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

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

DOI 10.11648/j.acm.20150406.11
Published in Applied and Computational Mathematics (Volume 4, Issue 6, December 2015)
Page(s) 387-395
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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.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

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

References
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[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.
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    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

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    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

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    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

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  • @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}
    }
    

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    AU  - Md. Nazrul Islam
    AU  - Md. Mahafujur Rahaman
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    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
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Author Information
  • Department of Mathematics, Islamic University, Kushtia, Bangladesh

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

  • Department of Mathematics, Islamic University, Kushtia, Bangladesh

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