The approximation of solutions for second order nonlinear oscillators using the polynomial least square method
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Authors
Constantin Bota
- Dept. of Mathematics, Politehnica” University of Timişoara, P-ta Victoriei, 2, Timişoara, 300006, Romania.
Abstract
In this paper, polynomial least square method (PLSM) is applied to find approximate solution for nonlinear oscillator
differential equations. We illustrate that this method is very convenient and does not require linearization or small parameters.
Comparisons are made between the results of PLSM and other methods in order to prove the accuracy of the PLSM method.
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ISRP Style
Constantin Bota, The approximation of solutions for second order nonlinear oscillators using the polynomial least square method, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 1, 234--242
AMA Style
Bota Constantin, The approximation of solutions for second order nonlinear oscillators using the polynomial least square method. J. Nonlinear Sci. Appl. (2017); 10(1):234--242
Chicago/Turabian Style
Bota, Constantin. "The approximation of solutions for second order nonlinear oscillators using the polynomial least square method." Journal of Nonlinear Sciences and Applications, 10, no. 1 (2017): 234--242
Keywords
- Nonlinear oscillators
- approximate polynomial solution.
MSC
References
-
[1]
A. Barari, M. Omidvar, A. R. Ghotbi, D. D. Ganji, Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations, Acta Appl. Math., 104 (2008), 161–171.
-
[2]
A. Beléndez, E. Gimeno, T. Beléndez, A. Hernández , Rational harmonic balance based method for conservative nonlinear oscillators: application to the Duffing equation, Mech. Res. Comm., 36 (2009), 728–734.
-
[3]
A. Beléndez, A. Hernandez, T. Beléndez, M. L. Alvarez, S. Gallego, M. Ortuño, C. Neipp, Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire, J. Sound Vib., 302 (2007), 1018–1029.
-
[4]
S. Durmaz, S. Altay Demirbağ, M. O. Kaya, Approximate solutions for nonlinear oscillation of a mass attached to a stretched elastic wire, Comput. Math. Appl., 61 (2011), 578–585.
-
[5]
Y.-M. Fu, J. Zhang, L.-J. Wan, Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS), Curr. Appl. Phys., 11 (2011), 482–485.
-
[6]
S. S. Ganji, D. D. Ganji, M. G. Sfahani, S. Karimpour, Application of AFF and HPM to the systems of strongly nonlinear oscillation, Curr. Appl. Phys., 10 (2010), 1317–1325.
-
[7]
D. D. Ganji, M. Gorji, S. Soleimani, M. Esmaeilpour, Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method, J. Zhejiang Univ. Sci. A , 10 (2009), 1263–1268.
-
[8]
F. Geng, A piecewise variational iteration method for treating a nonlinear oscillator of a mass attached to a stretched elastic wire, Comput. Math. Appl., 62 (2011), 1641–1644.
-
[9]
S. Ghosh, A. Roy, D. Roy, An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1133–1153.
-
[10]
A. Golbabai, M. Fardi, K. Sayevand, Application of the optimal homotopy asymptotic method for solving a strongly nonlinear oscillatory system, Math. Comput. Modelling, 58 (2013), 1837–1843.
-
[11]
M. N. Hamdan, N. H. Shabaneh, On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass, J. Sound Vib., 199 (1997), 711–736.
-
[12]
J.-H. He, The homotopy perturbation method nonlinear oscillators with discontinuities, Appl. Math. Comput., 151 (2004), 287–292.
-
[13]
J.-H. He, Variational approach for nonlinear oscillators, Chaos Solitons Fractals, 34 (2007), 1430–1439.
-
[14]
N. Jamshidi, D. D. Ganji, Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire, Curr. Appl. Phys., 10 (2010), 484–48
-
[15]
H. Kaur, R. C. Mittal, V. Mishra, Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Model., 38 (2014), 4958–4971.
-
[16]
Y. Khan, M. Akbarzade, A. Kargar, Coupling of homotopy and the variational approach for a conservative oscillator with strong odd-nonlinearity, Sci. Iran., 19 (2012), 417–422.
-
[17]
I. Kovacic, M. J. Brennan (Ed.), The Duffing equation, Nonlinear oscillators and their behaviour, John Wiley & Sons, Ltd., Chichester (2011)
-
[18]
V. Marinca, N. Herişanu, Periodic solutions of Duffing equation with strong non-linearity, Chaos Solitons Fractals, 37 (2008), 144–149.
-
[19]
Y.-H. Qian, S.-K. Lai, W. Zhang, Y. Xiang, Study on asymptotic analytical solutions using HAM for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass, Numer. Algorithms, 58 (2011), 293–314.
-
[20]
M. Razzaghi, G. Elnagar, Numerical solution of the controlled Duffing oscillator by the pseudospectral method, J. Comput. Appl. Math., 56 (1994), 253–261.
-
[21]
S. Tellı, O. Kopmaz, Free vibrations of a mass grounded by linear and nonlinear springs in series, J. Sound Vib., 289 (2006), 689–710.
-
[22]
L. Xu, Application of He’s parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire, Phys. Lett. A, 368 (2007), 259–262.