Analytical accurate solutions of nonlinear oscillator systems via coupled homotopy-variational approach

Abstract

In this paper, the coupled homotopy-variational approach (CHVA) based on combining homotopy with the variational approach is applied to solve the nonlinear Duffing equation, and new frequency- amplitude relationships are obtained. The coupled method works very well for the entire range of initial amplitudes and calculates higher-order approximations. The results are compared with those obtained using other known analytical and numerical methods to substantiate the accuracy and efficiency of the approximate analytical approach. The method provides better results than other existing methods. The advantage of the coupled method is that the current second order approximations almost coincide with the corresponding exact solutions. Thus, the presented CHVA could be applied to other strongly nonlinear differential equations.

Keywords

• Nonlinear vibration
• Approximate solutions
• Homotopy perturbation method
• Variational iteration method
• Nonlinear stiffnesses
• Duffing equation