Application of Parameter-expanding Method to Strongly Nonlinear Oscillators

An Improved 'Heuristic' Approximation for the Period of a Nonlinear Pendulum: Linear Analysis of a Classical Nonlinear Problem

“…0 2 ), and the other terms are different. Since the first two terms of these series are identical, series (13) and (14) track series (12) closely for small amplitudes, but as the coefficients of the third (and subsequent) terms of series (13) and (14) are different to their counterparts in series (12), the difference between these approximate expressions for the period of the pendulum and the exact value increases as the amplitude of the oscillation increases. Even so, and as shown in the reference [26], equation (10) is a better approximation to the exact value than is equation (9), as can be seen by comparing the coefficients of the powers !…”

“…In particular the study of nonlinear oscillators is of great interest to many researchers [2,3]. There are several methods used to find approximate solutions to nonlinear oscillators, such as perturbation techniques [4][5][6][7][8][9][10][11][12][13][14] or harmonic balance based methods [15][16][17][18][19][20][21][22]. Surveys of the literature with numerous references and useful bibliographies have given in [3,23].…”

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

“…It is very difficult to solve nonlinear problems and, in general, it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. There are several methods used to find approximate solutions to nonlinear problems, such as perturbation techniques [1][2][3][4][5][6], harmonic balance based methods [6][7][8][9] or other techniques [10][11][12][13][14][15][16][17][18]. An excellent review on some asymptotic methods for strongly nonlinear equations can be found in detail in references [19] and [20].…”

Approximate Analytical Solutions for the Relativistic Oscillator Using a Linearized Harmonic Balance Method

“…In particular, physical and mechanical oscillatory systems are often governed by second order nonlinear differential equations and their study is of great interest to many researchers. There are several methods used to find approximate solutions to nonlinear oscillators, such as perturbation [1,2], variational [3,4], homotopy perturbation [5][6][7][8][9][10][11], standard and modified Lindstedt-Poincaré [2,[12][13][14][15][16][17][18], harmonic balance [2,[19][20][21][22][23][24][25], bookkeeping parameter [26], iteration perturbation [27], parameter expanding [28], parametrized perturbation [29], artificial parameter [30], linearized and quasilinearized harmonic balance [31][32][33][34] methods, etc. Surveys of the literature with numerous references and useful bibliographies may be found in [2,[35][36][37].…”