Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems

Md. Mahafujur Rahaman

doi:doi:10.11648/j.ajam.20150306.15

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Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems

Md. Mahafujur Rahaman

Published in American Journal of Applied Mathematics (Volume 3, Issue 6)

Received: 15 September 2015 Accepted: 2 November 2015 Published: 17 November 2015

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Abstract

With a view to obtaining the transient response of the system where triply eigenvalues are equal and another is distinct, we have considered a fourth order more critically damped nonlinear systems, and enquired into analytical approximate solution in this paper. We have also suggested that the results obtained by the proposed method correspond to the numerical solutions obtained by the fourth order Runge-Kutta method satisfactorily.

Published in	American Journal of Applied Mathematics (Volume 3, Issue 6)
DOI	10.11648/j.ajam.20150306.15
Page(s)	265-270
Creative Commons	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.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

KBM, Eigenvalues, More Critically Damped System, Nonlinearity, Runge-Kutta Method

References

[1]	Bogoliubov, N. N. and Mitropolskii, Y. A., Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961.
[2]	Krylov, N. N. and Bogoliubov, N. N., Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947.
[3]	Popov, I. P., “A Generalization of the Bogoliubov Asymptotic Method in the Theory of Nonlinear Oscillations (in Russian)”, Dokl. Akad. USSR, vol. 3, pp. 308-310, 1956.
[4]	Mendelson, K. S., “Perturbation Theory for Damped Nonlinear Oscillations”, J. Math. Physics, vol. 2, pp. 3413-3415, 1970.
[5]	Murty, I. S. N., and Deekshatulu, B. L., “Method of Variation of Parameters for Over-Damped Nonlinear Systems”, J. Control, vol. 9(3), pp. 259-266, 1969.
[6]	Sattar, M. A., “An asymptotic Method for Second Order Critically Damped Nonlinear Equations”, J. Frank. Inst., vol. 321, pp. 109-113, 1986.
[7]	Murty, I. S. N., “A Unified Krylov-Bogoliubov Method for Solving Second Order Nonlinear Systems”, Int. J. Nonlinear Mech. vol. 6, pp. 45-53, 1971.
[8]	Osiniskii, Z., “Longitudinal, Torsional and Bending Vibrations of a Uniform Bar with Nonlinear Internal Friction and Relaxation,” Nonlinear Vibration Problems, vol. 4, pp. 159-166, 1962.
[9]	Mulholland, R. J., “Nonlinear Oscillations of Third Order Differential Equation,” Int. J. Nonlinear Mechanics, vol. 6. Pp. 279-294, 1971.
[10]	Bojadziev, G. N., “Damped Nonlinear Oscillations Modeled by a 3-dimensional Differential System”, Acta Mechanica, vol. 48, pp. 193-201, 1983.
[11]	Sattar, M. A., “An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems”, Ganit, J. Bangladesh Math. Soc., vol. 13, pp. 1-8, 1993.
[12]	Shamsul, M. A., “Asymptotic Methods for Second Order Over-damped and Critically Damped Nonlinear Systems”, Soochow Journal of Math., vol. 27, pp. 187-200, 2001.
[13]	Shamsul, M. A., “On Some Special Conditions of Third Order Over-damped Nonlinear Systems”, Indian J. Pure Appl. Math., vol. 33, pp. 727-742, 2002.
[14]	Akbar, M. A., Paul, A. C. and Sattar, M. A., “An Asymptotic Method of Krylov-Bogoliubov for Fourth Order Over-damped Nonlinear Systems”, Ganit, J. Bangladesh Math. Soc., vol. 22, pp. 83-96, 2002.
[15]	Murty, I. S. N., “Deekshatulu, B. L. and Krishna, G., “On an Asymptotic Method of Krylov-Bogoliubov for Over-damped Nonlinear Systems”, J. Frank. Inst., vol. 288, pp. 49-65, 1969.
[16]	Akbar, M. A., Shamsul, M. A. and Sattar M. A. “Asymptotic Method for Fourth Order Damped Nonlinear Systems”, Ganit, J. Bangladesh Math. Soc., Vol. 23, pp. 41-49, 2003.
[17]	Akbar, M. A., Shamsul, M. A. and Sattar M. A., “Krylov-Bogoliubov-Mitropolskii Unified Method for Solving n-th Order Nonlinear Differential Equations Under Some Special Conditions Including the Case of Internal Resonance”, Int. J. Non-linear Mech, Vol. 41, pp. 26-42, 2006.
[18]	Akbar, M. A., Uddin, M. S., Islam, M. R. and Soma, A. A. “Krylov-Bogoliubov-Mitropolskii (KBM) Method for Fourth Order More Critically Damped Nonlinear Systems”, J. Mech. of Continua and Math. Sciences, Vol. 2(1), pp. 91-107, 2007.
[19]	Islam, M. R., Rahman, M. H. and Akbar, M. A. “An Analytical Approximate Solution of Fourth Order More Critically Damped Nonlinear Systems”, Indian Journal of Mathematics, Vol. 50(3), pp. 611-626, 2009.
[20]	Hakim, M. A. “On Fourth Order More Critically Damped Nonlinear Differential Systems”, Journal of Physical Science, Vol. 15, pp.113-127, 2011.
[21]	Akbar, M. A., Shamsul, M. A. and Sattar, M. A. “A Simple Technique for Obtaining Certain Over-damped Solutions of an n-th Order Nonlinear Differential Equation”, Soochow Journal of Mathematics, Vol. 31(2), pp. 291-299, 2005.

