A Perturbation Technique to Compute Initial Amplitude and Phase for the Krylov-Bogoliubov-Mitropolskii Method

A. K. Azad ； M. Alal Hosen ； M. Saifur Rahman ； M. S. Alam

《Tamkang Journal of Mathematics》 43卷4期 (2012/12) Pp. 563-575

並列摘要

Recently, a unified Krylov-Bogoliubov-Mitropolskii method has been presented 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 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. 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 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.