Application of He’s Parameter-Expansion Method to a Coupled Van Der Pol oscillators with Two Kinds of Time-delay Coupling
In this paper, the dynamics of a system of two van der Pol oscillators with delayed position and velocity is studied. We provide an approximate solution for this system using parameterexpansion method. Also, we obtain approximate values for frequencies of the system. The parameter-expansion method is more efficient than the perturbation method for this system because the method is independent of perturbation parameter assumption.
[1] J.H. He, Variational iteration method-a kind of non-linear analytical
technique: some examples, Int. J. Non-linear Mech. 34(4) (1999) 699-
708.
[2] M.T. Darvishi, F. Khani, A.A. Soliman, The numerical simulation for stiff
systems of ordinary differential equations, Comput. Math. Appl. 54(7-8)
(2007) 1055-1063.
[3] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifth-order
Korteweg-de Vries equations, Chaos, Solitons and Fractals 39 (2009)
2484-2490.
[4] S. Abbasbandy, M.T. Darvishi, A numerical solution of Burgers- equation
by modified Adomian method, Appl. Math. Comput. 163 (2005) 1265-
1272.
[5] S. Abbasbandy, M.T. Darvishi, A numerical solution of Burgers- equation
by time discretization of Adomian-s decomposition method, Appl. Math.
Comput. 170 (2005) 95-102.
[6] J.H. He, New interpretation of homotopy perturbation method, Int. J.
Mod. Phys. B 20(18) (2006) 2561-2568.
[7] J.H. He, Application of homotopy perturbation method to nonlinear wave
equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[8] J.H. He, Homotopy perturbation method for bifurcation of nonlinear
problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[9] M.T. Darvishi, F. Khani, Application of He-s homotopy perturbation
method to stiff systems of ordinary differential equations, Zeitschrift fur
Naturforschung A, 63a (1-2) (2008) 19-23.
[10] M.T. Darvishi, F. Khani, S. Hamedi-Nezhad, S.-W. Ryu, New modification
of the HPM for numerical solutions of the sine-Gordon and coupled
sine-Gordon equations, Int. J. Comput. Math. 87(4) (2010) 908-919.
[11] M.T. Darvishi, Preconditioning and domain decomposition schemes to
solve PDEs, Int-l J. of Pure and Applied Math. 1(4) (2004) 419-439.
[12] M.T. Darvishi, S. Kheybari, F. Khani, A numerical solution of the
Korteweg-de Vries equation by pseudospectral method using Darvishi-s
preconditionings, Appl. Math. Comput. 182(1) (2006) 98-105.
[13] M.T. Darvishi, M. Javidi, A numerical solution of Burgers- equation
by pseudospectral method and Darvishi-s preconditioning, Appl. Math.
Comput. 173(1) (2006) 421-429.
[14] M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation solution of a
generalized Hirota-Satsuma KdV equation, Int. J. Comput. Math. 84(4)
(2007) 541-551.
[15] M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation method
and Darvishi-s preconditionings to solve the generalized Burgers-Huxley
equation, Commun. Nonlinear Sci. Numer. Simul. 13(10) (2008) 2091-
2103.
[16] S.J. Liao, An explicit, totally analytic approximate solution for Blasius
viscous flow problems, Int. J. Non-Linear Mech. 34 (1999) 759-778.
[17] S.J. Liao, Beyond perturbation: introduction to the homotopy analysis
method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[18] S.J. Liao, On the homotopy analysis method for nonlinear problems,
Appl. Math. Comput. 147 (2004) 499-513.
[19] S.J. Liao, A new branch of solutions of boundary-layer flows over an
impermeable stretched plate, Int. J. Heat Mass Transfer 48 (2005) 2529-
2539.
[20] S.J. Liao, A general approach to get series solution of non-similarity
boundary-layer flows, Commun. Nonlinear Sci. Numer. Simul. 14(5)
(2009) 2144-2159.
