Homotopy perturbation and Adomain decomposition methods for solving nonlinear Boussinesq equations

Communications on Applied Nonlinear Analysis

Volumn 15, Issue 3, 2008, Pages 57-70

  Zayed, E M E ^a,c   Nofal, T A ^b   Gepreel, K A ^a  

^a Zagazig University   (Egypt)

^b Taif University ^*  (Egypt)

^c TAIF UNIVERSITY   (Saudi Arabia)

Author keywords

Adomain decomposition method; Homotopy perturbation method; Nonlinear Boussinesq equations; Travelling wave solutions

Indexed keywords

EID: 51349121599     PISSN: 1074133X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (52)

1. T.A.Abassy, M.A.El-Tawil and H.K.Saleh, The solution of KdV and mKdV equations using Adomain Pade approximation, Int. J. Nonlinear Sci. Numer. Simul., 5 (2004), 327-340.
2. H.A.Abdusalam, On an improved complex tanh-function method, Int. J. Nonlinear Sci. Numer. Simul., 6 (2005), 99-106.
3. G.Adomain and R.Rach, Analytic solution of nonlinear boundary value problems in several dimensions by decomposition, J. Math. Anal. Appl., 174 (1993), 118-137.
4. E.Babolian, J.Biazar and A.R.Vahidi, A new computational method for Laplace transforms by decomposition method, Appl. Math. Comput., 150 (2004), 841-846.
5. E.Babolian, J.Biazar and A.R.Vahidi, Solution of a system of nonlinear equations by Adomain decomposition method, Appl. Math. Comput., 150 (2004), 847-854.
6. S.F.Deng, Bäcklund transformation and soliton solutions for KP equation, Chaos, Solitons & Fractals, 25 (2005), 475-480.
7. L. De-Sheng, G. Feng and Z. Hong-Qing, Solving the (2+1) dimensional higher order Broer-Kaup system via a transformation and tanh- function method, Chaos, Solitons & Fractals, 20 (2004), 1021-1025.
8. Gh.E. Draganescu and V. Capalnasan, Nonlinear relaxation phenomena in polycrystalline solids, Int. J. Nonlinear Sci. Numer. Simul., 4 (2003), 219-226.
9. S.M. El-Sayed, The decomposition method for studying the Klein- Gordon equation, Chaos, Solitons & Fractals, 18 (2003), 1025-1030.
10. S.A.El-Wakil and M.A.Abdu, New applications of Adomain decomposition method, Chaos, Solitons & Fractals, 33 (2007), 513-522.
11. E.Fan, Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled mKdV equation, Phys. Letters A, 282 (2001), 18-22.
12. D.D.Ganji and M.Rafei, Solitary wave solution for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method, Phys. Letters A, 356 (2006), 131-137.
13. I.Hashim and M.S.H.Chowdhury, Adaptation of homotopy perturbation method for numeric- analytic solution of system of O.D.Es, Phys. Letters A, 372 (2008), 470-481.
14. J.H.He, Variational principles for some nonlinear PDEs with variable coefficients, Chaos, Solitons & Fractals, 19 (2004), 847-851.
15. J.H. He, Variational iteration method: a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech., 34 (1999), 699-708.
16. J.H. He, Modified Lindsted-Poincare methods for some strongly nonlinear oscillations, Part I: expansion of a constant, Int. J. Nonlinear Mech., 37 (2002), 309-314.
17. J.H. He, Modified Lindsted-Poincare methods for some strongly nonlinear oscillations, Part II: a new transformation, Int J. Nonlinear Mech., 37 (2002), 315-320.
18. J.H. He, Modified Lindsted-Poincare methods for some strongly nonlinear oscillations, Part III: double series expansion, Int. J. Nonlinear Sci. Numer. Simul., 2 (2001), 317-320.
19. J.H. He Gongcheng Yu Kexue Zhong de jinshi feixianxing feixi fangfa (in Chinese), Henan Science and Technology Press, Zhengzhou 2002.
20. J.H. He, Determination of limit cycles for strongly nonlinear oscillators, Phys. Rev. Lett., 90 (2003), 174301-174303.
21. J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput., 151 (2004), 287-292.
22. J.H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput., 156 (2004), 527-539.
23. J.H. He, Asymptolology by homotopy perturbation method, Appl. Math. Comput., 156 (2004), 591-596.
24. J.H. He, Homotopy perturbation method for bifurcation of nonlinear wave equations, Int. J. Nonlinear Sci. Numer. Simul., 6,(2005), 207-208.
25. J.H. He, Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals, 26 (2005), 695-700.
26. J.H. He, Homotopy perturbation technique, J. Comput. Math. Appl. Mech. Eng., 178 (1999), 257-262.
