Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach. (English) Zbl 1248.34038

Akbarzade, M.; Kargar, A.

Math. Methods Appl. Sci. 34, No. 17, 2089-2094 (2011).

Based on the Hamiltonian approach, the authors present some numerical experiments for the Duffing equation \(u''+u+\varepsilon u^3=0\) with the initial conditions \(u(0)=A\) and \(u'(0)=0\).

Reviewer: Rodica Luca Tudorache (Iaşi)

MSC:

34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A45	Theoretical approximation of solutions to ordinary differential equations

Keywords:

nonlinear oscillators; Hamiltonian; amplitude; frequency relationship

References:

[1]	He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Nonlinear Mechanics 35 (1) pp 37– (2000) · Zbl 1068.74618
[2]	Zengin, Application of parameter-expansion method to nonlinear oscillators with discontinuities, International Journal of Nonlinear Sciences And Numerical Simulation 9 (3) pp 267– (2008) · Zbl 06942345
[3]	He, Variational approach for nonlinear oscillators, Chaos, Solitons and Fractals 34 (5) pp 1430– (2007) · Zbl 1152.34327
[4]	Herisanu, A Modified Variational Iteration Method for Strongly Nonlinear Problems, Nonlinear Science Letters A 1 (2) pp 183– (2010)
[5]	Jamshidi, Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire, Current Applied Physics 10 (2) pp 484– (2010)
[6]	Fereidoon, Application of He’s homotopy perturbation method to nonlinear shock damper dynamics, Archive of Applied Mechanics 80 (6) pp 641– (2010) · Zbl 1271.70052
[7]	Esmaeilpour, Application of He’s methods for laminar flow in a porous-saturated pipe, Mathematical Methods in the Applied Sciences 33 (9) pp 1112– (2010) · Zbl 1357.76061
[8]	Momeni, Application of He’s energy balance method to Duffing-harmonic oscillators, International Journal of Computer Mathematics 88 (1) pp 135– (2011) · Zbl 1211.65092
[9]	Ganji, Approximate solutions to Van der Pol damped nonlinear oscillators by means of He’s energy balance method, International Journal of Computer Mathematics 87 (9) pp 2014– (2010) · Zbl 1195.65092
[10]	Mehdipour, Application of the energy balance method to nonlinear vibrating equations, Current Applied Physics 10 (1) pp 104– (2010)
[11]	He, Preliminary report on the energy balance for nonlinear oscillations, Mechanics Research Communications 29 (2-3) pp 107– (2002) · Zbl 1048.70011
[12]	Akbarzade, Analysis of Nonlinear Oscillators with un Force by He’S Energy Balance Method, Progress In Electromagnetics Research C 3 pp 57– (2008)
[13]	Afrouzi, He’s Energy Balance Method for Nonlinear Oscillators with Discontinuities, International Journal of Nonlinear Sciences and Numerical Simulation 10 pp 301– (2009) · Zbl 06942402
[14]	Ganji, Determination of the frequency-amplitude relation for nonlinear oscillators with fractional potential using He’s energy balance method, Progress In Electromagnetics Research C 5 pp 21– (2008)
[15]	Ganji, Comparison of Energy Balance Period with Exact Period for Arising Nonlinear Oscillator Equations (He’s energy balance period for nonlinear oscillators with and without discontinuities), Acta Applicandae Mathematicae 108 (2) pp 353– (2009) · Zbl 1194.34061
[16]	Ganji, Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max-min approach, Applied Mathematical Modelling 34 (9) pp 2676– (2010) · Zbl 1195.34054
[17]	Ganji, Application of amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back, Mathematical Methods in the Applied Sciences 33 (2) pp 157– (2010) · Zbl 1387.34053
[18]	He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B 20 (10) pp 1141– (2006) · Zbl 1102.34039
[19]	Ganji, Approximate analytical solutions to non-linear oscillators using He’s amplitude-frequency formulation, International Journal of Mathematical Analysis 32 (4) pp 1591– (2010) · Zbl 1301.34049
[20]	He, Application of He Chengtian’s interpolation to Bethe equation, Computers and Mathematics with Applications 58 (11-12) pp 2427– (2009) · Zbl 1189.65025
[21]	He, Hamiltonian approach to nonlinear oscillators, Physics Letters A 374 (23) pp 2312– (2010) · Zbl 1237.70036
[22]	He J-H Non-perturbative methods for strongly nonlinear problems Dissertation 2006 3-86624-113-5
[23]	He, Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons and Fractals 26 (3) pp 827– (2005) · Zbl 1093.34520
[24]	Marinca, Application of modified homotopy perturbation method to nonlinear oscillations, Archives of Mechanics 58 (3) pp 241– (2006) · Zbl 1108.70012
[25]	Beléndez, A novel rational harmonic balance approach for periodic solutions of conservative nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation 10 (1) pp 13– (2008) · Zbl 1401.34038
[26]	Beléndez, Rational harmonic balance based method for conservative nonlinear oscillators: Application to the Duffing equation, Mechanics Research Communications 36 (6) pp 728– (2009) · Zbl 1258.70039