[21] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation,
Comput. Math. Appl. 58 (2009) 360-368.
[22] A. Aziz, F. Khani, M.T. Darvishi, Homotopy analysis method for
variable thermal conductivity heat flux gage with edge contact resistance,
Zeitschrift fuer Naturforschung A, 65a(10) (2010) 771-776.
[23] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution
equations using Exp-function method, Chaos, Solitons and Fractals 34
(2007) 1421-1429.
[24] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations,
Chaos, Solitons and Fractals 30(3) (2006) 700-708.
[25] J.H. He, X.H. Wu, Construction of solitary solution and compacton-like
solution by variational iteration method, Chaos, Solitons and Fractals 29
(2006) 108-113.
[26] F. Khani, S. Hamedi-Nezhad, M.T. Darvishi, S.-W. Ryu, New solitary
wave and periodic solutions of the foam drainage equation using the Expfunction
method, Nonlin. Anal.: Real World Appl. 10 (2009) 1904-1911.
[27] B.-C. Shin, M.T. Darvishi, A. Barati, Some exact and new solutions
of the Nizhnik-Novikov-Vesselov equation using the Exp-function method,
Comput. Math. Appl. 58(11/12) (2009) 2147-2151.
[28] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear
equations, Chaos, Solitons and Fractals 38(3) (2008) 903-910.
[29] X. Li., J.C. Ji, C.H. Hansen, Dynamics of two delay coupled van der Pol
oscillators, Mechanics Research Communications 33 (2006) 614-627.
[30] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlinear
Sci. Numer. Simul. 2 (2001) 257-264.
[31] M.T. Darvishi, A. Karami, B.-C. Shin, Application of He-s parameterexpansion
method for oscillators with smooth odd nonlinearities, Phys.
Lett. A 372(33) (2008) 5381-5384.
[32] B.-C. Shin, M.T. Darvishi, A. Karami, Application of He-s parameterexpansion
method to a nonlinear self-excited oscillator system, Int. J.
Nonlin. Sci. Num. Simulation 10(1) (2009) 137-143.
[33] S.T. Mohyud-Din, M.A. Noor, K.I. Noor, Parameter-expansion techniques
for strongly nonlinear oscillators, Int. J. Nonlin. Sci. Num.
Simulation 10(5) (2009) 581-583.
[34] N.H. Sweilam, R.F. Al-Bar, Implementation of the parameter-expansion
method for the coupled van der Pol oscillators, Int. J. Nonlin. Sci. Num.
Simulation 10(2) (2009) 259-264.
[35] N.H. Sweilam, M.M. Khader, Application of He-s parameter-expansion
method for the nonlinear differential equations, Int. J. Nonlin. Sci. Num.
Simulation 10(2) (2009) 265-272.
[36] F.O. Zengin, M.O. Kaya, S.A. Demirbag, Application of parameterexpansion
method to nonlinear oscillators with discontinuities, Int. J.
Nonlin. Sci. Num. Simulation 9(3) (2008) 267-270.
[37] L. Xu, He-s parameter-expanding methods for strongly nonlinear oscillators,
J. Comput. Appl. Math. 207(1) (2007) 148-154.
[38] L. Xu, Application of He-s parameter-expansion method to an oscillation
of a mass attached to a stretched elastic wire, Phys. Lett. A 368(3-4)
(2007) 259-262.
[39] L. Xu, Determination of limit cycle by He-s parameter-expanding
method for strongly nonlinear oscillators, J. of Sound and Vibration
302(1-2) (2007) 178-184.
[40] L.N. Zhang, L. Xu, Determination of the limit cycle by He-s parameterexpansion
for oscillators in a u(3)/(1+u(2)) potential, Zeitschrift fur
Naturforschung Section A 62(7-8) (2007) 396-398.
[1] J.H. He, Variational iteration method-a kind of non-linear analytical
technique: some examples, Int. J. Non-linear Mech. 34(4) (1999) 699-
708.
