the great potential of the OHAM.

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

[2]  Nicolae Herisanu,et al.  An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate , 2009, Appl. Math. Lett..

[3]  Ji-Huan He.  Homotopy Perturbation Method for Bifurcation of Nonlinear Problems , 2005 .

[4]  Y. K. Cheung,et al.  An Elliptic Lindstedt--Poincaré Method for Certain Strongly Non-Linear Oscillators , 1997 .

[5]  S. Liao,et al.  Application of Homotopy Analysis Method in Nonlinear Oscillations , 1998 .

[6]  Vasile Marinca,et al.  A NEW ANALYTICAL APPROACH TO NONLINEAR VIBRATION OF AN ELECTRICAL MACHINE , 2008 .

[7]  Mohd. Salmi Md. Noorani,et al.  The homotopy analysis method for Cauchy reaction–diffusion problems , 2008 .

[8]  N. Herisanu,et al.  Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer , 2008 .

[9]  Ji-Huan He,et al.  Preliminary report on the energy balance for nonlinear oscillations , 2002 .

[10]  G. Adomian.  A review of the decomposition method in applied mathematics , 1988 .

[11]  J. Ramos.  An artificial parameter Linstedt–Poincaré method for the periodic solutions of nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable , 2008 .

[12]  Juan I. Ramos,et al.  Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method , 2008 .

[13]  Ji-Huan He.  An approximate solution technique depending on an artificial parameter: A special example , 1998 .

[14]  Ivana Kovacic,et al.  Parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity , 2007 .

[15]  S. Abbasbandy.  The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation , 2007 .

[16]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[17]  Shijun Liao,et al.  A Second-Order Approximate Analytical Solution of a Simple Pendulum by the Process Analysis Method , 1992 .

[18]  N. Herisanu,et al.  Optimal homotopy asymptotic method with application to thin film flow , 2008 .

[19]  Matthew P. Cartmell,et al.  Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems , 2003 .

[20]  S. Liao.  An approximate solution technique not depending on small parameters: A special example , 1995 .

[21]  Alfredo Aranda,et al.  Improved LindstedtPoincar method for the solution of nonlinear problems , 2003, math-ph/0303052.

[22]  T. Hayat,et al.  Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations , 2008 .

[23]  R. Mahmood,et al.  Thin Film Flow of a Third Grade Fluid on a Moving Belt by He's Homotopy Perturbation Method , 2006 .

[24]  V. Marinca,et al.  Application of modified homotopy perturbation method to nonlinear oscillations , 2006 .

[25]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[26]  J. Awrejcewicz,et al.  Asymptotic approaches in nonlinear dynamics : new trends and applications , 1998 .

[27]  Augusto Beléndez,et al.  Application of He's Homotopy Perturbation Method to the Relativistic (An)harmonic Oscillator. I: Comparison between Approximate and Exact Frequencies , 2007 .

[28]  L. Cvetićanin.  Analyses of oscillators with non-polynomial damping terms , 2008 .