Explicit and exact solutions to cubic Duffing and double-well Duffing equations. (English) Zbl 1217.34004

Summary: A kind of explicit exact solution of nonlinear differential equations is obtained using a new approach applied in this case to look for exact solutions of the Duffing and double-well Duffing equations. The new proposed procedure is applied by using a quotient trigonometric function expansion method. The method can also be easily applied to solve other nonlinear differential equations.


34A05 Explicit solutions, first integrals of ordinary differential equations
Full Text: DOI


[1] Nayfeh, A.H.; Mook, D.T., Nonlinear oscillations, (1979), Wiley New York
[2] Hagedorn, P., Nonlinear oscillations, (1988), Clarendon Press Oxford · Zbl 0709.70512
[3] Beléndez, A.; Bernabeu, G.; Francés, J.; Méndez, D.I.; Marini, S., An accurate closed-form approximate solution for the quintic Duffing oscillator equation, Math. comput. modelling, 52, 637-641, (2010) · Zbl 1201.34019
[4] Belendez, A.; Hernandez, A.; Belendez, T., Application of he’s homotopy perturbation method to the Duffing-harmonic oscillator, Int. J. nonlinear sci. numer. simul., 8, 1, 79-88, (2007) · Zbl 1119.70017
[5] Peng, L., Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments, Math. comput. modelling, 45, 378-386, (2007) · Zbl 1177.34090
[6] Ramos, J.I., On lindstedt – poincaré techniques for the quintic Duffing equation, Appl. math. comput., 193, 303-310, (2007) · Zbl 1193.65142
[7] Kovacic, I.; Brennan, M.J.; Lineton, B., On the resonance response of an asymmetric Duffing oscillator, Int. J. non-linear mech., 43, 858-867, (2008)
[8] Marinca, V.; Herişanu, N., Periodic solutions of Duffing equation with strong non-linearity, Chaos solitons fractals, 37, 144-149, (2008) · Zbl 1156.34322
[9] Cveticanin, L., The approximate solving methods for the cubic Duffing equation based on Jacobi elliptic functions, Int. J. nonlinear sci. numer. simul., 10, 1491-1516, (2009)
[10] Thompsen, J.J., Vibrations and stability, (2003), Springer Berlin
[11] Litak, G.; Borowiec, M., Oscillators with asymmetric single and double well potential: transition to chaos revisited, Acta mech., 184, 47-59, (2006) · Zbl 1099.70018
[12] Wu, B.S.; Sun, W.P.; Lim, C.W., Analytical approximations to the double-well Duffing oscillators in large amplitude oscillations, J. sound. vibration, 307, 953-960, (2007)
[13] Ablowitz, M.J.; Clarkson, P.A., Soliton, nonlinear evolution equations and inverse scattering, (1991), Cambridge University Press New York · Zbl 0762.35001
[14] Hirota, R., Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. rev. lett., 27, 1192-1194, (1971) · Zbl 1168.35423
[15] Miura, M.R., Bäcklund transformation, (1978), Springer Berlin
[16] Weiss, J.; Tabor, M.; Carnevale, G., The painleve property for partial differential equations, J. math. phys., 24, 522-526, (1983) · Zbl 0514.35083
[17] Khalfallah, M., Exact travelling wave solutions of the boussinesq – burgers equation, Math. comput. modelling, 49, 666-671, (2009) · Zbl 1165.35445
[18] He, J.H.; Wu, X.H., Exp-function method for nonlinear wave equations, Chaos solitons fractals, 30, 700-708, (2006) · Zbl 1141.35448
[19] Zhang, S., Exp-function method: solitary, periodic and rational wave solutions of nonlinear evolution equations, Nonl. sci. lett. A, 1, 143-146, (2010)
[20] Wu, X.H.; He, L.H., Solitary solutions, periodic solutions and compaction-like solutions using the exp-function method, Comput. math. appl., 54, 966-986, (2007) · Zbl 1143.35360
[21] Kabir, M.M.; Khajeh, A., New explicit solutions for the Vakhnenko and a generalized form of the nonlinear heat conduction equations via exp-function method, Int. J. nonlinear sci. num., 10, 1307-1318, (2009)
[22] Dai, C.Q.; Zhang, J.F., Application of he’s exp-function method to the stochastic \(m K \operatorname{d} V\) equation, Int. J. nonlinear sci. num., 10, 675-680, (2009)
[23] Liu, S.K.; Fu, Z.T.; Liu, S.D.; Zhao, Q., Jacobi elliptic expansion method and periodic wave solutions of nonlinear wave equations, Phys lett. A, 289, 69-74, (2001) · Zbl 0972.35062
[24] Lai, S.; Lv, X.; Shuai, M., The Jacobi elliptic function solutions to a generalized benjamin – bona – mahony equation, Math. comput. modelling, 49, 369-378, (2009) · Zbl 1165.35447
[25] Elhanbaly, A.; Abdou, M., Exact travelling wave solutions for two nonlinear evolution equations using the improved F-expansion method, Math. comput. modelling, 46, 1265-1276, (2007) · Zbl 1173.35413
[26] Sirendaoreji, New exact traveling wave solutions for the Kawahara and modified Kawahara equation, Chaos solitons fractals, 19, 147-150, (2004) · Zbl 1068.35141
[27] Wazwaz, A.M., Exact solutions for the fourth order nonlinear schrodinger equations with cubic and power law nonlinearities, Math. comput. modelling, 43, 802-808, (2006) · Zbl 1136.35458
[28] Yusufoglu, E.; Bekir, A., Exact solutions of coupled nonlinear klein – gordon equations, Math. comput. modelling, 48, 1694-1700, (2008) · Zbl 1187.35141
[29] Ugurlu, Y.; Kaya, D., Exact and numerical solutions of generalized drinfeld – sokolov equations, Phys. lett. A, 372, 2867-2873, (2008) · Zbl 1220.74027
[30] Kudryashov, N.A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos solitons fractals, 24, 1217-1231, (2005) · Zbl 1069.35018
[31] Vitanov, N.K., Application of simplest equations of Bernoulli and Riccati kind for obtaining exact travelling-wave solutions for a class of PDEs with polynomial nonlinearity, Commun. nonlinear sci numer. simul., 15, 2050-2060, (2010) · Zbl 1222.35062
[32] Fu, Z.; Liu, S.; Liu, S., New exact solution to the \(K \operatorname{d} V\)-burgers – kuramato equation, Chaos solitons fractals, 23, 609-616, (2005)
[33] Cai, X.-C.; Li, M.-S., Periodic solutions of Jacobi elliptic equation by he’s perturbation method, Comput. math. appl., 54, 1210-1212, (2007) · Zbl 1267.65099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.