On Some New Approximate Solutions of Fourth Order Nonlinear Differential Equations

Abstract

Most of the perturbation methods are developed to find periodic solutions of nonlinear systems; transients are not considered. First Krylov and BogoLiubov introduced a perturbation method to discuss the transients in the second order autonomous systems with small nonlinearities. The method is well known as an "asymptotic averaging method" in the theory of nonlinear oscillations. Later, the method has been amplified and justified by BogoLiubov and Mitropolskii. In this dissertation, we have modified and extended the Krylov-. BogoLiubov-Mitropolskii (KBM) method to investigate some fourth order nonlinear systems. First a fourth order over-damped nonlinear autonomous differential system is considered and a new perturbation solution is developed. Then a method is developed to find asymptotic solution of damped oscillatory nonlinear systems. We then again solve the fourth order over­damped nonlinear systems under some special conditions. Later, unified KBM method is used to obtain the approximate solution of the fourth order ordinary differential equation with small nonlinearities, when a pair of eigen-values of the unperturbed equation is a multiple (I. e., double, triple etc.) of the other pair or pairs. In case of oscillatory processes some of the natural frequencies of the unperturbed equation may be in integral ratio and thus internal resonance is introduced, which is an interesting and important part of nonlinear vibrations. Modified and compact form of KBM method is used to find approximate solutions of fourth order nonlinear systems with large damping. The methods are illustrated by several examples.