zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.
MSC:
65-03Historical (numerical analysis)
34-03Historical (ordinary differential equations)
65L99Numerical methods for ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
01A25Chinese mathematics
References:
[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)
[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)
[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)
[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)