Decomposition method using He’s polynomials for Kdv Boussinesq and coupled Kdv Boussinesq equations

Abstract

This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations.

Keywords

• Decomposition method,He’s polynomials,cubic Boussinesq equation

Kaynakça

1. Wazwaz, A. The variational iteration method for rational solutions for KdV, K(2;2), Burgers, and cubic Boussinesq equations, Journal of Computational and Applied Mathematics, 207(1), pp. 18–23 (2007).
2. J. Ahmad, M. Mushtaq and N.Sajjad, Exact Solution Of Whitham-Broer-Kaup Shallow Water Wave Equations, Journal of Science and Arts, 1(30), pp. 5-12, 2015.
3. Girgis, L., Zerrad, E. and Biswas, A. Solitary wave solutions of the Peregrine equation, International Journal of Oceans and Oceanography 4(1), pp. 45–54 (2010).
4. G. Domairry, A. G. Davodi, and A. G. Davodi,vSolutions for the double sine-Gordon equation by Exp-function, tanh, and extended tanh methods, Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 384–398, 2010.
5. Jawad, A.J.M., Petkovic, M.D. and Biswas, A. Soliton solutions of Burgers equations and perturbed Burgers equation, Applied Mathematics and Computation, 216(11), pp. 3370–3377 (2010).
6. Jawad, A.J.M., Petkovic, M.D. and Biswas, A. Soliton solutions of a few nonlinear wave equations, Applied Mathematics and Computation, 216(9), pp. 2649–2658 (2010).
7. Li, B., Chen, Y. and Zhang, H.Q. Explicit exact solutions for some nonlinear partial differential equations with nonlinear terms of any order, Czechoslovak Journal of Physics, 53(4), pp. 283–295 (2003)
8. Malfliet, W. Solitary wave solutions of nonlinear wave equations, American Journal of Physics, 60, pp. 650–654 (1992).

9. Mousa, M.M. and Kaltayev, A. Constructing approximate and exact solutions for Boussinesq equations using homotopy perturbation Pade technique, World Academy of Science, Engineering and Technology, 38, pp. 3730–3740 (2009).
10. Ugurlu, Y. and Kaya, D. Exact and numerical solutions of generalized Drinfeld–Sokolov equations, Physics Letters A, 372(16), pp. 2867–2873 (2008).
11. Wu, L., Chen, S. and Pang, C. Traveling wave solutions for generalized Drinfeld–Sokolov equations, Applied Mathematical Modeling, 33(11), pp. 4126–4130 (2009).
12. Ganji, D.D., Nourollahi, M. and Rostamian, M. A comparison of variational iteration method with Adomian’s decomposition method in some highly nonlinear equations, International Journal of Science and Technology, 2(2), pp. 179–188 (2007)
13. Wolfram. The Mathematica, fifth ed., Wolfram Media/Cambridge University Press, Champaign, IL 61820, USA Book (2003).
14. He, J.H. Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation, 156, pp. 527–539 (2004).
15. Malfliet, W. The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, Journal of Computational and Applied Mathematics, 164–165, pp. 529–541 (2004).
16. Wazwaz, A., Partial Differential Equations and Solitary Waves Theory, Springer Verlag, New York, NY, USA (2009).
17. Wazwaz, A. The Cole–Hopf transformation and multiple soliton solutions for the integrable sixth-order Drinfeld–Sokolov–Satsuma–Hirota equation, Applied Mathematics and Computation, 207(1), pp. 248–255 (2009).
18. Kara, A.H. and Mahomed, F.M. ‘Noether-type symmetries and conservation laws via partial Lagrangians’, Nonlinear Dynamics, 45(3–4), pp. 367–383 (2006).
19. V. S. Manoranjan, A. R. Mitchell, and J. Ll. Morris, Numerical solutions of the good Boussinesq equation, Society for Industrial and Applied Mathematics, vol. 5, no. 4, pp. 946–957, 1984.
20. S. S. Ray, Soliton solutions for time fractional coupled modified KdV equations using new coupled fractional reduced differential transform method, Journal of Mathematical Chemistry, vol. 51, no. 8, pp. 2214–2229, 2013.
21. S. Zhou and X. Cheng, Numerical solution to couple nonlinear Schrodinger equations on unbounded domains, Mathematics and Computers in Simulation, vol. 80, no. 12, pp. 2362–2373, 2010.
22. H. Hu, L. Liu, and L. Zhang, New analytical position, negaton and complexiton solutions of a coupled KdV-mKdV system, Applied Mathematics and Computation, vol. 219, no. 11,pp. 5743– 5749, 2013.
23. A. G. Bratsos, A second order numerical scheme for the solution of the one-dimensional Boussinesq equation, Numerical Algorithms, vol. 46, no. 1, pp. 45–58, 2007.
24. A. G. Bratsos, Ch. Tsitouras, and D. G. Natsis, Linearized numerical schemes for the Boussinesq equation, Applied Numerical Analysis and Computational Mathematics, vol. 2, no. 1, pp. 34–53, 2005.
25. M. S. Ismail and A. G. Bratsos, A predictor-corrector scheme for the numerical solution of the Boussinesq equation, Journal of Applied Mathematics & Computing, vol. 13, no. 1-2, pp. 11–27, 2003.
26. A. Mohebbi and Z. Asgari, Efficient numerical algorithms for the solution of good Boussinesq equation in water wave propagation, Computer Physics Communications, vol. 182, no. 12, pp. 2464–2470, 2011.
27. A.-M. Wazwaz, Multiple-soliton solutions for the Boussinesq equation, Applied Mathematics and Computation, vol. 192, no. 2, pp. 479–486, 2007.
28. B. S. Attili, The Adomian decomposition method for solving the Boussinesq equation arising in water wave propagation, Numerical Methods for Partial Differential Equations, vol. 22, no. 6, pp. 1337–1347, 2006.
29. A.-M. Wazwaz, New travelling wave solutions to the Boussinesq and the Klein-Gordon equations, Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 5, pp. 889–901, 2008.
30. A. M.Wazwaz, Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos, Solitons and Fractals, vol. 12, no. 8, pp. 1549–1556, 2001.
31. A. Yildirim and S. T. Mohyud-Din, A variational approach for soliton solutions of good Boussinesq equation, Journal of King Saud University, vol. 22, no. 4, pp. 205–208, 2010.
32. A. Biswas, D. Milovic, and A. Ranasinghe, Solitary waves of Boussinesq equation in a power law media, Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11,pp. 3738–3742, 2009.