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An ancient Chinese mathematical algorithm and its application to nonlinear oscillators. (English) Zbl 1219.65002
Summary: An ancient Chinese mathematical method is briefly introduced, and its application to nonlinear oscillators is elucidated where He’s amplitude-frequency formulation is outlined. Three examples are given to show the extremely simple solution procedure and remarkably accurate solutions.

65-03Historical (numerical analysis)
34-03Historical (ordinary differential equations)
65L99Numerical methods for ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
01A25Chinese mathematics
Full Text: DOI
[1] He, J. H.: Ancient chinese algorithm: the ying buzu shu (method of surplus and deficiency) vs. Newton iteration method, Appl. math. Mech. (English ed.) 23, 1407-1412 (2002) · Zbl 1023.01002 · doi:10.1007/BF02438379
[2] He, J. H.: Application of he chengtian’s interpolation to Bethe equation, Comput. math. Appl. 58, 2427-2430 (2009) · Zbl 1189.65025 · doi:10.1016/j.camwa.2009.03.027
[3] He, J. H.; Tang, H.: Rebuild of King fang 40 BC musical scales by he’s inequality, Appl. math. Comput. 168, 909-914 (2005) · Zbl 1160.00307 · doi:10.1016/j.amc.2004.09.016
[4] He, J. H.: Some interpolation formulas in chinese ancient mathematics, Appl. math. Comput. 152, 367-371 (2004) · Zbl 1046.01002 · doi:10.1016/S0096-3003(03)00559-9
[5] He, J. H.: Zu-geng’s axiom vs. Cavalieri’s theory, Appl. math. Comput. 152, 9-15 (2004) · Zbl 1091.01006 · doi:10.1016/S0096-3003(03)00529-0
[6] He, J. H.: Solution of nonlinear equations by an ancient chinese algorithm, Appl. math. Comput. 151, 293-297 (2004) · Zbl 1049.65039 · doi:10.1016/S0096-3003(03)00348-5
[7] He, J. H.: He chengtian’s inequality and its applications, Appl. math. Comput. 151, 887-891 (2004) · Zbl 1043.01004
[8] He, J. H.: Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern phys. B 20, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[9] Geng, L.; Cai, X. C.: He’s frequency formulation for nonlinear oscillators, European J. Phys. 28, 923-931 (2007) · Zbl 1162.70019 · doi:10.1088/0143-0807/28/5/016
[10] He, J. H.: Comment on ’he’s frequency formulation for nonlinear oscillators’, European J. Phys. 29, No. 4, L19-L22 (2008)
[11] He, J. H.: An improved amplitude--frequency formulation for nonlinear oscillators, Int. J. Nonlinear sci. Numer. simul. 9, 211-212 (2008)
[12] He, J. H.: An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering, Internat. J. Modern phys. B 22, 3487-3578 (2008) · Zbl 1149.76607 · doi:10.1142/S0217979208048668
[13] Cai, X. C.; Wu, W. Y.: He’s frequency formulation for the relativistic harmonic oscillator, Comput. math. Appl. 58, 2358-2359 (2009) · Zbl 1189.65162 · doi:10.1016/j.camwa.2009.03.024
[14] Zhang, H. L.: Application of he’s amplitude--frequency formulation to a nonlinear oscillator with discontinuity, Comput. math. Appl. 58, 2197-2198 (2009) · Zbl 1189.65181 · doi:10.1016/j.camwa.2009.03.018
[15] Zhang, Y. N.; Xu, F.; Deng, L. L.: Exact solution for nonlinear Schrödinger equation by he’s frequency formulation, Comput. math. Appl. 58, 2449-2451 (2009) · Zbl 1189.81064 · doi:10.1016/j.camwa.2009.03.015
[16] Fan, J.: He’s frequency--amplitude formulation for the Duffing harmonic oscillator, Comput. math. Appl. 58, 2473-2476 (2009) · Zbl 1189.65163 · doi:10.1016/j.camwa.2009.03.049
[17] Zhao, L.: He’s frequency--amplitude formulation for nonlinear oscillators with an irrational force, Comput. math. Appl. 58, 2477-2479 (2009) · Zbl 1189.65185 · doi:10.1016/j.camwa.2009.03.041
[18] Ren, Z. F.; Liu, G. Q.; Kang, Y. X.: Application of he’s amplitude--frequency formulation to nonlinear oscillators with discontinuities, Phys. scr. 80, 045003 (2009) · Zbl 05651743
[19] J.H. He, Non-perturbative methods for strongly nonlinear problems, de-Verlag im Internet GmbH, Berlin, 2006.
[20] Acton, J. R.; Squire, P. T.: Solving equations with physical understanding, (1985)
[21] Öziş, T.; Yıldırım, A.: Determination of frequency formulation relation for a Duffing-harmonic oscillator by the energy balance method, Comput. math. Appl. 54, 1184-1187 (2007) · Zbl 1147.34321 · doi:10.1016/j.camwa.2006.12.064
[22] Mohyud-Din, S. T.; Noor, M. A.; Noor, K. I.: Parameter-expansion techniques for strongly nonlinear oscillators, Int. J. Nonlinear sci. Numer. simul. 10, 581-583 (2009) · Zbl 1177.65169
[23] Marinca, V.; Herisanu, N.: Periodic solutions for some strongly nonlinear oscillations by he’s variational iteration method, Comput. math. Appl. 54, 1188-1196 (2007) · Zbl 1267.65101