27. M.Inc, On numerical soliton solutions for the Kaup-Kupershmidt equation and convergence analysis of the decomposition method, Appl. Math. Comput., 172 (2006), 72-85.
28. M.Javidi and A.Y.Jalilian, Exact solitary wave solutions of Boussinesq equation by VIM., Chaos, Solitons & Fractals, 36 (2008), 1256-1260.
29. M.Javidi and A.Y.Jalilian, Exact and numerical solitary wave solutions of generalized Zakharov equation by VIM, Chaos, Solitons & Fractals, 36 (2008), 309-313.
30. D. Kaya and S.M. El-Sayed, An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons & Fractals, 17 (2003), 869-877.
31. D. Kaya, An explicit and numerical solutions of some fifth- order KdV equation by decomposition method, Appl. Math. Comput., 144 (2003), 353-363.
32. H.M. Liu, Generalized variational principles for ion acoustic plasma waves by He's semi-inverse method, Chaos, Solitons & Fractals, 23, (2005), 573-576.
33. H.M. Liu, Variational approach to nonlinear Electrochemical system, Int. J. Nonlinear Sci. Numer. Simul. 5, (2004), 95-96.
34. H.M. Liu, Approximate period of nonlinear oscillators with discontinuous by modified Lindstedt-Poincare method, Chaos, Solitons & Fractals, 23, (2005), 577-579.
35. D.Lu, Jacobi elliptic function solution for two variant Boussinesq equations, Chaos, Solitons & Fractals, 24, (2005), 1373-1385.
36. S.Ma and Q.Lu, Dynamical biforcation for a predator-prey Meta population model with delay, Int. J. Nonlinear Sci. Numer. Simul., 6 (2005), 13-18.
37. A.M.Meson and F.Vericat, Variational analysis for the multi-fractal spectra of local entropies and Lyapunov exponents, Chaos, Solitons & Fractals, 19, (2004), 1031-1038.
38. O.Pashaev and G.Tanoglu, Vector shoch soliton and Hirota bilinear method, Chaos, Solitons & Fractals, 26, (2005), 95-105.
39. A.Sadighi and D.D.Ganji, Analytic treatment of linear and nonlinear Schrodinger equation: A study with homtopy perturbation and Adomain decomposition methods, Phys. Letters A, 372 (2008), 465-469.
40. J.Shen and W.Xu, Bifurcations of smooth and non-smooth travelling wave solutions of the Degasperis-Procesi equation, Int. J. Nonlinear Sci. Numer. Simul., 5 (2004), 397-402.
41. M.I.Syam, Adomain decomposition method for approximating the solution of the KdV equation, Appl. Math. Comput., 162 (2005), 1465-1473.
42. G. Tsigaridas, A. Fragos, I. Polyzos and M. Fakis et al., Evolution of near soliton initial conditions in nonlinear wave equations through their Bäcklund transforms, Chaos, Solitons & Fractals, 23 (2005), 1841-1854.
43. V.O. Vakhnenko, E.J. Parkes and A.J. Morrison, A Bäcklund transformation and the inverse scattering transform method for the generalizied Vakhnenko equation, Chaos, Solitons & Fractals, 17 (2003), 683-692.
44. A.M.Wazwaz, A comparison between Adomain decomposition method and Taylor series method in the series solutions, Appl. Math. Comput., 97 (1998), 37-44.
45. A.M.Wazwaz and S.M.El-Sayed, A new modification of Adomain decomposition method for linear and nonlinear operators, Appl. Math. Comput., 122 (2001), 393-405.
46. A.M.Wazwaz, A study on linear and nonlinear Schrodinger equations by VIM, Chaos, Solitons & Fractals 37, (2008), 1136-1142.
47. A.M.Wazwaz, The decomposition method applied to systems of PDEs and to the reaction-diffusion Brusselator model, Appl. Math. Comput., 110 (2000), 251-264.
48. Q.Wang, Homotopy perturbation method for fractional KdV-Burger equation, Chaos, Solitons & Fractals, 35 (2008), 843-850.
49. E.M.E.Zayed, Hassan A. Zedan and Khaled A. Gepreel "Group analysis and modified extended Tanh- function to find the invariant solutions and soliton solutions for nonlinear Euler equations" Int. J. Nonlinear Sci. Numer. Simul., 5 (2004), 221-234.
50. Y.Zhang and J.Xu, Classification and computation of non-resonant double Hopf bifurcation, Int. J. Nonlinear Sci. Numer. Simul., 6 (2005), 63-68.
51. Z.Zhang and Q.Bi, Bifurcations of a generalized Camassa -Holm equation, Int. J. Nonlinear Sci. Numer. Simul., 6 (2005), 81-86.
52. Y.Zheng and Y.Fu, Effect of damage on bifurcation and chaos of viscoelastic plates, Int. J. Nonlinear Sci. Numer. Simul., 6 (2005), 87-91.