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Md. Mahafujur Rahaman. (2015). Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems. American Journal of Applied Mathematics, 3(6), 265-270. https://doi.org/10.11648/j.ajam.20150306.15

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ACS Style

Md. Mahafujur Rahaman. Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems. Am. J. Appl. Math. 2015, 3(6), 265-270. doi: 10.11648/j.ajam.20150306.15

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AMA Style

Md. Mahafujur Rahaman. Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems. Am J Appl Math. 2015;3(6):265-270. doi: 10.11648/j.ajam.20150306.15

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@article{10.11648/j.ajam.20150306.15,
  author = {Md. Mahafujur Rahaman},
  title = {Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems},
  journal = {American Journal of Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {265-270},
  doi = {10.11648/j.ajam.20150306.15},
  url = {https://doi.org/10.11648/j.ajam.20150306.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.15},
  abstract = {With a view to obtaining the transient response of the system where triply eigenvalues are equal and another is distinct, we have considered a fourth order more critically damped nonlinear systems, and enquired into analytical approximate solution in this paper. We have also suggested that the results obtained by the proposed method correspond to the numerical solutions obtained by the fourth order Runge-Kutta method satisfactorily.},
 year = {2015}
}

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TY  - JOUR
T1  - Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems
AU  - Md. Mahafujur Rahaman
Y1  - 2015/11/17
PY  - 2015
N1  - https://doi.org/10.11648/j.ajam.20150306.15
DO  - 10.11648/j.ajam.20150306.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 265
EP  - 270
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20150306.15
AB  - With a view to obtaining the transient response of the system where triply eigenvalues are equal and another is distinct, we have considered a fourth order more critically damped nonlinear systems, and enquired into analytical approximate solution in this paper. We have also suggested that the results obtained by the proposed method correspond to the numerical solutions obtained by the fourth order Runge-Kutta method satisfactorily.
VL  - 3
IS  - 6
ER  -

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Author Information

Md. Mahafujur Rahaman

Department of Computer Science & Engineering, Z.H. Sikder University of Science & Technology, Shariatpur, Bangladesh

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Md. Mahafujur Rahaman. (2015). Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems. American Journal of Applied Mathematics, 3(6), 265-270. https://doi.org/10.11648/j.ajam.20150306.15

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ACS Style

Md. Mahafujur Rahaman. Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems. Am. J. Appl. Math. 2015, 3(6), 265-270. doi: 10.11648/j.ajam.20150306.15

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AMA Style

Md. Mahafujur Rahaman. Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems. Am J Appl Math. 2015;3(6):265-270. doi: 10.11648/j.ajam.20150306.15

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@article{10.11648/j.ajam.20150306.15,
  author = {Md. Mahafujur Rahaman},
  title = {Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems},
  journal = {American Journal of Applied Mathematics},
  volume = {3},
  number = {6},
  pages = {265-270},
  doi = {10.11648/j.ajam.20150306.15},
  url = {https://doi.org/10.11648/j.ajam.20150306.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.15},
  abstract = {With a view to obtaining the transient response of the system where triply eigenvalues are equal and another is distinct, we have considered a fourth order more critically damped nonlinear systems, and enquired into analytical approximate solution in this paper. We have also suggested that the results obtained by the proposed method correspond to the numerical solutions obtained by the fourth order Runge-Kutta method satisfactorily.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Krylov-Bogoliubov-Mitropolskii Method for Fourth Order More Critically Damped Nonlinear Systems
AU  - Md. Mahafujur Rahaman
Y1  - 2015/11/17
PY  - 2015
N1  - https://doi.org/10.11648/j.ajam.20150306.15
DO  - 10.11648/j.ajam.20150306.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 265
EP  - 270
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20150306.15
AB  - With a view to obtaining the transient response of the system where triply eigenvalues are equal and another is distinct, we have considered a fourth order more critically damped nonlinear systems, and enquired into analytical approximate solution in this paper. We have also suggested that the results obtained by the proposed method correspond to the numerical solutions obtained by the fourth order Runge-Kutta method satisfactorily.
VL  - 3
IS  - 6
ER  -

Copy | Download