New Approximate Solution of Non-Linear Differential Systems

Abstract

Most of the perturbation methods are developed to find periodic solutions of nonlinear systems; transients are not considered. At first, Krylov and Bogoliubov introduced a perturbation method which is well known as “asymptotic averaging method” to discuss the transients in the second order autonomous systems with small nonlinearities. Later, this method has been amplified and justified by Bogoliubov and Mitropolskii. Mitropolskii has extended the method for slowly varying coefficients to determine the steady state periodic motions and transient processes. In this dissertation, we have modified and extended the KBM method to investigate some second order nonlinear systems. Firstly, a second order time dependent nonlinear differential system is considered. Then a new perturbation technique is developed to find an asymptotic solution of nonlinear systems in presence of an external force. Finally, this technique is used to obtain an asymptotic solution of a time dependent nonlinear differential system with slowly varying coefficients using the extended KBM method. These methods are illustrated with several examples.