# zbMATH — the first resource for mathematics

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method. (English) Zbl 1154.65349
Summary: He’s homotopy perturbation method has been adapted to calculate higher-order approximate periodic solutions for a nonlinear oscillator with discontinuity for which the elastic force term is an anti-symmetric and quadratic term. We find that He’s homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Just one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 0.73% for all values of oscillation amplitude, while this relative error is as low as 0.040% when the second iteration is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance method reveals that the former is very effective and convenient.

##### MSC:
 65L99 Numerical methods for ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations
Full Text:
##### References:
 [1] He, J.H., A new perturbation technique which is also valid for large parameters, J. sound vibration, 229, 1257-1263, (2000) · Zbl 1235.70139 [2] He, J.H., Modified lindstedt – poincare methods for some non linear oscillations. part III: double series expansion, Int. J. nonlinear sci. numer. simul., 2, 317-320, (2001) · Zbl 1072.34507 [3] Özis, T.; Yildirim, A., Determination of periodic solution for a $$u^{1 / 3}$$ force by he’s modified lindstedt – poincaré method, J. sound vibration, 301, 415-419, (2007) · Zbl 1242.70044 [4] Amore, P.; Fernández, F.M., Exact and approximate expressions for the period of anharmonic oscillators, European J. phys., 26, 589-601, (2005) [5] Amore, P.; Raya, A.; Fernández, F.M., Alternative perturbation approaches in classical mechanics, European J. phys., 26, 1057-1063, (2005) · Zbl 1080.70014 [6] Mickens, R.E., Oscillations in planar dynamics systems, (1996), World Scientific Singapore · Zbl 0840.34001 [7] Beléndez, A.; Hernández, A.; Márquez, A.; Beléndez, T.; Neipp, C., Analytical approximations for the period of a simple pendulum, European J. phys., 27, 539-551, (2006) [8] Beléndez, A.; Hernández, A.; Beléndez, T.; Álvarez, M.L.; Gallego, S.; Ortuño, M.; Neipp, C., Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire, J. sound vibration, 302, 1018-1029, (2007) [9] Itovich, G.R.; Moiola, J.L., On period doubling bifurcations of cycles and the harmonic balance method, Chaos solitons fractals, 27, 647-665, (2005) · Zbl 1083.37041 [10] Shou, D.H.; He, J.H., Application of parameter-expanding method to strongly nonlinear oscillators, Int. J. nonlinear sci. numer. simul., 8, 1, 121-124, (2007) [11] 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 [12] He, J.H., Variational approach for nonlinear oscillators, Chaos solitons fractals, 34, 1430-1439, (2007) · Zbl 1152.34327 [13] Tatari, M.; Dehghan, M., The use of he’s variational iteration method for solving a fokker – planck equation, Phys. scripta, 74, 310-316, (2006) · Zbl 1108.82033 [14] Tatari, M.; Dehghan, M., Solution of problems in calculus of variations via he’s variational iteration method, Phys. lett. A, 362, 401-406, (2007) · Zbl 1197.65112 [15] Batiha, B.; Noorani, M.S.M.; Hashim, I., Numerical simulation of the generalized Huxley equation by he’s variational iteration method, Appl. math. comput., 186, 1322-1325, (2007) · Zbl 1118.65367 [16] Beléndez, A.; Hernández, A.; Beléndez, T.; Márquez, A.; Neipp, C., An improved ‘heuristic’ approximation for the period of a nonlinear pendulum: linear analysis of a classical nonlinear problem, Int. J. nonlinear sci. numer. simul., 8, 3, 329-334, (2007) · Zbl 1119.70017 [17] Geng, Lei; Cai, Xu-Chu, He’s frequency formulation for nonlinear oscillators, European J. phys., 28, 923-931, (2007) · Zbl 1162.70019 [18] Xu, L., He’s parameter-expanding methods for strongly nonlinear oscillators, J. comput. appl. math., 207, 148-154, (2007) · Zbl 1120.65084 [19] J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation, De-Verlag im Internet GmbH, Berlin, 2006 [20] He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat. J. modern phys. B, 20, 1141-1199, (2006) · Zbl 1102.34039 [21] He, J.H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. math. comput., 151, 287-292, (2004) · Zbl 1039.65052 [22] 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. nonlinear sci. numer. simul., 7, 1, 109-117, (2006) [23] Cveticanin, L., Homotopy-perturbation for pure nonlinear differential equation, Chaos solitons fractals, 30, 1221-1230, (2006) · Zbl 1142.65418 [24] Abbasbandy, S., Application of he’s homotopy perturbation method for Laplace transform, Chaos solitons fractals, 30, 1206-1212, (2006) · Zbl 1142.65417 [25] Beléndez, A.; Hernández, A.; Beléndez, T.; Márquez, A., Application of the homotopy perturbation method to the nonlinear pendulum, European J. phys., 28, 93-104, (2007) · Zbl 1119.70017 [26] Beléndez, A.; Hernández, A.; Beléndez, T.; Fernández, E.; Álvarez, M.L.; Neipp, C., Application of he’s homotopy perturbation method to the Duffing-harmonic oscillator, Int. J. nonlinear sci. numer. simul., 8, 1, 79-88, (2007) [27] Ganji, D.D.; Sadighi, A., Application of he’s homotopy-perturbation method to nonlinear coupled systems of reaction – diffusion equations, Int. J. nonlinear sci. numer. simul., 7, 4, 411-418, (2006) [28] Abbasbandy, S., A numerical solution of Blasius equation by adomian’s decomposition method and comparison with homotopy perturbation method, Chaos solitons fractals, 31, 257-260, (2007) [29] Siddiqui, A.; Mahmood, R.; Ghori, Q., Thin film flow of a third grade fluid on moving a belt by he’s homotopy perturbation method, Int. J. nonlinear sci. numer. simul., 7, 1, 15-26, (2006) [30] He, J.H., Homotopy perturbation method for solving boundary value problems, Phys. lett. A, 350, 87-88, (2006) · Zbl 1195.65207 [31] Rafei, M.; Ganji, D.D., Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, Int. J. nonlinear sci. numer. simul., 7, 3, 321-328, (2006) [32] Ganji, D.D., The application of he’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. lett., 355, 337-341, (2006) · Zbl 1255.80026 [33] Chowdhury, M.S.H.; Hashim, I., Application of homotopy-perturbation method to nonlinear population dynamics models, Phys. lett. A, 368, 251-258, (2007) · Zbl 1209.65107 [34] Ariel, P.D.; Hayat, T., Homotopy perturbation method and axisymmetric flow over a stretching sheet, Int. J. nonlinear sci. numer. simul., 7, 4, 399-406, (2006) [35] Chowdhury, M.S.H.; Hashim, I., Solutions of time-dependent emden – fowler type equations by homotopy-perturbation method, Phys. lett. A, 368, 305-313, (2007) · Zbl 1209.65106 [36] Chowdhury, M.S.H.; Hashim, I., Solutions of a class of singular second-order IVPs by homotopy-perturbation method, Phys. lett. A, 365, 439-447, (2007) · Zbl 1203.65124 [37] Shakeri, F.; Dehghan, M., Inverse problem of diffusion by he’s homotopy perturbation method, Phys. scripta, 75, 551-556, (2007) · Zbl 1110.35354 [38] Dehghan, M.; Shakeri, F., Solution of a partial differential equation subject to temperature overspecification by he’s homotopy perturbation method, Phys. scripta, 75, 778-787, (2007) · Zbl 1117.35326 [39] Chowdhury, M.S.H.; Hashim, I., Solutions of emden – fowler equations by homotopy-perturbation method, Nonlinear anal. B: real world appl., (2007) · Zbl 1209.65106 [40] Cveticanin, L., Application of homotopy-perturbation to non linear partial differential equations, Chaos solitons fractals, (2007) · Zbl 1197.65200 [41] Mickens, R.E.; Ramadhani, I., Investigations of an anti-symmetric quadratic non linear oscillator, J. sound vibration, 155, 190-193, (1992) [42] Wang, S.-Q.; He, J.H., Nonlinear oscillator with discontinuity by parameter-expansion method, Chaos solitons fractals, (2007) [43] Beléndez, A.; Pascual, C.; Gallego, S.; Ortuño, M.; Neipp, C., Application of a modified he’s homotopy perturbation method to obtain higher-order approximations of a $$x^{1 / 3}$$ force nonlinear oscillator, Phys. lett. A, (2007) · Zbl 1209.65083 [44] Mickens, R.E.; Mixon, M., Application of generalized harmonic balance method to an anti-symmetric quadratic non linear oscillator, J. sound vibration, 159, 546-568, (1992) · Zbl 0925.70234 [45] (), Beta Function and Incomplete Beta Function, §6.2 and 6.6, pp. 258 and 263 [46] (), Gamma (Factorial) Function and Incomplete Gamma Function, §6.1 and 6.5, pp. 258 and 263
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.