Approximate Solution Techniques for Fourth Order Nonlinear Ordinary Differential Systems

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 second order weakly nonlinear differential systems. Later, the method of KB has been improved and justified by Bogoliubov and Mitropolskii in 1967. In literature, this method is known as the Krylov-Bogoliubov-Mitropolskii (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 restrictions. Ji-Huan He has developed a homotopy perturbation method for second order strongly nonlinear differential systems without damping. Recently, Uddin el al. have developed approximate analytical technique for second order strongly nonlinear differential systems with damping combining He's homotopy perturbation technique and the extended form of the KBM method. In this thesis, an analytical approximate technique will be presented by combining the He's hornotopy perturbation technique and the extended form of the KBM method for solving certain type of fourth order strongly nonlinear differential systems with small damping and cubic nonlinearity. Also, the KBM method will be modified and elaborated to find out the solutions of fourth order weakly and near critically damped nonlinear differential systems by imposing some restrictions on the eigen values. To justify the presented methods, the approximate solutions have been compared to those solutions obtained by the fourth order Runge-Kuttu method.