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

Abstract

A new analytical approximate expression for the period of the simple pendulum is obtained by using a heuristic but pedagogical derivation. This formula depends on two parameters obtained by comparing term-by-term the power-series expansions of the approximate and exact expressions for the period. This formula is compared with others in the literature and the numerical results obtained show that the published approximations are not nearly as good as the new expression proposed in this paper.

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 homotopy perturbation method [14][15][16][17][18][19][20][21][22][23][24][25] is a general method for solving the differential and integral equations. This method is explained in references 7 and 8 for solving a nonlinear differential equation like Equation (14).…”

“…Substitution of the power series (19) and (20) into Equation (18), and collecting terms of the same power of , p gives the following set of linear equations: …”

“…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.…”

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

“…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 to 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][7], harmonic balance based methods [8][9][10][11][12] or other techniques [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].…”

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].…”