A new perturbation solution for systems with strong quadratic and cubic nonlinearities

The new perturbation algorithm combining the method of multiple scales (MS) and Lindstedt–Poincare techniques is applied to an equation with quadratic and cubic nonlinearities. Approximate analytical solutions are found using the classical MS method and the new method. Both solutions are contrasted with the direct numerical solutions of the original equation. For the case of strong nonlinearities, solutions of the new method are in good agreement with the numerical results, whereas the amplitude and frequency estimations of classical MS yield high errors. For strongly nonlinear systems, exact periods match well with the new technique while there are large discrepancies between the exact and classical MS periods. Copyright © 2009 John Wiley & Sons, Ltd.