Nonlinear oscillator with discontinuity by generalized harmonic balance method

The homotopy perturbation method is used to obtain the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x^1/3. We find this method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies 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 frequency of less than 0.60% for small and large values of oscillation amplitude, while this relative error is as low as 0.050% for the second iteration. Comparison of the results obtained using this method with those obtained by different harmonic balance methods reveals that the former is more effective and convenient for these types of nonlinear oscillators.This work was supported by the "Ministerio de Ciencia e Innovación", Spain, under project FIS2008-05856-C02-02

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

The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this paper predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre’s formula to approximately obtain this mean are used.This work has been supported by the “Ministerio de Ciencia e Innovación” of Spain, under projects FIS2008-05856-C02-01 and FIS2008-05856-C02-02

Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable

The second-order harmonic balance method is used to construct three approximate frequency-amplitude relations for a conservative nonlinear singular oscillator in which the restoring force is inversely proportional to the dependent variable. Two procedures are used to solve the nonlinear differential equation approximately. In the first the differential equation is rewritten in a form that does not contain the expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first one and the discrepancy between the approximate frequency and the exact one is lower than 1.28%.This work was supported by the "Ministerio de Educación y Ciencia", Spain, under project FIS2005-05881-C02-02, and by the "Generalitat Valenciana", Spain, under project ACOMP/2007/020

Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method

He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). 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 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient.This work was supported by the “Ministerio de Educación y Ciencia”, Spain, under project FIS2005-05881-C02-02, and by the “Generalitat Valenciana”, Spain, under project ACOMP/2007/020

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

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

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

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.This work was supported by the “Ministerio de Educación y Ciencia”, Spain, under project FIS2005-05881-C02-02 and the “Generalitat Valenciana”, Spain, under project ACOMP/ 2007/020