Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods

Abstract

A modified He's homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. 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, and so on. 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. The approximate formulas obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient.

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

“…For small values of the amplitude A it is possible to take into account the following power series expansions [15,16] …”

“…It is important to point out that the exact behaviour of the approximate frequency when A tends to zero is not obtained when other approximate methods as used including the harmonic balance method [17][18][19], the homotopy perturbation method [15,16], the energy balance method [20], the variational iteration method [21], a modified iteration procedure [22] or the Ritz procedure [23].…”

A Novel Rational Harmonic Balance Approach for Periodic Solutions of Conservative Nonlinear Oscillators

“…In order to determine an improved approximation we use a generalized, rational form given by the following expression [61,62] …”

Approximate Solutions for Conservative Nonlinear Oscillators by He’s Homotopy Method

“…In previous papers [37,38] we considered the simplest case, N = 1 (n = 0, 1), in (27) - (29), however, now we are going to improve He's homotopy perturbation method taking into account the approximation N = 2 (n = 0, 1, 2). We obtain…”