×

A numerical solution of the Klein-Gordon equation and convergence of the decomposition method. (English) Zbl 1084.65101

The authors use the Adomian decomposition method (ADM) for solving a nonlinear hyperbolic equation: the Klein-Gordon equation. The Adomian method allows to find the solution without discretization or linearization. The convergence is proved by using a result given by T. Mavoungou and Y. Cherruault [Kybernetes 21, No. 6, 13–25 (1992; Zl 0801.35007)]. Practically the solution is obtained with only initial conditions. Numerical results are given proving the efficacy of the ADM technique. It would be interesting to treat partial differential equations with both initial and boundary conditions.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 0801.35007
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston, MA · Zbl 0802.65122
[2] Adomian, G., A review of the decomposition method in applied mathematics, J. math. anal. appl., 135, 501-544, (1988) · Zbl 0671.34053
[3] Jiminez, B.S.; Vazquez, L., Analysis of four numerical schemes for a nonlinear klein – gordon equation, Appl. math. comput., 35, 61-94, (1990) · Zbl 0697.65090
[4] Ben-Yu, G.; Xun, L.; Vazquez, L., A Legendre spectral method for solving the nonlinear klein – gordon equation, Comput. appl. math., 15, 19-36, (1996) · Zbl 0856.65117
[5] Li, X.; Guo, B.Y., A Legendre spectral method for solving the nonlinear klein – gordon equation, J. comput. math., 15, 105-126, (1997) · Zbl 0876.65073
[6] Vu-Quoc, L.; Li, S., Invariant-conserving finite difference algorithms for the nonlinear klein – gordon equation, Comput. methods appl. mech. engrg., 107, 314-391, (1993) · Zbl 0790.65101
[7] Wong, Y.S.; Chang, Q.; Gong, L., An initial-boundary value problem of a nonlinear klein – gordon equation, Appl. math. comput., 84, 77-93, (1997) · Zbl 0884.65091
[8] 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
[9] Ablowitz, M.J.; Herbst, B.M.; Schober, C.M., Numerical simulation of quasi-periodic solutions of the sine-Gordon equation, Physica D, 87, 37-47, (1995) · Zbl 1194.65121
[10] Khaliq, A.Q.M.; Abukhoider, B.; Sheng, Q.; Ismail, M.S., A predictor-corrector scheme for the sine-Gordon equation, Numer. methods partial diff. equat.: int. J., 16, 133-146, (2000) · Zbl 0951.65089
[11] Argyris, J.; Haase, M.; Heinrich, J.C., Finite element approximations to the two-dimensional sine-Gordon solutions, Comput. methods appl. mech. engrg., 86, 1-26, (1991) · Zbl 0762.65073
[12] D. Kaya, A closed form solution technique for solving nonlinear Klein-Gordon equations, in: Proceeding of the First International Conference on Nonlinear Analysis Nonlinear Modeling, 2001, pp. 28-36
[13] El-Sayed, S.M., The decomposition method for studying the klein – gordon equation, Chaos solitons fractals, 18, 1025-1030, (2003) · Zbl 1068.35069
[14] D. Kaya, A numerical solution of the sine-Gordon equation using the modified decomposition method, Appl. Math. Comput., in press · Zbl 1022.65114
[15] Drazin, P.G.; Johnson, R.S., Solutions: an introduction, (1989), Cambridge University Press Cambridge · Zbl 0661.35001
[16] Cherruault, Y., Convergence of Adomian’s method, Kybernetes, 18, 31-38, (1989) · Zbl 0697.65051
[17] Rèpaci, A., Nonlinear dynamical systems: on the accuracy of Adomian’s decomposition method, Appl. math. lett., 3, 35-39, (1990) · Zbl 0719.93041
[18] Cherruault, Y.; Adomian, G., Decomposition methods: A new proof of convergence, Math. comput. model., 18, 103-106, (1993) · Zbl 0805.65057
[19] Ngarhasta, N.; Some, B.; Abbaoui, K.; Cherruault, Y., New numerical study of Adomian method applied to a diffusion model, Kybernetes, 31, 61-75, (2002) · Zbl 1011.65073
[20] Mavoungou, T.; Cherruault, Y., Convergence of Adomian’s method and applications to nonlinear partial differential equations, Kybernetes, 21, 13-25, (1992) · Zbl 0801.35007
[21] Elwakil, S.A.; El-Labany, S.K.; Zahran, M.A.; Sabry, R., Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. lett. A, 299, 179-188, (2002) · Zbl 0996.35043
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.