Modified variational iteration method for analytical solutions of nonlinear oscillators

Abstract

The second-order nonlinear oscillators have rich dynamics. We proposed a novel analytical method based on both variational iteration method and Adomian method. The variational iteration method is used to establish an equivalent integral system. So then Adomian polynomials are adopted to linearize the strong nonlinear terms in nonlinear oscillators and analytical solutions are obtained successively.

Existence and control of weighted pseudo almost periodic solutions for abstract nonlinear vibration systems

“…xample 8 Consider the following nonlinear oscillator system which generalizes the one in He et al 28 x…”

A complement to period/frequency estimation of a nonlinear oscillator

“…Solving equations (13), (16) and (21) simultaneously, a closed solution can be obtained. It is obvious that a%A, the result is more accurate than that by the homotopy perturbation method.…”

“…However, all formulae cannot obtain a more accurate solution. Some famous analytical methods, e.g., the homotopy perturbation method [4][5][6][7][8][9][10][11][12][13] and the variational iteration method, [14][15][16] can continue their solution processes until a needed accuracy is obtained. This paper tries to give an effective modification so that any accurate solutions can be obtained.…”

Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations

“…This method is recently further modified for vector functions by introducing matrix Lagrange multiplier. 18,19 The main characteristic of this method is the presence of the elements of flexibility and ability in solving nonlinear and linear problems. This method is independent of the complexities of Adomianit polynomials.…”