Homotopy perturbation and Elzaki transform for solving Sine-Gorden and Klein-Gorden equations

Iranian Journal of Numerical Analysis and Optimization

مقاله 3، دوره 3، شماره 2 - شماره پیاپی 4، مهر 2013، صفحه 33-46   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v3i2.24079

نویسندگان

E. Hesameddini^* ؛ N. Abdollahy

Department of Mathematics, Faculty of Basic Sciences, Shiraz University of Technology P.O.Box 71555-313, Shiraz, Iran.

چکیده

In this paper, the homotopy perturbation method (HPM) and Elzaki transform is employed to obtain the approximate analytical solution of the Sine Gorden and the Klein Gorden equations. The nonlinear terms can be handled by the use of homotopy perturbation method. The proposed homotopy perturbation method is applied to reformulate the first and the second order initial value problems which leads to the solution in terms of transformed variable, and the series solution that can be obtained by making use of the inverse transformation.

کلیدواژه‌ها

Homotopy-perturbation method؛ Elzaki transform؛ Sine-Gorden equation؛ Klein-Gorden equation

مراجع

1. Caudrey P., Elibeck I. and Gibbon J. The sine-gorden equation as a model classical field theory, Nuovo cimento, 25 (1975), 496–511.
2. Chowdhury, M. S. H. and Hashim, I. Application of homotopy perturbation method to Kline-Gorden and Sine-Gorden equations, Chaos, Solitons Fractals, 39 (2009), 1928-1935.
3. Christiansen P. L. and Lomdahl, P. S. Numerical solution of 2+1 dimensional Sine-Gordon solitons, Phys 2D (1981), 482-494.
4. Deeba, E. and Khuri, S. A decomposition method for solving the nonlinear Kline-gorden equation, J Comput Phys, 124 (1996), 442-448.
5. Dodd P., Elibeck I. and Gibbon J. salitons and nonlinear wave equation, London Academic, 1982.
6. El-Sayad S. The decomposition method for studying the Kline-Gordon equation, Chaos, Soliton Fractals, 18 (2003), 1025–1030.
7. Elzaki T. M. and Hilal, E. M. A. Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations, Mathematical Theory and Modling, 2 (2012), 3, 33-42.
8. Guo, B. Y., Pascual, P. J., Rodriguez, M. J. and Vzquez, L. Numerical solution of the Sine-Gorden equation, Appl Math comput, 18 (1986), 1–14.
9. He J. A copuling method of homotopy technique and perturbation technique for nonlinear problem, Int J Nonlinear Mech, 35 (2000), 37-43.

10. He J. Homotopy-perturbation method for solving boundary value problem, Phys Lett A, 350 (2006), 87-88.
11. He J. Non-perturbative methods for strongly nonlinear problems, Germanay: Die Deutsche bibliothek, 2006.
12. He J. Some asymptotic method for strongly nonlinear equations, Int J Mod Phys B, 20 (2006), 1141-1199.
13. He J. Varuational iteration method-a kind of non-linear analytical technique: some examples, Int J Nonlinear mech, 34 (1999), 699–708.
14. Herbst B. and Ablowitz M. Numerical homoclinic instabilities in the Sine Gorden equation, Quaest Math, 15 (1992), 345–630.
15. Hesameddini, E. and Latifizadeh, H. A new vision to the He’s homotopy perturbation method, Nonlinear sci. simul., 10 (2009), 1389–1398.
16. Hirota, R. Exact three-soliton solution of the two-dimensional Sine Gorden equation, J Phys Soc Jpn, 35 (1973), 15–66.
17. Hu H. C. and Lou, S. Y. New guasi-periodic waves of the (2+1)dimensional Sine-Gorden system, Phys Lett A, 341 (2005), 422–426.
18. Kaliappan, P. and Lakshmanan, M Kadomstev-Petviashvili two dimensional Sine-Gorden equations: reduction to painleve transcendents, J Phys A, 12 (1979), 249–252.
19. Kaya D. A numerical solution of the Sine-Gorden equation using the modified decomposition method, Appl Math Comput, 143 (2003, 309–317.
20. Kaya D. and El-Sayed S. A numerical solution of the Kline-Gordon equation and convergence of the decomposition method, Appl Math Comput, 156 (2004), 341–353.
21. Leibbrandt, G. New exact solutions of the classical Sine-Gorden equation in 2+1 and 3+1 dimensionals, Phys Rev Lett, 41 (1978), 435–438.
22. Mohyud-Din, S. T. and Yildirim A. Variatonal Iteration Method for solving Kline-Gordon equation, JAMSI, 6 (2010), 1, 99–106.
23. Wazwaz A. The tanh method: exact solution of Sine-Gordon and the sinh-Gordon equations, Appl Math Comput, 167 (2005), 1196–1210.
24. Xin, J. X. Modeling light bullets with the two-dimensional Sine-Gordon equation, Phys D, 135 (2000), 345