[2] M.T. Darvishi, F. Khani, A.A. Soliman, The numerical simulation for stiff
systems of ordinary differential equations, Comput. Math. Appl. 54(7-8)
(2007) 1055-1063.
[3] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifth-order
Korteweg-de Vries equations, Chaos, Solitons and Fractals 39 (2009)
2484-2490.
[4] S. Abbasbandy, M.T. Darvishi, A numerical solution of Burgers- equation
by modified Adomian method, Appl. Math. Comput. 163 (2005) 1265-
1272.
[5] S. Abbasbandy, M.T. Darvishi, A numerical solution of Burgers- equation
by time discretization of Adomian-s decomposition method, Appl. Math.
Comput. 170 (2005) 95-102.
[6] J.H. He, New interpretation of homotopy perturbation method, Int. J.
Mod. Phys. B 20(18) (2006) 2561-2568.
[7] J.H. He, Application of homotopy perturbation method to nonlinear wave
equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[8] J.H. He, Homotopy perturbation method for bifurcation of nonlinear
problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[9] M.T. Darvishi, F. Khani, Application of He-s homotopy perturbation
method to stiff systems of ordinary differential equations, Zeitschrift fur
Naturforschung A, 63a (1-2) (2008) 19-23.
[10] M.T. Darvishi, F. Khani, S. Hamedi-Nezhad, S.-W. Ryu, New modification
of the HPM for numerical solutions of the sine-Gordon and coupled
sine-Gordon equations, Int. J. Comput. Math. 87(4) (2010) 908-919.
[11] M.T. Darvishi, Preconditioning and domain decomposition schemes to
solve PDEs, Int-l J. of Pure and Applied Math. 1(4) (2004) 419-439.
[12] M.T. Darvishi, S. Kheybari, F. Khani, A numerical solution of the
Korteweg-de Vries equation by pseudospectral method using Darvishi-s
preconditionings, Appl. Math. Comput. 182(1) (2006) 98-105.
[13] M.T. Darvishi, M. Javidi, A numerical solution of Burgers- equation
by pseudospectral method and Darvishi-s preconditioning, Appl. Math.
Comput. 173(1) (2006) 421-429.
[14] M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation solution of a
generalized Hirota-Satsuma KdV equation, Int. J. Comput. Math. 84(4)
(2007) 541-551.
[15] M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation method
and Darvishi-s preconditionings to solve the generalized Burgers-Huxley
equation, Commun. Nonlinear Sci. Numer. Simul. 13(10) (2008) 2091-
2103.
[16] S.J. Liao, An explicit, totally analytic approximate solution for Blasius
viscous flow problems, Int. J. Non-Linear Mech. 34 (1999) 759-778.
[17] S.J. Liao, Beyond perturbation: introduction to the homotopy analysis
method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[18] S.J. Liao, On the homotopy analysis method for nonlinear problems,
Appl. Math. Comput. 147 (2004) 499-513.
[19] S.J. Liao, A new branch of solutions of boundary-layer flows over an
impermeable stretched plate, Int. J. Heat Mass Transfer 48 (2005) 2529-
2539.
[20] S.J. Liao, A general approach to get series solution of non-similarity
boundary-layer flows, Commun. Nonlinear Sci. Numer. Simul. 14(5)
(2009) 2144-2159.
[21] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation,
Comput. Math. Appl. 58 (2009) 360-368.
[22] A. Aziz, F. Khani, M.T. Darvishi, Homotopy analysis method for
variable thermal conductivity heat flux gage with edge contact resistance,
Zeitschrift fuer Naturforschung A, 65a(10) (2010) 771-776.
[23] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution
equations using Exp-function method, Chaos, Solitons and Fractals 34
(2007) 1421-1429.
[24] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations,
Chaos, Solitons and Fractals 30(3) (2006) 700-708.
