Approximate Solutions of Second and Fourth Order Ordinary Differential Systems with Strong Generalized Nonlinearity

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 weakly nonlinear differential systems for oscillatory, damped oscillatory, over damped, critically damped and more critically damped cases by imposing some special restrictions. Ji-Huan He has developed a homotopy perturbation method for solving second order strongly nonlinear differential systems without damping. Uddin ci al. have presented an approximate analytical technique for second order strongly nonlinear differential systems with damping by combining He's homotopy perturbation technique and the extended form of the KBM method. Recently, Uddin ci al. have developed an analytical approximate technique for solving a certain type of fourth order strongly nonlinear oscillatory differential systems with small damping and cubic nonlinearity by combining He's homotopy perturbation and the extended form of the KBM methods. In this thesis, approximate analytical techniques shall be presented by combining the He's homotopy perturbation technique and the extended form of the KBM method for solving the second and fourth order nonlinear ordinary differential systems with strong generalized nonlinearity. To justify the presented methods, the approximate solutions have been compared to those solutions obtained by the fourth order Runge-Kutta method.