×

The variational iteration method for studying the Klein-Gordon equation. (English) Zbl 1152.65475

Summary: We use He’s variational iteration method for solving linear and nonlinear Klein-Gordon equations. Also, the results are compared with those obtained by Adomian’s decomposition method (ADM). The results reveal that the method is very effective and simple.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[2] He, J. H., A new approach to nonlinear partial differential equations, Comm. Nonlinear Sci. Numer. Simul., 2, 230-235 (1997)
[3] He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., 167, 57-68 (1998) · Zbl 0942.76077
[4] He, J. H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Engrg., 167, 69-73 (1998) · Zbl 0932.65143
[5] He, J. H., Variational iteration method—A kind of non-linear analytical technique: Some examples, Internat. J. Non-Linear Mech., 34, 699-708 (1999) · Zbl 1342.34005
[6] He, J. H., Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114, 115-123 (2000) · Zbl 1027.34009
[7] He, J. H., Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fractals, 19, 847-851 (2004) · Zbl 1135.35303
[8] Abdou, M. A.; Soliman, A. A., New applications of variational iteration method, Physica D, 211, 1-8 (2005) · Zbl 1084.35539
[9] Abdou, M. A.; Soliman, A. A., Variational iteration method for solving Burgers’ and coupled Burgers’ equations, J. Comput. Appl. Math., 181, 245-251 (2005) · Zbl 1072.65127
[10] Momani, S.; Abuasad, S., Application of He’s variational iteration method to Helmholtz equation, Chaos Solitons Fractals, 27, 1119-1123 (2006) · Zbl 1086.65113
[11] Odibat, Z. M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Internat. J. Nonlinear Sci. Numer. Simul, 7, 27-34 (2006) · Zbl 1401.65087
[12] He, J. H., Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern Phys. B, 20, 1141-1199 (2006) · Zbl 1102.34039
[13] Abulwafa, E. M.; Abdou, M. A.; Mahmoud, A. A., The solution of nonlinear coagulation problem with mass loss, Chaos Solitons Fractals, 26, 313-330 (2006) · Zbl 1101.82018
[14] Bildik, N.; Konuralp, A., The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Internat. J. Nonlinear Sci. Numer. Simul., 7, 65-70 (2006) · Zbl 1401.35010
[15] Dodd, R. K.; Eilbeck, I. C.; Gibbon, J. D.; Morris, H. C., Solitons and Nonlinear Wave Equations (1982), Academic: Academic London · Zbl 0496.35001
[16] Khaliq, A. Q.; Abukhodair, B.; Sheng, Q.; Ismail, M. S., A predictor-corrector scheme for sine-Gordon equation, Numer. Methods Partial Differential Equations, 16, 133-146 (2000) · Zbl 0951.65089
[17] Lynch, M. A.M., Large amplitude instability in finite difference approximations to the Klein-Gordon equation, Appl. Numer. Math., 31, 173-182 (1999) · Zbl 0937.65098
[18] Lu, X.; Schmid, R., Symplectic integration of sine-Gordon type systems, Math. Comput. Simulation, 50, 255-263 (1999)
[19] Kaya, D., An implementation of the ADM for generalized one-dimensional Klein-Gordon equation, Appl. Math. Comput., 166, 426-433 (2005) · Zbl 1074.65118
[20] 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
[21] El-Sayed, S. M., The decomposition method for studying the Klein-Gordon equation, Chaos Solitons Fractals, 18, 1025-1030 (2003) · Zbl 1068.35069
[22] Wazwaz, A. M., The modified decomposition method for analytic treatment of differential equations, Appl. Math. Comput., 173, 165-176 (2006) · Zbl 1089.65112
[23] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Methods (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[24] Wazwaz, A. M., A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102, 77-86 (1997) · Zbl 0928.65083
[25] Wazwaz, A. M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111, 53-69 (2000) · Zbl 1023.65108
[26] Zhao, Xueqin; Zhi, Hongyan; Yu, Yaxuan; Zhang, Hongqing, A new Riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations, Appl. Math. Comput., 172, 24-39 (2006) · Zbl 1088.65095
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.