A numerical solution of the sine-Gordon equation using the modified decomposition method. (English) Zbl 1022.65114

Summary: The decomposition method for solving the sine-Gordon equation has been implemented. By using a number of initial values, the explicit and numerical solutions of the equation are calculated in the form of convergent power series with easily computable components. The present method performs extremely well in terms of accuracy, efficiency, simplicity, stability and reliability.


65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
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
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[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] Herbst, B.M.; Ablowitz, M.J., Numerical homoclinic instabilities in the sine-Gordon equation, Quaest. math., 15, 345-363, (1992) · Zbl 0785.65086
[4] 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
[5] Wei, G.W., Discrete singular convolution for the sine-Gordon equation, Physica D, 137, 247-259, (2000) · Zbl 0944.35087
[6] Khaliq, A.Q.M.; Abukhodair, B.; Sheng, Q.; Ismail, M.S., A predictor-corrector scheme for the sine-Gordon equation, Numer. methods partial differ. equations, 16, 133-146, (2000) · Zbl 0951.65089
[7] D. Kaya, A closed form solution technique for solving nonlinear Klein-Gordon equations, in: The First International Conference on Nonlinear Analysis and Nonlinear Modeling, 2001, pp. 28-36
[8] Drazin, P.G.; Johnson, R.S., Solutions: an introduction, (1989), Cambridge University Press Cambridge · Zbl 0661.35001
[9] Cherruault, Y., Convergence of adomian’s method, Kybernetics, 18, 31-38, (1989) · Zbl 0697.65051
[10] Rèpaci, A., Nonlinear dynamical systems: on the accuracy of adomian’s decomposition method, Appl. math. lett., 3, 35-39, (1990) · Zbl 0719.93041
[11] Cherruault, Y.; Adomian, G., Decomposition methods: a new proof of convergence, Math. comput. modeling, 18, 103-106, (1993) · Zbl 0805.65057
[12] Wazwaz, A.M., A reliable modification of Adomian decomposition method, Appl. math. comput., 102, 77-86, (1999) · Zbl 0928.65083
[13] Ablowitz, M.J.; Herbst, B.M.; Schober, C., On the numerical solution of the sine-Gordon equation. II. performance of numerical shemes, J. comput. phys., 131, 354-367, (1997) · Zbl 0874.65076
[14] Wolfram, S., Mathematica for windows, (1993), Wolfram Research
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