
A perturbation technique to compute initial amplitude and phase for the KrylovBogoliubovMitropolskii methodDOI: 10.5556/j.tkjm.43.2012.563575 Keywords: Nonlinear Differential Equation , Perturbation Technique Abstract: Recently, a unified KrylovBogoliubovMitropolskii 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 secondorder 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 eigenvalues 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., undamped, underdamped and overdamped.