[25] J.H. He, X.H. Wu, Construction of solitary solution and compacton-like
solution by variational iteration method, Chaos, Solitons and Fractals 29
(2006) 108-113.
[26] F. Khani, S. Hamedi-Nezhad, M.T. Darvishi, S.-W. Ryu, New solitary
wave and periodic solutions of the foam drainage equation using the Expfunction
method, Nonlin. Anal.: Real World Appl. 10 (2009) 1904-1911.
[27] B.-C. Shin, M.T. Darvishi, A. Barati, Some exact and new solutions
of the Nizhnik-Novikov-Vesselov equation using the Exp-function method,
Comput. Math. Appl. 58(11/12) (2009) 2147-2151.
[28] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear
equations, Chaos, Solitons and Fractals 38(3) (2008) 903-910.
[29] X. Li., J.C. Ji, C.H. Hansen, Dynamics of two delay coupled van der Pol
oscillators, Mechanics Research Communications 33 (2006) 614-627.
[30] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlinear
Sci. Numer. Simul. 2 (2001) 257-264.
[31] M.T. Darvishi, A. Karami, B.-C. Shin, Application of He-s parameterexpansion
method for oscillators with smooth odd nonlinearities, Phys.
Lett. A 372(33) (2008) 5381-5384.
[32] B.-C. Shin, M.T. Darvishi, A. Karami, Application of He-s parameterexpansion
method to a nonlinear self-excited oscillator system, Int. J.
Nonlin. Sci. Num. Simulation 10(1) (2009) 137-143.
[33] S.T. Mohyud-Din, M.A. Noor, K.I. Noor, Parameter-expansion techniques
for strongly nonlinear oscillators, Int. J. Nonlin. Sci. Num.
Simulation 10(5) (2009) 581-583.
[34] N.H. Sweilam, R.F. Al-Bar, Implementation of the parameter-expansion
method for the coupled van der Pol oscillators, Int. J. Nonlin. Sci. Num.
Simulation 10(2) (2009) 259-264.
[35] N.H. Sweilam, M.M. Khader, Application of He-s parameter-expansion
method for the nonlinear differential equations, Int. J. Nonlin. Sci. Num.
Simulation 10(2) (2009) 265-272.
[36] F.O. Zengin, M.O. Kaya, S.A. Demirbag, Application of parameterexpansion
method to nonlinear oscillators with discontinuities, Int. J.
Nonlin. Sci. Num. Simulation 9(3) (2008) 267-270.
[37] L. Xu, He-s parameter-expanding methods for strongly nonlinear oscillators,
J. Comput. Appl. Math. 207(1) (2007) 148-154.
[38] L. Xu, Application of He-s parameter-expansion method to an oscillation
of a mass attached to a stretched elastic wire, Phys. Lett. A 368(3-4)
(2007) 259-262.
[39] L. Xu, Determination of limit cycle by He-s parameter-expanding
method for strongly nonlinear oscillators, J. of Sound and Vibration
302(1-2) (2007) 178-184.
[40] L.N. Zhang, L. Xu, Determination of the limit cycle by He-s parameterexpansion
for oscillators in a u(3)/(1+u(2)) potential, Zeitschrift fur
Naturforschung Section A 62(7-8) (2007) 396-398.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:56125", author = "Mohammad Taghi Darvishi and Samad Kheybari", title = "Application of He’s Parameter-Expansion Method to a Coupled Van Der Pol oscillators with Two Kinds of Time-delay Coupling", abstract = "In this paper, the dynamics of a system of two van der Pol oscillators with delayed position and velocity is studied. We provide an approximate solution for this system using parameterexpansion method. Also, we obtain approximate values for frequencies of the system. The parameter-expansion method is more efficient than the perturbation method for this system because the method is independent of perturbation parameter assumption.
", keywords = "Parameter-expansion method, coupled van der pol oscillator, time-delay system.", volume = "5", number = "2", pages = "121-6", }
