Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations

Abstract

In this paper, homotopy perturbation method (HPM) and variational iteration method (VIM) are applied to solve nonlinear oscillator differential equations. Illustrative examples reveal that these methods are very effective and convenient for solving nonlinear differential equations. Moreover, the methods do not require linearization or small perturbation. Comparisons are also made between the exact solutions and the results of the homotopy perturbation method and variational iteration method in order to prove the precision of the results obtained from both methods mentioned.

A New Approach to Van der Pol’s Oscillator Problem

“…Such an analysis does not exist in the literature and the results obtained are new. These results are in good agreement with those given in [12] as well as those obtained by the numerical and the Adomian decomposition methods. This analysis therefore provides further support for the validity of LDM.…”

Non-linear vibration of Euler-Bernoulli beams

“…In recent years, some promising approximate analytical solutions have been proposed, such as Frequency Amplitude Formulation [13], Variational Iteration [5,6,14,17], Homotopy-Perturbation [3,4,7,24], Parametrized-Perturbation [18], Max-Min [15,19,29], Differential Transform Method [16], Adomian Decomposition Method [22], Energy Balance [23,30], etc.…”

The approximation of solutions for second order nonlinear oscillators using the polynomial least square method

“…The exact solution of this equation is u(t) = sin(t). Using the polynomial least square method we performed the following 7-th degree approximate polynomial solution of equation (3.1): Table 1 presents the comparison between the absolute errors (as the difference in absolute value between the approximate solution and the exact solution) corresponding to the approximate solution obtained using the homotopy perturbation method (HPM), to the approximate solution obtained by variational iteration method (VIM) [1] and to our approximate solution obtained by PLSM. It is easy to see that the approximate solution given by PLSM is much closer to the exact solution than the previous ones: the approximate solution given by HPM, and the approximate solution given by VIM [1].…”

“…Table 2 presents the comparison between the absolute errors corresponding to the approximate solution obtain by HPM, to the approximate solution given by VIM [1] and our approximate solution obtained by PLSM. …”