Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems

Md. Nazrul Islam; Md. Mahafujur Rahaman; M. Abul Kawser

doi:doi:10.11648/j.acm.20150406.11

| Peer-Reviewed

Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems

Md. Nazrul Islam, Md. Mahafujur Rahaman, M. Abul Kawser

Published in Applied and Computational Mathematics (Volume 4, Issue 6)

Received: 26 August 2015 Accepted: 19 September 2015 Published: 29 September 2015

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In oscillatory problems, the method of Krylov–Bogoliubov–Mitropolskii (KBM) is one of the most used techniques to obtain analytical approximate solution of nonlinear systems with a small non-linearity. This article modifies the KBM method to examine the solutions of fifth order critically damped nonlinear systems with four pairwise equal eigenvalues and one distinct eigenvalue, in which the latter eigenvalue is much larger than the former four pairwise eigenvalues. This paper suggests that the results obtained in this study correspond accurately to the numerical solutions obtained by the fourth order Runge-Kutta method. This paper, therefore, concludes that the modified KBM method provides highly accurate results, which can be applied for different kinds of nonlinear differential systems.

Published in	Applied and Computational Mathematics (Volume 4, Issue 6)
DOI	10.11648/j.acm.20150406.11
Page(s)	387-395
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), 2024. Published by Science Publishing Group

Keywords

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

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., “A Unified Krylov-Bogoliubov Method for Solving Second Order Nonlinear Systems”, Int. J. Nonlinear Mech. vol. 6, pp. 45-53, 1971.
[6]	Sattar, M. A., “An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems”, Ganit, J. Bangladesh Math. Soc., vol. 13, pp. 1-8, 1993.
[7]	Bojadziev, G. N., “Damped Nonlinear Oscillations Modeled by a 3-dimensional Differential System”, Acta Mechanica, vol. 48, pp. 193-201, 1983.
[8]	Shamsul, M. A. and Sattar, M. A., “An Asymptotic Method for Third Order Critically Damped Nonlinear Equations”, J. Mathematical and Physical Sciences, vol. 30, pp. 291-298, 1996.
[9]	Islam, M. R. and Akbar, M. A., “A New Asymptotic Solution for Third Order More Critically Damped Nonlinear Systems”, IAENG Journal of Applied Mathematics, v. 39(1), 2009.
[10]	Shamsul, M. A. and Sattar, M. A., “A Unified Krylov-Bogoliubov-Mitropolskii Method for Solving Third Order Nonlinear Systems”, Indian J. pure appl. Math., vol. 28, pp. 151-167, 1997.
[11]	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.
[12]	Islam, M. R., Uddin, M. S., Akbar, M. A., Huda, M. A. and Hossain, S. M. S., “A New Technique for Fourth Order Critically Damped Nonlinear Systems with Some Conditions”, Bull. Cal. Math. Soc., vol. 100(5), pp. 501-514, 2008.
[13]	Rahaman, M. M. and Rahman, M. M., “Analytical Approximate Solutions of Fifth Order More Critically Damped Systems in the case of Smaller Triply Repeated Roots”, IOSR Journals of Mathematics, vol. 11(2), pp. 35-46, 2015.
[14]	Rahaman, M. M. and Kawser, M. A., “Asymptotic Solution of Fifth Order Critically Damped Non-linear Systems with Pair Wise Equal Eigenvalues and Another is Distinct.”, Journal of Research in Applied Mathematics, vol. 2(3), pp. 01-15, 2015.
[15]	Shamsul, M. A., “Asymptotic Method for Certain Third-order Non-oscillatory Nonlinear Systems”, J. Bangladesh Academy of Sciences, vol. 27, pp. 141-148, 2003.
[16]	Sattar, M. A., “An asymptotic Method for Second Order Critically Damped Nonlinear Equations”, J. Frank. Inst., vol. 321, pp. 109-113, 1986.
[17]	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.
[18]	Shamsul, M. A., “A Unified Krylov-Bogoliubov-Mitropolskii Method for Solving n-th Order Nonlinear Systems”, J. Frank. Inst., vol. 339, pp. 239-248, 2002.
[19]	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.
[20]	Shamsul, M. A., “Bogoliubov's Method for Third Order Critically Damped Nonlinear Systems”, Soochow J. Math., vol. 28, pp. 65-80, 2002.
[21]	Shamsul, M. A., “Method of Solution to the n-th Order Over-damped Nonlinear Systems Under Some Special Conditions”, Bull. Cal. Math. Soc., vol. 94, pp. 437-440, 2002.
[22]	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.
[23]	Shamsul, M. A., “On Some Special Conditions of Over-damped Nonlinear Systems”, Soochow J. Math., vol. 29, pp. 181-190, 2003.
[24]	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.

