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Year 2020, Volume: 3 Issue: 2, 53 - 60, 22.06.2020
https://doi.org/10.32323/ujma.597667

Abstract

References

  • [1] A.H. Nayfeh, Perturbation Method, Wiley, New York (1973).
  • [2] A.H. Nayfeh, D.T. Mook, Nonlinear oscillations, Wiley, New York (1979).
  • [3] N.M. Krylov, N.N. Bogolyubov, Introduction to non-linear mechanics, Princeton Univ. Press., (1947).
  • [4] S.E. Jones, Remarks on the perturbation process for certain conservative systems, Int. J. Non-Linear Mech., 13 (1978), 125-128.
  • [5] T.D. Burton, A perturbation method for certain nonlinear oscillators, Int. J. Non-Linear Mech., 19 (1984), 397-407.
  • [6] Y.K. Cheung, S.H. Chen, S.L. Lau,A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators, Int. J. Non-Linear Mech., 26 (1991), 367-378.
  • [7] J.H. He, Homoptopy perturbation method for bifurcation and nonlinear problems, Int. J. Non-linear Sci. Numerical Simulation, 6 (2005), 207-208.
  • [8] B.S. Wu, C.W, Lim, Large amplitude nonlinear oscillations of a general conservative system, Int. J. Non-Linear Mech., 39 (2004), 859-807.
  • [9] M.S. Alam, M.E. Haque, M.B. Hossain, A new analytical technique to find periodic solutions of nonlinear systems, Int. J. Non-Linear Mech., 42 (2007), 1035-1045.
  • [10] J.H. He, Preliminary reports on the energy balance for nonlinear oscillations, Mechanics Research Communications, 29 (2002), 107-111.
  • [11] R.E. Mickens, Iteration procedure for determining approximate solutions to nonlinear oscillator equation, J. Sound Vib., 116 (1987), 185-188.
  • [12] G. Veronis, A note on the method of multiple time-scales, Q. Appl. Math., (1980), 363-368.
  • [13] T.D. Burton, Z. Rahman, On the multi-scale analysis of strongly non-linear forced oscillators, Int. J. Non-Linear Mech., 21 (1986), 135-146.
  • [14] M.S. Alam, I.A. Yeasmin, M.S. Ahamed, Generalization of the modified Lindstedt-Poincare method for solving some strongly nonlinear oscillators, Ain Shams Engg. J., 10 (2019), 195-201.
  • [15] R. B. Dingle, Asymptotic expansions: Their derivation and interpretation, London Academic Press, (1973).
  • [16] E. J. Hinch, Perturbation methods, Cambridge University Press, (1991).
  • [17] F. Say, Optimal successive complementary expansion for singular differential equations, Math Meth Appl. Sci., (2020), 1-10.
  • [18] A. D. Dean, Exponential asymptotics and homoclinic snaking, Ph.D. Thesis, University of Nottingham, 2012.
  • [19] H. Poincare, Sur les integrales irr´eguli`eres, Acta Mathematica, 8 (1886), 295-344.
  • [20] V. Marinca, N. Herisanu, A modified iteration perturbation method for some nonlinear oscillation problems, Acta Mechanica, 184 (1-4) (2006), 231-242.

Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators

Year 2020, Volume: 3 Issue: 2, 53 - 60, 22.06.2020
https://doi.org/10.32323/ujma.597667

Abstract

The modified Lindstedt-Poincare method has been extended to obtain a faster convergent solution of nonlinear oscillators. First of all a classical type Lindstedt-Poincare solution has been determined and then a conversion formula has been used to find the desired solution. The solution has been compared and justified by corresponding numerical solution.

