The modified homotopy perturbation method for solving strongly nonlinear oscillators

In this paper we propose a reliable algorithm for the solution of nonlinear oscillators. Our algorithm is based upon the homotopy perturbation method (HPM), Laplace transforms, and Pade approximants. This modified homotopy perturbation method (MHPM) utilizes an alternative framework to capture the periodic behavior of the solution, which is characteristic of oscillator equations, and to give a good approximation to the true solution in a very large region. The current results are compared with those derived from the established Runge-Kutta method in order to verify the accuracy of the MHPM. It is shown that there is excellent agreement between the two sets of results. Results also show that the numerical scheme is very effective and convenient for solving strongly nonlinear oscillators.

[1]  G. A. Baker.  Essentials of Padé approximants , 1975 .

[2]  Jianping Cai,et al.  An equivalent nonlinearization method for strongly nonlinear oscillations , 2005 .

[3]  Nicolai Minorsky,et al.  Introduction to non-linear mechanics : topological methods, analytical methods, nonlinear resonance, relaxation oscillations , 1947 .

[4]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[5]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[6]  Ji-Huan He,et al.  Addendum:. New Interpretation of Homotopy Perturbation Method , 2006 .

[7]  Shaher Momani.  ANALYTICAL APPROXIMATE SOLUTIONS OF NONLINEAR OSCILLATORS BY THE MODIFIED DECOMPOSITION METHOD , 2004 .

[8]  C. Dang,et al.  An aftertreatment technique for improving the accuracy of Adomian's decomposition method , 2002 .

[9]  Augusto Beléndez,et al.  Application of He's Homotopy Perturbation Method to the Duffing-Harmonic Oscillator , 2007 .

[10]  Ji-Huan He.  New interpretation of homotopy perturbation method , 2006 .

[11]  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 .

[12]  Augusto Beléndez,et al.  Application of He's Homotopy Perturbation Method to the Relativistic (An)harmonic Oscillator. II: A More Accurate Approximate Solution , 2007 .

[13]  Augusto Beléndez,et al.  An Improved 'Heuristic' Approximation for the Period of a Nonlinear Pendulum: Linear Analysis of a Classical Nonlinear Problem , 2007 .

[14]  Zacharias A. Anastassi,et al.  Special Optimized Runge-Kutta Methods for IVPs with Oscillating Solutions , 2004 .

[15]  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 .

[16]  Robert E. O'Malley,et al.  Introduction to Perturbation Methods (M. H. Holmes) , 1996, SIAM Rev..

[17]  D. Roy,et al.  An adaptation of adomian decomposition for numeric–analytic integration of strongly nonlinear and chaotic oscillators , 2007 .

[18]  Lan Xu,et al.  He's parameter-expanding methods for strongly nonlinear oscillators , 2007 .

[19]  Ji-Huan He.  A coupling method of a homotopy technique and a perturbation technique for non-linear problems , 2000 .

[20]  Ji-Huan He,et al.  Application of Parameter-expanding Method to Strongly Nonlinear Oscillators , 2007 .

[21]  Augusto Beléndez,et al.  Application of a modified He's homotopy perturbation method to obtain higher-order approximations of an x1/3 force nonlinear oscillator , 2007 .

[22]  Livija Cveticanin,et al.  Homotopy–perturbation method for pure nonlinear differential equation , 2006 .

[23]  N. Bogolyubov,et al.  Asymptotic Methods in the Theory of Nonlinear Oscillations , 1961 .

[24]  Vedat Suat Ertürk,et al.  Solutions of non-linear oscillators by the modified differential transform method , 2008, Comput. Math. Appl..

[25]  Vimal Singh,et al.  Perturbation methods , 1991 .