Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The solution obtained for different initial conditions for a second order nonlinear system show a good coincidence with those obtained by numerical method. The method is illustrated by an example.
| Published in | Applied and Computational Mathematics (Volume 1, Issue 1) |
| DOI | 10.11648/j.acm.20120101.11 |
| Page(s) | 1-5 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Nonlinear System, Varying Coefficient, Unperturbed Equation, Damped Oscillatory System
| [1] | N.N, Krylov and N.N., Bogoliubov, Introduction to Nonlinear Mechanics. Princeton University Press, New Jersey, 1947. |
| [2] | N. N, Bogoliubov and Yu. Mitropolskii, Asymptotic Methods in the Theory of nonlinear Oscillations, Gordan and Breach, New York, 1961. |
| [3] | Yu.,Mitropolskii, "Problems on Asymptotic Methods of Non-stationary Oscillations" (in Russian), Izdat, Nauka, Moscow, 1964. |
| [4] | I. P. Popov, "A generalization of the Bogoliubov asymptotic method in the theory of nonlinear oscillations", Dokl.Akad. Nauk SSSR 111, 1956, 308-310 (in Russian). |
| [5] | G.,Bojadziev, and J. Edwards, "On Some Asymptotic Methods for Non-oscillatory and Oscillatory Processes", Nonlinear Vibration Problems, 20, 1981, pp69-79. |
| [6] | I.S.N. Murty, "A Unified Krylov-Bogoliubov Method for Second Order Nonlinear Systems", Int. J. nonlinear Mech., 6, 1971, pp45-53. |
| [7] | M.,Shamsul Alam, "Unified Krylov-Bogoliubov-Mitropolskii Method for Solving n-th Order Nonlinear System with Slowly Varying Coefficients", Journal of Sound and Vibration, 256, 2003, pp987-1002. |
| [8] | Hung Cheng and Tai Tsun Wu, "An Aging Spring, Studies in Applied Mathematics", 49, 1970, pp183-185. |
| [9] | K.C, Roy, M. Shamsul Alam, "Effects of Higher Approximation of Krylov- Bogoliubov-Mitropolskii Solution and Matched Asymptotic Solution of a Differential System with Slowly Varying Coefficients and Damping Near to a Turning Point", Vietnam Journal of Mechanics, VAST, 26, 2004, pp182-192. |
| [10] | Pinakee Dey., Harun or Rashid, M. Abul Kalam Azad and M S Uddin, "Approximate Solution of Second Order Time Dependent Nonlinear Vibrating Systems with Slowly Varying Coefficients", Bull. Cal. Math. Soc, 103, (5), 2011, pp371-38. |
APA Style
Pinakee Dey, Babul Hossain, Musa Miah, Mohammad Mokaddes Ali. (2012). Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions. Applied and Computational Mathematics, 1(1), 1-5. https://doi.org/10.11648/j.acm.20120101.11
ACS Style
Pinakee Dey; Babul Hossain; Musa Miah; Mohammad Mokaddes Ali. Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions. Appl. Comput. Math. 2012, 1(1), 1-5. doi: 10.11648/j.acm.20120101.11
AMA Style
Pinakee Dey, Babul Hossain, Musa Miah, Mohammad Mokaddes Ali. Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions. Appl Comput Math. 2012;1(1):1-5. doi: 10.11648/j.acm.20120101.11
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Download@article{10.11648/j.acm.20120101.11,
author = {Pinakee Dey and Babul Hossain and Musa Miah and Mohammad Mokaddes Ali},
title = {Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions},
journal = {Applied and Computational Mathematics},
volume = {1},
number = {1},
pages = {1-5},
doi = {10.11648/j.acm.20120101.11},
url = {https://doi.org/10.11648/j.acm.20120101.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20120101.11},
abstract = {Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The solution obtained for different initial conditions for a second order nonlinear system show a good coincidence with those obtained by numerical method. The method is illustrated by an example.},
year = {2012}
}
TY - JOUR T1 - Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions AU - Pinakee Dey AU - Babul Hossain AU - Musa Miah AU - Mohammad Mokaddes Ali Y1 - 2012/12/30 PY - 2012 N1 - https://doi.org/10.11648/j.acm.20120101.11 DO - 10.11648/j.acm.20120101.11 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 1 EP - 5 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20120101.11 AB - Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The solution obtained for different initial conditions for a second order nonlinear system show a good coincidence with those obtained by numerical method. The method is illustrated by an example. VL - 1 IS - 1 ER -
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