×

Nonlinear oscillator with discontinuity by parameter-expansion method. (English) Zbl 1210.70023

Summary: The parameter-expansion method is applied to a nonlinear oscillator with discontinuity. One iteration is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. Comparison of the obtained solution with the exact one shows that the method is very effective and convenient.

MSC:

70K99 Nonlinear dynamics in mechanics
34A36 Discontinuous ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] He, J. H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl Math Comput, 151, 287-292 (2004) · Zbl 1039.65052
[2] Liu, H. M., Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method, Chaos, Solitons & Fractals, 23, 2, 577-579 (2005) · Zbl 1078.34509
[3] He JH. Non-perturbative methods for strongly nonlinear problems. Dissertation. Berlin: de-Verlag im Internet GmbH; 2006.; He JH. Non-perturbative methods for strongly nonlinear problems. Dissertation. Berlin: de-Verlag im Internet GmbH; 2006.
[4] He, J. H., Homotopy perturbation method for solving boundary value problems, Phys Lett A, 350, 87-88 (2006) · Zbl 1195.65207
[5] He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, Int J Non-linear Sci Numer Simul, 6, 207-208 (2005) · Zbl 1401.65085
[6] He, J. H., Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons & Fractals, 26, 827-833 (2005) · Zbl 1093.34520
[7] Cai, X. C.; Wu, W. Y.; Li, M. S., Approximate period solution for a kind of nonlinear oscillator by He’s perturbation method, Int J Non-linear Sci Numer Simul, 7, 109-112 (2006)
[8] Ariel, P. D.; Hayat, T.; Asghar, S., Homotopy perturbation method and axisymmetric flow over a stretching sheet, Int J Non-linear Sci Numer Simul, 7, 399-406 (2006)
[9] Ganji, D. D.; Sadighi, A., Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, Int J Non-linear Sci Numer Simul, 7, 411-418 (2006)
[10] Cveticanin, L., Homotopy-perturbation method for pure nonlinear differential equation, Chaos, Solitons & Fractals, 30, 1221-1230 (2006) · Zbl 1142.65418
[11] He, J. H., Variational iteration method – a kind of non-linear analytical technique, Int J Non-linear Mech, 34, 4, 699-708 (1999) · Zbl 1342.34005
[12] He, J. H.; Wu, X. H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons & Fractals, 29, 108-113 (2006) · Zbl 1147.35338
[13] Odibat, Z. M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int J Non-linear Sci Numer Simul, 7, 27-34 (2006) · Zbl 1401.65087
[14] Yusufoglu, E., Variational iteration method for construction of some compact and noncompact structures of Klein-Gordon equations, Int J Non-linear Sci Numer Simul, 8, 2, 152-158 (2007)
[15] He, J. H., Determination of limit cycles for strongly nonlinear oscillators, Phys Rev Lett, 90, 174301 (2003)
[16] D’Acunto, M., Determination of limit cycles for a modified van der Pol oscillator, Mech Res Commun, 33, 93-98 (2006) · Zbl 1192.70026
[17] D’Acunto, M., Self-excited systems: analytical determination of limit cycles, Chaos, Solitons & Fractals, 30, 719-724 (2006) · Zbl 1142.70010
[18] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int J Mod Phys B, 20, 1141-1199 (2006) · Zbl 1102.34039
[19] He, J. H., Bookkeeping parameter in perturbation methods, Int J Non-linear Sci Numer Simul, 2, 257-264 (2001) · Zbl 1072.34508
[20] Shou, D. H., Application of parameter-expanding method to strongly nonlinear oscillators, Int J Non-linear Sci Numer Simul, 8, 113-116 (2007)
[21] He, J. H., Modified Lindstedt-Poincare methods for some strongly non-linear oscillations. Part I: expansion of a constant, Int J Non-linear Mech, 37, 309-314 (2002) · Zbl 1116.34320
[22] He, J. H., Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations. Part III: double series expansion, Int J Non-linear Sci Numer Simul, 2, 317-320 (2001) · Zbl 1072.34507
[23] He, J. H., Modified Lindstedt-Poincare methods for some strongly non-linear oscillations. Part II: a new transformation, Int J Non-linear Mech, 37, 315-320 (2002) · Zbl 1116.34321
[24] Xu, L., He’s parameter-expanding methods for strongly nonlinear oscillators, J Comput Appl Math, 207, 1, 148-154 (2007) · Zbl 1120.65084
[25] Öziş, T.; Yıldırım, A., Determination of periodic solution for a \(u^{1/3}\) force by He’s modified Lindstedt-Poincaré method, J Sound Vib, 301, 415-419 (2007) · Zbl 1242.70044
[26] Öziş, T.; Yıldırım, A., Determination of limit cycles by a modified straightforward expansion for nonlinear oscillators, Chaos, Solitons & Fractals, 32, 445-448 (2007)
[27] Öziş, T.; Yıldırım, A., Traveling wave solution of Korteweg-de Vries equation using He’s homotopy perturbation method, Int J Non-linear Sci Numer Simul, 8, 2, 239-242 (2007)
[28] Öziş, T.; Yıldırım, A., A comparative study of He’s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities, Int J Non-linear Sci Numer Simul, 8, 2, 243-248 (2007)
[29] He, J. H., New interpretation of homotopy perturbation method, Int J Mod Phys B, 20, 2561-2568 (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.