대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제26권4호
  • /
  • Pages.695-708
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  • 2011
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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A GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-TH ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING

  • Azad, M. Abul Kalam (Department of Mathematics Rajshahi University of Engineering and Technology (RUET)) ;
  • Alam, M. Shamsul (Department of Mathematics Rajshahi University of Engineering and Technology (RUET)) ;
  • Rahman, M. Saifur (Department of Mathematics Rajshahi University of Engineering and Technology (RUET)) ;
  • Sarker, Bimolendu Shekhar (Computer Center Rajshahi University of Engineering and Technology (RUET))
  • 투고 : 2010.05.23
  • 발행 : 2011.10.31
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초록

Based on the multiple-time-scale (MTS) method, a general formula has been presented for solving an n-th, n = 2, 3, ${\ldots}$, order ordinary differential equation with strong linear damping forces. Like the solution of the unified Krylov-Bogoliubov-Mitropolskii (KBM) method or the general Struble's method, the new solution covers the un-damped, under-damped and over-damped cases. The solutions are identical to those obtained by the unified KBM method and the general Struble's method. The technique is a new form of the classical MTS method. The formulation as well as the determination of the solution from the derived formula is very simple. The method is illustrated by several examples. The general MTS solution reduces to its classical form when the real parts of eigen-values of the unperturbed equation vanish.

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참고문헌

  1. G. N. Bojadziev, Damped forced nonlinear vibrations of systems with delay, J. Sound Vibration 46 (1976), 113-120. https://doi.org/10.1016/0022-460X(76)90821-X
  2. G. N. Bojadziev, Two variables expansion method applied to the study of damped non-linear oscillations, Nonlinear Vibration Problems 21 (1981), 11-18.
  3. G. N. Bojadziev, Damped nonlinear oscillations modeled by a 3-dimensional differential system, Acta Mech. 48 (1983), no. 3-4, 193-201. https://doi.org/10.1007/BF01170417
  4. N. N. Bogoliubov and Yu. A. Mitropolskii, Asymptotic Methods in the Theory of Non- linear Oscillations, Gordan and Breach, New York, 1961.
  5. A. Hassan, The KBM derivative expansion method is equivalent to the multiple-time- scales method, J. Sound Vibration 200 (1997), no. 4, 433-440. https://doi.org/10.1006/jsvi.1996.0710
  6. B. Z. Kaplan, Use of complex variables for the solution of certain nonlinear systems, J. Computer Methods in Applied Mechanics and Engineering 13 (1978), 281-291. https://doi.org/10.1016/0045-7825(78)90063-4
  7. N. N. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947.
  8. I. S. N. Murty, A unified Krylov-Bogoliubov method for solving second order nonlinear systems, Int. J. Nonlinear Mech. 6 (1971), 45-53. https://doi.org/10.1016/0020-7462(71)90033-3
  9. I. S. N. Murty, B. L. Deekshatulu, and G. Krisna, On asymptotic method of Krylov- Bogoliubov for overdamped nonlinear systems, J. Frank Inst. 288 (1969), 49-64. https://doi.org/10.1016/0016-0032(69)00203-1
  10. A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York-London-Sydney, 1973.
  11. A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley-Interscience [John Wiley & Sons], New York, 1981.
  12. I. P. Popov, A generalization of the asymptotic method of N. N. Bogolyubov in the theory of non-linear oscillations, Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 308-311.
  13. R. A. Rink, A procedure to obtain the initial amplitude and phase for the Krylov- Bogoliubov method, J. Franklin Inst. 303 (1977), 59-65. https://doi.org/10.1016/0016-0032(77)90075-8
  14. M. Shamsul Alam, A unified Krylov-Bogoliubov-Mitropolskii method for solving n-order nonlinear systems, J. Franklin Inst. 339 (2002), 239-248. https://doi.org/10.1016/S0016-0032(02)00020-0
  15. M. Shamsul Alam, A unified Krylov-Bogoliubov-Mitropolskii method for solving n-th order nonlin- ear systems with varying coefficients, J. Sound and Vibration 265 (2003), 987-1002. https://doi.org/10.1016/S0022-460X(02)01239-7
  16. 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. https://doi.org/10.1016/j.ijnonlinmec.2003.08.008
  17. M. Shamsul Alam, M. Abul Kalam Azad, and M. A. Hoque. A general Struble's technique for solving an n-th order weakly nonlinear differential system with damping, Int. J. Nonlinear Mech. 41 (2006), 905-918. https://doi.org/10.1016/j.ijnonlinmec.2006.08.001