Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities

Abstract

A modified He's homotopy perturbation method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn(x). The He's homotopy perturbation method is modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second order linear differential equation. We find this modified 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 1.6% for all values of oscillation amplitude, while this relative error is 0.65% for the second iteration and 0.24% when the third-order approximation is considered. Comparison of the result obtained using this method with the exact ones reveals that this modified method 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.…”

Application of the homotopy perturbation method to seven - order Sawada – Kotara equations

“…One of the semi-exact methods is the homotopy perturbation method (HPM) [7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Approximate Analytical Solutions to Nonlinear Oscillations of Non-Natural Systems Using He's Energy Balance Method

“…The traditional perturbation methods have many shortcomings, and they are not valid for strongly nonlinear equations. To overcome the shortcomings, many new techniques have been appeared in open literature [3][4][5][6][7][8][9][10][11][12][13][14], such as Nonperturbative methods [3], homotopy perturbation method [4][5][6][7][33][34][35], perturbation techniques [8], Lindstedt-Poincaré method [9,10], parameter-expansion method [11,12] and Parameterized perturbation method [13,14].…”