Cite This Article

Plain Text BibTeX RIS

APA Style

Md. Nazrul Islam, Md. Mahafujur Rahaman, M. Abul Kawser. (2015). Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems. Applied and Computational Mathematics, 4(6), 387-395. https://doi.org/10.11648/j.acm.20150406.11

Copy | Download

ACS Style

Md. Nazrul Islam; Md. Mahafujur Rahaman; M. Abul Kawser. Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems. Appl. Comput. Math. 2015, 4(6), 387-395. doi: 10.11648/j.acm.20150406.11

Copy | Download

AMA Style

Md. Nazrul Islam, Md. Mahafujur Rahaman, M. Abul Kawser. Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems. Appl Comput Math. 2015;4(6):387-395. doi: 10.11648/j.acm.20150406.11

Copy | Download

@article{10.11648/j.acm.20150406.11,
  author = {Md. Nazrul Islam and Md. Mahafujur Rahaman and M. Abul Kawser},
  title = {Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {6},
  pages = {387-395},
  doi = {10.11648/j.acm.20150406.11},
  url = {https://doi.org/10.11648/j.acm.20150406.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150406.11},
  abstract = {In oscillatory problems, the method of Krylov–Bogoliubov–Mitropolskii (KBM) is one of the most used techniques to obtain analytical approximate solution of nonlinear systems with a small non-linearity. This article modifies the KBM method to examine the solutions of fifth order critically damped nonlinear systems with four pairwise equal eigenvalues and one distinct eigenvalue, in which the latter eigenvalue is much larger than the former four pairwise eigenvalues. This paper suggests that the results obtained in this study correspond accurately to the numerical solutions obtained by the fourth order Runge-Kutta method. This paper, therefore, concludes that the modified KBM method provides highly accurate results, which can be applied for different kinds of nonlinear differential systems.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems
AU  - Md. Nazrul Islam
AU  - Md. Mahafujur Rahaman
AU  - M. Abul Kawser
Y1  - 2015/09/29
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150406.11
DO  - 10.11648/j.acm.20150406.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 387
EP  - 395
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150406.11
AB  - In oscillatory problems, the method of Krylov–Bogoliubov–Mitropolskii (KBM) is one of the most used techniques to obtain analytical approximate solution of nonlinear systems with a small non-linearity. This article modifies the KBM method to examine the solutions of fifth order critically damped nonlinear systems with four pairwise equal eigenvalues and one distinct eigenvalue, in which the latter eigenvalue is much larger than the former four pairwise eigenvalues. This paper suggests that the results obtained in this study correspond accurately to the numerical solutions obtained by the fourth order Runge-Kutta method. This paper, therefore, concludes that the modified KBM method provides highly accurate results, which can be applied for different kinds of nonlinear differential systems.
VL  - 4
IS  - 6
ER  -

Copy | Download

Author Information

Md. Nazrul Islam

Department of Mathematics, Islamic University, Kushtia, Bangladesh
Md. Mahafujur Rahaman

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

Department of Mathematics, Islamic University, Kushtia, Bangladesh

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Md. Nazrul Islam, Md. Mahafujur Rahaman, M. Abul Kawser. (2015). Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems. Applied and Computational Mathematics, 4(6), 387-395. https://doi.org/10.11648/j.acm.20150406.11

Copy | Download

ACS Style

Md. Nazrul Islam; Md. Mahafujur Rahaman; M. Abul Kawser. Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems. Appl. Comput. Math. 2015, 4(6), 387-395. doi: 10.11648/j.acm.20150406.11

Copy | Download

AMA Style

Md. Nazrul Islam, Md. Mahafujur Rahaman, M. Abul Kawser. Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems. Appl Comput Math. 2015;4(6):387-395. doi: 10.11648/j.acm.20150406.11

Copy | Download

@article{10.11648/j.acm.20150406.11,
  author = {Md. Nazrul Islam and Md. Mahafujur Rahaman and M. Abul Kawser},
  title = {Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {6},
  pages = {387-395},
  doi = {10.11648/j.acm.20150406.11},
  url = {https://doi.org/10.11648/j.acm.20150406.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150406.11},
  abstract = {In oscillatory problems, the method of Krylov–Bogoliubov–Mitropolskii (KBM) is one of the most used techniques to obtain analytical approximate solution of nonlinear systems with a small non-linearity. This article modifies the KBM method to examine the solutions of fifth order critically damped nonlinear systems with four pairwise equal eigenvalues and one distinct eigenvalue, in which the latter eigenvalue is much larger than the former four pairwise eigenvalues. This paper suggests that the results obtained in this study correspond accurately to the numerical solutions obtained by the fourth order Runge-Kutta method. This paper, therefore, concludes that the modified KBM method provides highly accurate results, which can be applied for different kinds of nonlinear differential systems.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems
AU  - Md. Nazrul Islam
AU  - Md. Mahafujur Rahaman
AU  - M. Abul Kawser
Y1  - 2015/09/29
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150406.11
DO  - 10.11648/j.acm.20150406.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 387
EP  - 395
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150406.11
AB  - In oscillatory problems, the method of Krylov–Bogoliubov–Mitropolskii (KBM) is one of the most used techniques to obtain analytical approximate solution of nonlinear systems with a small non-linearity. This article modifies the KBM method to examine the solutions of fifth order critically damped nonlinear systems with four pairwise equal eigenvalues and one distinct eigenvalue, in which the latter eigenvalue is much larger than the former four pairwise eigenvalues. This paper suggests that the results obtained in this study correspond accurately to the numerical solutions obtained by the fourth order Runge-Kutta method. This paper, therefore, concludes that the modified KBM method provides highly accurate results, which can be applied for different kinds of nonlinear differential systems.
VL  - 4
IS  - 6
ER  -

Copy | Download