Application of homotopy-perturbation method to Klein-Gordon and sine-Gordon equations. (English) Zbl 1197.65164

Summary: The homotopy-perturbation method (HPM) is employed to obtain approximate analytical solutions of the Klein-Gordon and sine-Gordon equations. An efficient way of choosing the initial approximation is presented. Comparisons with the exact solutions, the solutions obtained by the Adomian decomposition method (ADM) and the variational iteration method (VIM) show the potential of HPM in solving nonlinear partial differential equations.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.


65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35L71 Second-order semilinear hyperbolic equations
35A35 Theoretical approximation in context of PDEs
Full Text: DOI


[1] Ablowitz, M. J.; Herbst, B. M.; Schober, C., Constance on the numerical solution of the sine-Gordon equation. I: Integrable discretizations and homoclinic manifolds, J Comput Phys, 126, 299-314 (1996) · Zbl 0866.65064
[2] Ablowitz, M. J.; Herbst, B. M.; Schober, C., Homotopy-perturbation method and axisymmetric flow over a stretching sheet, Int J Nonlinear Sci Numer Simul, 7, 4, 399-406 (2006)
[4] Caudrey, P. J.; Eilbeck, I. C.; Gibbon, J. D., The sine-Gordon equation as a model classical field theory, Nuovo Cimento, 25, 497-511 (1975)
[6] Chowdhury, M. S.H.; Hashim, I., Solutions of a class of singular second-order IVPs by homotopy-perturbation method, Phys Lett A, 365, 439-447 (2007) · Zbl 1203.65124
[8] Cveticanin, L., The homotopy-perturbation method applied for solving complex-valued differential equations with strong cubic nonlinearity, J Sound Vib, 285, 1171-1179 (2005) · Zbl 1238.65085
[9] Deeba, E. Y.; Khuri, S. A., A decomposition method for solving the nonlinear Klein-Gordon equation, J Comput Phys, 124, 442-448 (1996) · Zbl 0849.65073
[10] Dodd, R. K.; Eilbeck, I. C.; Gibbon, J. D., Solitons and nonlinear wave equations (1982), Academic: Academic London · Zbl 0496.35001
[11] El-Sayed, S., The decomposition method for studying the Klein-Gordon equation, Chaos, Solitons & Fractals, 18, 1025-1030 (2003) · Zbl 1068.35069
[12] Ganji, D. D.; Rajabi, A., Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, Int Commun Heat Mass, 33, 391-400 (2006)
[13] Ganji, D. D.; Sadighi, A., Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, J Comput Appl Math., 207, 1, 24-34 (2004) · Zbl 1120.65108
[14] He, J. H., Variational iteration method – a kind of non-linear analytical technique: some examples, Int J Nonlinear Mech, 34, 699-708 (1999) · Zbl 1342.34005
[15] He, J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int J Nonlinear Mech, 35, 1, 37-43 (2000) · Zbl 1068.74618
[16] He, J. H., Homotopy-perturbation method for solving boundary value problems, Phys Lett A, 350, 87-88 (2006) · Zbl 1195.65207
[17] He, J. H., New interpretation of homotopy-perturbation method, Int J Mod Phys B, 20, 18, 1-7 (2006)
[18] He, J. H., Non-perturbative methods for strongly nonlinear problems (2006), Die Deutsche bibliothek: Die Deutsche bibliothek Germany
[19] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int J Mod Phys B, 20, 1141-1199 (2006) · Zbl 1102.34039
[20] Herbst, B. M.; Ablowitz, M. J., Numerical homoclinic instabilities in the sine-Gordon equation, Quaest Math, 15, 345-363 (1992) · Zbl 0785.65086
[21] Kaya, D., A numerical solution of the sine-Gordon equation using the modified decomposition method, Appl Math Comput, 143, 309-317 (2003) · Zbl 1022.65114
[22] Kaya, D.; El-Sayed, S. M., A numerical solution of the Klein-Gordon equation and convergence of the decomposition method, Appl Math Comput, 156, 341-353 (2004) · Zbl 1084.65101
[23] Mo, J. Q.; Lin, W. T., Homotopy-perturbation method of equatorial eastern Pacific for the El Nino-Southern Oscillation mechanism, Chin Phys, 14, 875-878 (2005)
[24] Noor, M. A.; Mohyud-Din, S. T., An efficient algorithm for solving fifth-order boundary value problems, Math Comput Model, 45, 7-8, 954-964 (2007) · Zbl 1133.65052
[26] Wazwaz, A. M., The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Appl Math Comput, 167, 1196-1210 (2005) · Zbl 1082.65585
[27] Wazwaz, A. M., The modified decomposition method for analytic treatment of differential equations, Appl Math Comput, 173, 165-176 (2006) · Zbl 1089.65112
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.