References

  • [1] A.H. Nayfeh, Perturbation Method, Wiley, New York (1973).
  • [2] A.H. Nayfeh, D.T. Mook, Nonlinear oscillations, Wiley, New York (1979).
  • [3] N.M. Krylov, N.N. Bogolyubov, Introduction to non-linear mechanics, Princeton Univ. Press., (1947).
  • [4] S.E. Jones, Remarks on the perturbation process for certain conservative systems, Int. J. Non-Linear Mech., 13 (1978), 125-128.
  • [5] T.D. Burton, A perturbation method for certain nonlinear oscillators, Int. J. Non-Linear Mech., 19 (1984), 397-407.
  • [6] Y.K. Cheung, S.H. Chen, S.L. Lau,A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators, Int. J. Non-Linear Mech., 26 (1991), 367-378.
  • [7] J.H. He, Homoptopy perturbation method for bifurcation and nonlinear problems, Int. J. Non-linear Sci. Numerical Simulation, 6 (2005), 207-208.
  • [8] B.S. Wu, C.W, Lim, Large amplitude nonlinear oscillations of a general conservative system, Int. J. Non-Linear Mech., 39 (2004), 859-807.
  • [9] M.S. Alam, M.E. Haque, M.B. Hossain, A new analytical technique to find periodic solutions of nonlinear systems, Int. J. Non-Linear Mech., 42 (2007), 1035-1045.
  • [10] J.H. He, Preliminary reports on the energy balance for nonlinear oscillations, Mechanics Research Communications, 29 (2002), 107-111.
  • [11] R.E. Mickens, Iteration procedure for determining approximate solutions to nonlinear oscillator equation, J. Sound Vib., 116 (1987), 185-188.
  • [12] G. Veronis, A note on the method of multiple time-scales, Q. Appl. Math., (1980), 363-368.
  • [13] T.D. Burton, Z. Rahman, On the multi-scale analysis of strongly non-linear forced oscillators, Int. J. Non-Linear Mech., 21 (1986), 135-146.
  • [14] M.S. Alam, I.A. Yeasmin, M.S. Ahamed, Generalization of the modified Lindstedt-Poincare method for solving some strongly nonlinear oscillators, Ain Shams Engg. J., 10 (2019), 195-201.
  • [15] R. B. Dingle, Asymptotic expansions: Their derivation and interpretation, London Academic Press, (1973).
  • [16] E. J. Hinch, Perturbation methods, Cambridge University Press, (1991).
  • [17] F. Say, Optimal successive complementary expansion for singular differential equations, Math Meth Appl. Sci., (2020), 1-10.
  • [18] A. D. Dean, Exponential asymptotics and homoclinic snaking, Ph.D. Thesis, University of Nottingham, 2012.
  • [19] H. Poincare, Sur les integrales irr´eguli`eres, Acta Mathematica, 8 (1886), 295-344.
  • [20] V. Marinca, N. Herisanu, A modified iteration perturbation method for some nonlinear oscillation problems, Acta Mechanica, 184 (1-4) (2006), 231-242.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Md. Nazmul Sharif 0000-0003-4234-6651

M. S. Alam This is me 0000-0002-6325-6797

İ. A. Yeasmin This is me 0000-0002-0392-8547

Publication Date June 22, 2020
Submission Date July 28, 2019
Acceptance Date March 25, 2020
Published in Issue Year 2020 Volume: 3 Issue: 2

Cite

APA Sharif, M. N., Alam, M. S., & Yeasmin, İ. A. (2020). Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators. Universal Journal of Mathematics and Applications, 3(2), 53-60. https://doi.org/10.32323/ujma.597667
AMA Sharif MN, Alam MS, Yeasmin İA. Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators. Univ. J. Math. Appl. June 2020;3(2):53-60. doi:10.32323/ujma.597667
Chicago Sharif, Md. Nazmul, M. S. Alam, and İ. A. Yeasmin. “Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators”. Universal Journal of Mathematics and Applications 3, no. 2 (June 2020): 53-60. https://doi.org/10.32323/ujma.597667.
EndNote Sharif MN, Alam MS, Yeasmin İA (June 1, 2020) Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators. Universal Journal of Mathematics and Applications 3 2 53–60.
IEEE M. N. Sharif, M. S. Alam, and İ. A. Yeasmin, “Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators”, Univ. J. Math. Appl., vol. 3, no. 2, pp. 53–60, 2020, doi: 10.32323/ujma.597667.
ISNAD Sharif, Md. Nazmul et al. “Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators”. Universal Journal of Mathematics and Applications 3/2 (June 2020), 53-60. https://doi.org/10.32323/ujma.597667.
JAMA Sharif MN, Alam MS, Yeasmin İA. Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators. Univ. J. Math. Appl. 2020;3:53–60.
MLA Sharif, Md. Nazmul et al. “Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators”. Universal Journal of Mathematics and Applications, vol. 3, no. 2, 2020, pp. 53-60, doi:10.32323/ujma.597667.
Vancouver Sharif MN, Alam MS, Yeasmin İA. Faster Convergent Modified Lindstedt-Poincare Solution of Nonlinear Oscillators. Univ. J. Math. Appl. 2020;3(2):53-60.

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