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Journal of Engineering Mathematics
Erschienen in: Journal of Engineering Mathematics 1/2016

29.07.2015

Exact solutions of the Liénard- and generalized Liénard-type ordinary nonlinear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator

verfasst von: Tiberiu Harko, Shi-Dong Liang

Erschienen in: Journal of Engineering Mathematics | Ausgabe 1/2016

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Abstract

We investigate the connection between the linear harmonic oscillator equation and some classes of second-order nonlinear ordinary differential equations of Liénard and generalized Liénard type, which physically describe important oscillator systems. By means of a method inspired by quantum mechanics, and which consists of the deformation of the phase space coordinates of the harmonic oscillator, we generalize the equation of motion of the classical linear harmonic oscillator to several classes of strongly nonlinear differential equations. The first integrals, and a number of exact solutions of the corresponding equations, are explicitly obtained. The devised method can be further generalized to derive explicit general solutions of nonlinear second-order differential equations unrelated to the harmonic oscillator. Applications of the obtained results for the study of the traveling wave solutions of the reaction–convection–diffusion equations, and of the large amplitude-free vibrations of a uniform cantilever beam, are also presented.
Vorheriger Artikel Analytical model for the temperature field around a nonuniform three-dimensional moving heat source: friction stir welding modelling
Nächster Artikel Nonlinear dynamic responses of viscoelastic fiber–metal-laminated beams under the thermal shock

