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

Authors

  • M. Saifur Rahman

DOI:

https://doi.org/10.5556/j.tkjm.43.2012.853

Keywords:

Nonlinear Differential Equation, Perturbation Technique

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.

Author Biography

M. Saifur Rahman

Department of Mathematics, Rajshahi University of Engineering and Technology (RUET), Rajshahi 6204,Bangladesh.

References

M. Shamsul Alam, A unified Krylov-Bogoliubov-Mitropolskii method for solving n-order nonlinear systems, J. Franklin Inst., 339(2002), 239-248.

N. N. Krylov and N. N. Bogoliubov, Introduction to nonlinear mechanics, Princeton University Press, placeStateNew Jersey, 1947.

N. N. Bogoliubov and Yu. Mitropolskii, Asymptotic methods in the theory of nonlinear oscillations, Gordan and Breach, placeStateNew York, 1961.

I. S. N. Murty, B. L. Deekshatulu and G. Krisna, On asymptotic method of Krylov-Bogoliubov for over-damped nonlinear systems, J. Frank Inst., 288(1969), 49--64.

I. P. Popov, A generalization of the Bogoliubov asymptotic methods in the theory of nonlinear oscillation (in Russian), Dokl. Akad. Nauk. SSSR., 111(1956), 308--310.

I. S. N. Murty, B. L. Deekshatulu and G. Krisna, On asymptotic method of Krylov-Bogoliubov for over-damped nonlinear systems, J. Frank Inst., 288(1969), 49--64.

R. A. Rink, A procedure to obtain the initial amplitude and phase for the Krylov-Bogoliubov method, J. Franklin Inst., 303 (1977), 59--65.

M. Shamsul Alam, A modified and compact form of Krylov-Bogoliubov-Mitropolskii unified KBM method for solving an n-th order nonlinear differential equation,Int. J. Nonlinear Mech., 39 (2004), 1343--1357.

M. Shamsul Alam and M. A. Sattar, A unified Krylov-Bogoliubov-Mitropolskii method for solving third-order nonlinear system, Indian J. Pure Appl. Math., 28, 151--167.

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Published

2012-12-31

How to Cite

Rahman, M. S. (2012). A perturbation technique to compute initial amplitude and phase for the Krylov-Bogoliubov-Mitropolskii method. Tamkang Journal of Mathematics, 43(4), 563-575. https://doi.org/10.5556/j.tkjm.43.2012.853

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Papers