Generalization of the modified Lindstedt-Poincare method for solving some strong nonlinear oscillators

Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators

“…5.2 that the values of these coefficients are opposite in sign when 19 32 < k. But all are positive when k ≤ 19 32 and δ 2 vanishes when k = 19 32 . Thus for k = 1 and k = 19 32 , we have obtained respectively a 0 ω(k = 1) ω(k = Table 1: Comparison of the approximate frequencies obtained by present method with the numerical and other existing frequencies (Alam et al [14] and Cheung et al [6], k = 1) for the Duffing oscillator (where Er(%) denotes absolute percentage error). Comparing these results, we easily expect that α-series (Eq.…”

“…We have introduced k in the first approximate solution and consequently k appear in the second, third and fourth approximations. Choosing a suitable value of k, we can find a series of ω which converge faster than that of obtained by Cheung et al [6] and Alam et al [14].…”

“…From Fig. 5 Table 3: Comparison of the approximate frequencies obtained by present method with the numerical and other existing frequencies (Alam et al [14] and Cheung et al [6], k = 1) for the oscillatorẍ + x + εẋ 2 x = 0 (where Er(%) denotes absolute percentage error). For the nonlinear oscillatorẍ + x + εẍx 2 = 0, we have obtained…”

“…However, all the approximate solutions obtained by approaches of [4]- [6] are effective for Duffing oscillator with cubical nonlinearity. The aim of this article is to present a new form of the modified Lindstedt-Poincare method of Cheung et al [6] based on the conversion formula presented by Alam et al [14] by introducing a parameter k. The solutions obtained for various nonlinear oscillators nicely agree with corresponding numerical solutions and provide better results than other existing solutions. Besides the classical perturbation methods, many approximate techniques have been presented for solving the stronger nonlinear oscillators.…”

“…For different values of the unknown constant k we have calculated some results and presented in Table 4 together with corresponding numerical results and other existing frequencies (Alam et al [14] and Cheung et al [6], k = 1). From Table 4 it is clear that frequency of the oscillator depends on the parameter k and comparing various results suitable value of k can be determined.…”