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Literatur
1.
Liénard A (1928) Étude des oscillations entretenues. Revue générale de l’électricité 23:901–912, 946–954
2.
3.
Andronov AA, Leontovich EA, Gordon II, Maier AG (1973) Qualitative theory of second order dynamic systems. Wiley, New YorkMATH
4.
Chandrasekar VK, Senthilvelan M, Kundu A, Lakshmanan M (2006) A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators. J Phys A 39:9743–9754ADSMathSciNetCrossRefMATH
5.
6.
Zou L, Chen X, Zhang W (2008) Local bifurcations of critical periods for cubic Linard equations with cubic damping. J Comput Appl Math 222:404–410ADSMathSciNetCrossRefMATH
7.
Pandey SN, Bindu PS, Senthilvelan M, Lakshmanan M (2009) A group theoretical identification of integrable cases of the Linard-type equation \(x{^{\prime \prime }}+f(x)x+g(x)\)=0: I. Equations having nonmaximal number of Lie point symmetries. J Math Phys 50:082702–082702-19ADSMathSciNetCrossRefMATH
8.
Pandey SN, Bindu PS, Senthilvelan M, Lakshmanan M (2009) A group theoretical identification of integrable equations in the Linard-type equation \(x{^{\prime \prime }}+f(x)x+g(x)\)=0: II. Equations having maximal Lie point symmetries. J Math Phys 50:102701–102701-25ADSMathSciNetCrossRefMATH
9.
10.
Messias M, Alves G, Márcio R (2011) Time-periodic perturbation of a Liénard equation with an unbounded homoclinic loop. Physica D 240:1402–1409ADSMathSciNetCrossRefMATH
11.
Mickens RE (2010) Truly nonlinear oscillations: harmonic balance, parameter expansions, iteration, and averaging methods. World Scientific, LondonCrossRefMATH
12.
13.
DiBenedetto E (2011) Classical mechanics: theory and mathematical modeling. Springer, New YorkCrossRefMATH
14.
Mak MK, Harko T (2012) New integrability case for the Riccati equation. Appl Math Comput 218:10974–10981MathSciNetMATH
15.
Mak MK, Harko T (2013) New further integrability cases for the Riccati equation. Appl Math Comput 219:7465–7471MathSciNetMATH
16.
Harko T, Lobo FSN, Mak MK (2014) Analytical solutions of the Riccati equation with coefficients satisfying integral or differential conditions with arbitrary functions. Univ J Appl Math 2:109–118
17.
Euler N (1997) Transformation properties of \(\frac{{\rm d}^2x}{{\rm d}t^2} +f_1 (t)\frac{{\rm d}x}{{\rm d}t} +f_2 (t) x+f_3 (t) x^n = 0\). J Nonlinear Math Phys 4:310–337ADSMathSciNetCrossRefMATH
18.
Harko T, Lobo FSN, Mak MK (2013) Integrability cases for the anharmonic oscillator equation. J Pure Appl Math 10:115–129
19.
Harko T, Lobo FSN, Mak MK (2014) A class of exact solutions of the Linard type ordinary non-linear differential equation. J Eng Math 89:193–205MathSciNetCrossRef
20.
Harko T, Lobo FSN, Mak MK (2013) A Chiellini type integrability condition for the generalized first kind Abel differential equation. Univ J Appl Math 1:101–104
21.
Mancas SC, Rosu HC (2013) Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations. Phys Lett A 377:1234–1238MathSciNetCrossRefMATH
22.
Rosu HC, Cornejo-Perez O, Chen P (2012) Nonsingular parametric oscillators Darboux-related to the classical harmonic oscillator. Europhys Lett 100:60006ADSCrossRef
23.
24.
Chandrasekar VK, Senthilvelan M, Lakshmanan M (2005) Unusual Liénard-type nonlinear oscillator. Phys Rev E 72:066203ADSCrossRef
25.
Chandrasekar VK, Senthilvelan M, Lakshmanan M (2007) On the general solution for the modified Emden-type equation \(\ddot{x} + \alpha x\dot{x}+ \beta x^3 = 0\). J Phys A 40:4717–4727ADSMathSciNetCrossRefMATH
26.
Chandrasekar VK, Sheeba JH, Pradeep G, Divyasreea RS, Lakshmanan M (2012) A class of solvable coupled nonlinear oscillators with amplitude independent frequencies. Phys Lett A 376:2188–2194ADSCrossRefMATH
27.
28.
Durga Devi A, Gladwin Pradeep R, Chandrasekar VK, Lakshmanan M (2013) Method of generating N-dimensional isochronous nonsingular Hamiltonian systems. J Nonlinear Math Phys 20:78–93MathSciNetCrossRef
29.
Chithiika Ruby V, Senthilvelan M, Lakshmanan M (2012) Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator. J Phys A 45:382002ADSMathSciNetCrossRefMATH
30.
Gubbiottia G, Nucci MC (2014) Noether symmetries and the quantization of a Liénard-type nonlinear oscillator. J Nonlinear Math Phys 21:248–264MathSciNetCrossRef
31.
32.
Dasarathy BV (1975) Equivalent linear models for non-linear non-autonomous systems. J Sound Vib 42:447–452ADSCrossRefMATH
33.
34.
Sawada K, Osawa T (1978) On exactly soluble nonlinear ordinary differential equations of the Liénard type. J Phys Soc Jpn 44:1730–1732ADSMathSciNetCrossRefMATH
35.
Merzbacher E (1998) Quantum mechanics. Wiley, New YorkMATH
36.
37.
Arcos-Olalla R, Reyes MA, Rosu HC (2012) An alternative factorization of the quantum harmonic oscillator and two-parameter family of self-adjoint operators. Phys Lett A 376:2860–2865ADSMathSciNetCrossRef
38.
Ibragimov NH (2009) A practical course in differential equations and mathematical modelling. World Scientific, LondonCrossRef
39.
Gilding BH, Kersner R (1996) The characterization of reaction–convection–diffusion processes by travelling waves. J Differ Equ 124:27–79ADSMathSciNetCrossRefMATH
40.
Mei Z (2000) Numerical bifurcation analysis for reaction–diffusion equations. Springer, BerlinCrossRefMATH
41.
Wilhelmsson H, Lazzaro E (2001) Reaction–diffusion problems in the physics of hot plasmas. Institute of Physics Publishing, BristolCrossRef
42.
Gilding BH, Kersner R (2004) Travelling waves in nonlinear diffusion convection reaction. Birkhäuser, BaselCrossRefMATH
43.
Harko T, Mak MK (2015) Travelling wave solutions of the reaction–diffusion mathematical model of glioblastoma growth: an Abel equation based approach. Math Biosci Eng 12:41–69MathSciNetCrossRefMATH
44.
Solovyova A, Schuck P, Costenaro L, Ebel C (2001) Non-ideality by sedimentation velocity of halophilic malate dehydrogenase in complex solvents. Biophys J 81:1868–1880CrossRef
45.
Follain N, Valleton J-M, Lebrun L, Alexandre B, Schaetzel P, Metayer M, Marais S (2010) Simulation of kinetic curves in mass transfer phenomena for a concentration-dependent diffusion coefficient in polymer membranes. J Membr Sci 349:195–207CrossRef
46.
Joannés S, Mazé L, Bunsell AR (2014) A concentration-dependent diffusion coefficient model for water sorption in composite. Compos Struct 108:111–118CrossRef
47.
Hamdan MN, Dado MHF (1997) Large aplitude free vibrations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. J Sound Vib 206:151–168ADSCrossRef
48.
He JH (2005) Homotopy perturbation method for bifurcation of nonlinear problems. Int J Nonlinear Sci Numer Simul 6:207–208MathSciNet
49.
Yildirim A (2010) Determination of periodic solutions for nonlinear oscillators with fractional powers by He’s modified Lindstedt–Poincaré method. Meccanica 45:1–6MathSciNetCrossRefMATH
50.
Ramos JI (2009) An artificial parameter Linstedt–Poincaré method for oscillators with smooth odd nonlinearities. Chaos Solitons Fractals 41:380–393ADSCrossRefMATH
51.
Herisanu N, Marinca V (2010) Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. Meccanica 45:847–855MathSciNetCrossRefMATH
52.
Akbarzade M, Khanb Y (2012) Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: analytical solutions. Math Comput Model 55:480–489MathSciNetCrossRefMATH
Metadaten
Titel
Exact solutions of the Liénard- and generalized Liénard-type ordinary nonlinear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator
verfasst von
Tiberiu Harko
Shi-Dong Liang
Publikationsdatum
29.07.2015
Verlag
Springer Netherlands
Erschienen in
Journal of Engineering Mathematics / Ausgabe 1/2016
Print ISSN: 0022-0833
Elektronische ISSN: 1573-2703
DOI
https://doi.org/10.1007/s10665-015-9812-z

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