Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

Abstract

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An improved homotopy analysis method with accelerated convergence for nonlinear problems

“…There are various types of nonlinear problems in the engineering field, most of which are strongly nonlinear and do not have an analytical solution. To date, many powerful methods have been developed to solve nonlinear differential equations, such as the Krylov-Bogoliubov-Mitropolsky (KBM) [1][2][3], the method of harmonic balance [4][5][6], Adomian's decomposition method (ADM) [7,8] and the small parameter method [9][10][11]. However, these methods cannot provide us with a simple way to adjust and control the convergence region and rate of approximate solution series.…”

An analytical technique to find approximate solutions of nonlinear damped oscillatory systems

“…Certainly the later process [26] is laborious. Yamgoue and Kofane [27] also investigated damped oscillations of some strongly nonlinear planar Hamiltonian systems (see also [28]). In this article, approximate solutions of nonlinear differential systems with damping are determined combining Krylov-Bogoliubov-Mitropolskii (KBM) method [2,3] and a new HB method [29].…”