Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method. (English) Zbl 1202.34072

Summary: A new approach combining the features of the homotopy concept with an efficient computational algorithm which provides a simple and rigorous procedure to control the convergence of the solution is proposed to find accurate analytical explicit solutions for some oscillators with discontinuities and a fractional power restoring force which is proportional to \(\mathrm{sgn}(x)\). A very fast convergence to the exact solution was proved, since the second-order approximation lead to very accurate results. Comparisons with numerical results are presented to show the effectiveness of this method. Four numerical applications prove the accuracy of the method, which works very well for the whole range of initial amplitudes. The obtained results prove the validity and efficiency of the method, which can be easily extended to other strongly nonlinear problems.


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
65L99 Numerical methods for ordinary differential equations
Full Text: DOI


[1] Nayfeh, A. H.; Mook, D. T., Nonlinear Oscillations (1979), John Willey and Sons: John Willey and Sons New York
[2] Hagedorn, P., Nonlinear Oscillations (1988), Clarendon: Clarendon Oxford · Zbl 0709.70512
[3] Mickens, R. E., Oscillations in Planar Dynamic Systems (1996), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 0840.34001
[4] He, J. H., Modified Lindstedt-Poincare methods for some strongly non-linear oscillations, Internat. J. Non-Linear Mech., 37, 309-320 (2002)
[5] Ramos, J. I., An artificial parameter-Lindstedt-Poincare method for oscillators with smooth odd nonlinearities, Chaos Solitons Fractals, 41, 380-393 (2009) · Zbl 1198.65150
[6] Cheung, Y. K.; Chen, S. H.; Lau, S. L., A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators, Internat. J. Non-Linear Mech., 26, 367-378 (1991) · Zbl 0755.70021
[7] Wu, B. S.; Li, P. S., A method for obtaining approximate analytic periods for a class of nonlinear oscillators, Meccanica, 36, 167-176 (2001) · Zbl 1008.70016
[8] Shou, D. H.; He, J. H., Application of parameter-expanding method to strongly nonlinear oscillators, Int. J. Nonlinear Sci. Numer. Simul., 8, 121-124 (2007)
[9] He, J. H.; Wu, G. C.; Austin, F., The variational iteration method which should be followed, Nonlinear Sci. Lett. A., 1, 1, 1-30 (2010)
[10] Belendez, A.; Hernandez, A.; Belendez, T., Application of He’s homotopy perturbation method to the Duffing-harmonic oscillator, Int. J. Nonlinear Sci. Numer. Simul., 8, 79-88 (2007) · Zbl 1119.70017
[11] Liao, S. J., Beyond Perturbation-Introduction in the Homotopy Analysis Method (2003), Chapman and Hall/CRC
[12] Herişanu, N.; Marinca, V., An iteration procedure with application to Van der Pol oscillator, Int. J. Nonlinear Sci. Numer. Simul., 10, 353-361 (2009)
[13] Pakdemirli, M.; Karahan, M. M.F., A new perturbation solution for systems with strong quadratic and cubic nonlinearities, Math. Methods Appl. Sci., 33, 6, 704-712 (2010) · Zbl 1193.34116
[14] Ramos, J. I., Linearized Galerkin and artificial parameter techniques for the determination of periodic solutions of nonlinear oscillators, Appl. Math. Comput., 196, 483-493 (2008) · Zbl 1135.65344
[15] Wazwaz, A. M., The modified Adomian decomposition method for solving linear and nonlinear boundary value problems of tenth-order and twelfth-order, Int. J. Nonlinear Sci. Numer. Simul., 1, 17-24 (2000) · Zbl 0966.65058
[16] Zengin, F. O.; Kaya, M. O.; Demirbag, S. A., Application of parameter-expansion method to nonlinear oscillators with discontinuities, Int. J. Nonlinear Sci. Numer. Simul., 9, 267-270 (2008)
[17] Belendez, A.; Pascual, C.; Ortuno, M.; Belendez, T.; Gallego, S., Application of a modified He’s homotopy perturbation to obtain higher-order approximations to a nonlinear oscillator with discontinuities, Nonlinear Anal. RWA, 10, 601-610 (2009) · Zbl 1167.34327
[18] Ramos, J. I., Piecewise-linearized methods for oscillators with fractional-power nonlinearities, J. Sound Vib., 300, 502-521 (2007) · Zbl 1241.34045
[19] Lim, C. W.; Wu, B. S., Accurate higher-order approximations to frequencies of nonlinear oscillators with fractional powers, J. Sound Vib., 281, 1157-1162 (2005) · Zbl 1236.34052
[20] Cveticanin, L., Oscillator with fraction order restoring force, J. Sound Vib., 320, 1064-1077 (2009)
[21] Kovacic, I., On the motion of non-linear oscillators with a fractional-order restoring force and time variable parameters, Phys. Lett. A, 373, 1839-1843 (2009) · Zbl 1229.70070
[22] Marinca, V.; Herişanu, N., Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, J. Sound Vib., 329, 1450-1459 (2010)
[23] Marinca, V.; Herişanu, N., Application of optimal asymptotic method for solving nonlinear equations arising in heat transfer, Int. Commun. Heat Mass Transfer, 35, 710-715 (2008)
[24] Herişanu, N.; Marinca, V.; Dordea, T.; Madescu, G., A new analytical approach to nonlinear vibration of an electrical machine, Proc. Rom. Acad. Ser. A, 9, 229-236 (2008)
[25] Marinca, V.; Herişanu, N.; Nemes, I., Optimal homotopy asymptotic method with application to thin film flow, Cent. Eur. J. Phys., 6, 648-653 (2008)
[26] Marinca, V.; Herişanu, N.; Bota, C.; Marinca, B., An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Appl. Math. Lett., 22, 245-251 (2009) · Zbl 1163.76318
[27] Ali, J.; Islam, S.; Islam, S.; Zaman, G., The solution of multipoint boundary value problems by the optimal homotopy asymptotic method, Comput. Math. Appl., 59, 2000-2006 (2010) · Zbl 1189.65154
[28] Joneidi, A. A.; Ganji, D. D.; Babaelahi, M., Micropolar flow in a porous channel with high mass transfer, Int. Commun. Heat Mass Transfer, 36, 1082-1088 (2009)
[29] Herişanu, N.; Marinca, V., A modified variational iteration method for strongly nonlinear problems, Nonlinear Sci. Lett. A, 1, 2, 183-192 (2010) · Zbl 1222.65089
[30] Marinca, V.; Herişanu, N., Optimal homotopy perturbation method for strongly nonlinear differential equations, Nonlinear Sci. Lett. A, 1, 3, 273-280 (2010) · Zbl 1222.65089
[31] He, J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178, 257-262 (1999) · Zbl 0956.70017
[32] He, J. H., Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 35, 1, 73-79 (2003) · Zbl 1030.34013
[33] He, J. H., New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B, 20, 18, 1141-1199 (2006)
[34] Amore, P.; Aranda, A., Improved Lindstedt-Poincare method for the solution of nonlinear problems, J. Sound Vib., 283, 1115-1136 (2005) · Zbl 1237.70097
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.