Research Article Received 6 February 2009 Published online 16 June 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/mma.1187 MOS subject classification: 34 A 34; 34 C 15; 34 E 10; 34 E 13; 65 L 05 A new perturbation solution for systems with strong quadratic and cubic nonlinearities Mehmet Pakdemirli ∗ † and Mustafa Mehmet Fatih Karahan Communicated by Cash 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. Keywords: perturbation methods; Lindstedt–Poincare method; multiple scales method; numerical solutions; systems with quadratic and cubic nonlinearities 1. Introduction Perturbations methods are widely used for over a century to determine approximate analytical solutions for mathematical models. Algebraic equations, integrals, differential equations, difference equations and integro-differential equations can be solved approxi- mately with these techniques. The direct expansion method (straightforward expansion) does not produce physically valid solutions for most of the cases and depending on the nature of the equation, many different perturbation techniques, such as Lindstedt– Poincare technique, renormalization method, method of multiple scales (MS), averaging methods, method of matched asymptotic expansions, etc., are developed within time. One of the deficiencies in applying perturbation methods is that a small parameter is needed in the equations or the small parameter should be introduced artificially to the equations. Nevertheless, the solved problem is a weak nonlinear problem and it becomes hard to obtain a valid approximate solution for strongly nonlinear systems. There have been a number of attempts recently to validate perturbation solutions for strongly nonlinear systems. Hu and Xiong [1] contrasted two different approaches of Lindstedt–Poincare methods using the duffing equation. First, they solved the equation with classical method and then they made a slight modification in the expansions. Instead of expanding the transformation frequency, they expanded the natural frequency and obtained solutions with excellent convergence properties for the duffing equation. The time histories of solutions agree with the numerical solutions for arbitrarily large perturbation parameters. In a similar paper, the approximate and exact frequencies are contrasted for the duffing equation [2]. The case of vanishing restoring force was also treated for the same equation [3]. The obtained periods are contrasted with the exact period with good convergence properties for large parameters. While a complete review of the attempts to validate perturbation solutions for strongly nonlinear oscillators is beyond the scope of this work, a partial list will be given. Among the many developed methods, Linearized perturbation method [4--6], parameter expanding method [7, 8], new time transformations as modifications of Lindstedt–Poincare method [9--11], iteration methods [12, 13] are some examples. Very recently, MS method is modified by incorporating the time transformation of Lindstedt–Poincare method [14]. One of the major advantages of MS method over the Lindstedt–Poincare method is that transient solutions can be found using the former whereas it is impossible to retrieve such solutions using the latter. However, by expanding the natural frequency in modified Lindstedt–Poincare method as in [1--3], convergent solutions for large parameters are possible. Combining both methods would augment the advantages Department of Mechanical Engineering, Celal Bayar University, 45140 Muradiye, Manisa, Turkey ∗ Correspondence to: M. Pakdemirli, Department of Mechanical Engineering, Celal Bayar University, 45140 Muradiye, Manisa, Turkey. † E-mail: mpak@bayar.edu.tr Contract/grant sponsor: The Scientific and Technological Research Council of Turkey (TUB ˙ ITAK); contract/grant number: 108M490 704 Copyright © 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2010, 33 704–712