Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

Abstract

In this paper He's homotopy perturbation method has been adapted to calculate higherorder approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is anti-symmetric and quadratic term. We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 0.73%for all values of oscillation amplitude, while this relative error is as low as 0.040% when the second iteration is considered. Comparison of the result obtained using this method with those obtained by harmonic balance method reveals that the former is very effective and convenient.

Application of a Modified Homotopy Perturbation Method for Calculation of Secular Axial Frequencies in a Nonlinear Ion Trap with Hexapole, Octopole and Decapole Superpositions

“…The new differential equation for 2 ( ) x t is difficult to solve because of an infinite number of harmonic terms in 1 ( ) x t . At this stage we consider the modification in He's homotopy perturbation method [26][27][28] to simplify the solution procedure. In this modified version of homotopy method the infinite series expansion for 1 ( ) x t in Equation (34) is truncated and an approximate equation…”

“…In a recent study [11], this technique was used in the framework of the parameter expanding or modified Lindstedt-Poincare method [12][13][14] for calculation of secular axial frequencies in a nonlinear ion trap with hexapole, octopole and decapole superpositions. In this paper, we use the same technique and the homotopy perturbation method [14][15][16][17][18][19][20][21][22][23][24][25] with a modification [26][27][28] for studying the asymmetric nonlinear oscillators. The modification in He's homotopy perturbation method is introduced by truncating the infinite series corresponding to the first order approximate solution before introducing this solution in the second order linear differential equation and so on.…”

“…Due to the inequality of angular frequency for negative amplitudes and positive amplitudes, we use the sign function for constructing the two auxiliary nonlinear equations. Then, the modified homotopy perturbation method [26][27][28] is used for solving the two auxiliary nonlinear differential equations and the ion secular frequencies are calculated. We compare the results of this paper with those obtained by using the modified Lindstedt-Poincare method [11] and with the exact results.…”

Solutions for Conservative Nonlinear Oscillators Using an Approximate Method Based on Chebyshev Series Expansion of the Restoring Force

“…By applying the first approximation based on the harmonic balance method (HBM) and a second-order RHBM, Mickens [2] achieved analytical approximate frequencies which relative errors were 0.7% and 0.07% [22], respectively. Both of them are less accurate than that calculated in this paper.…”

“…Beléndez et al [22] approximately solved this oscillator using the MHPM. They achieved relative errors of 0.7%, 0.05%, and 0.03% for the first-, second-, and third-order analytical approximations, respectively.…”