Analytical Solutions of Third and Fourth Order Weakly Nonlinear Ordinary Differential Systems with Damping and Slowly Varying Coefficients by the Unified KBM Method

Abstract

A perturbation method known as "the asymptotic averaging method" in the theory of nonlinear oscillations was first presented by Krylov and Bogoliubov (KB) in 1947. Primarily, the method was developed only to obtain the periodic solutions of the second order weakly nonlinear differential systems. Later, the method of KB has been improved and justified by Bogoliubov and Mitropolaskii in 1967. In literature, this method is known as the Krylov- Bogoliubov-Mitropolaskii (KBM) method. Now a days this method is used for obtaining the solutions of second, third and fourth order nonlinear differential systems for oscillatory, damped oscillatory, over damped, critically damped and more critically damped cases by imposing some proper restrictions, in this thesis, an analytical approximate technique is extended to find out the second approximate solutions of third order weakly nonlinear differential systems in the presence of strong linear damping and slowly varying coefficients based on the KBM method. Also, the KBM method is presented to fmd out the solutions of a fourth order weakly nonlinear differential systems in the presence of strong linear damping and slowly varying coefficients including some limitations. To justify the presented method, the approximate solutions have been compared to those solutions obtained by the fourth order Runge-Kutta method graphically.