A perturbation technique to compute initial amplitude and phase for the Krylov-Bogoliubov-Mitropolskii method

Abstract

Recently, a unified Krylov-Bogoliubov-Mitropolskii method has been presented (by Shamsul \cite{1}) for solving an $n$-th, $n=2$ or $n>2$, order nonlinear differential equation. Instead of amplitude(s) and phase(s), a set of variables is used in \cite{1} to obtain a general formula in which the nonlinear differential equations can be solved. By a simple variables transformation the usual form solutions (i.e., in terms of amplitude(s) and phase(s)) have been found. In this paper a perturbation technique is developed to calculate the initial values of the variables used in \cite{1}. By the noted transformation the initial amplitude(s) and phase(s) can be calculated quickly. Usually the conditional equations are nonlinear algebraic or transcendental equations; so that a numerical method is used to solve them. Rink \cite{7} earlier employed an asymptotic method for solving the conditional equations of a second-order differential equation; but his derived results were not so good. The new results agree with their exact values (or numerical results) nicely. The method can be applied whether the eigen-values of the unperturbed equation are purely imaginary, complex conjugate or real. Thus the derived solution is a general one and covers the three cases, i.e., un-damped, under-damped and over-damped.