Application of the homotopy perturbation method to the modified regularized long-wave equation

Numerical Methods for Partial Differential Equations

Volumn 26, Issue 2, 2010, Pages 399-411

  Achouri, Talha ^a   Omrani, Khaled ^a  

^a INSTITUT SUPÉRIEUR DES SCIENCES APPLIQUÉES ET DE TECHNOLOGIE DE SOUSSE   (Tunisia)

Author keywords

Conservation laws; Homotopy perturbation method; MRLW equation; Solitary waves solution

Indexed keywords

EID: 76649105494     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20441     Document Type: Article

References (44)

1. 1 D. H. Peregrine, Calculations of the development of an undular bore, J Fluid Mech 25 (1966), 321–330.
2. 2 T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos Trans Soc Lond A 272 (1972), 47–78.
3. 3 T. Achouri, N. Khiari, and K. Omrani, On the convergence of difference schemes for the Benjamin‐Bona‐Mahony (BBM) equation, Appl Math Comput 182 (2006), 999–1005.
4. 4 K. Omrani and M. Ayadi, Finite difference discretization of the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation, Numer Methods Partial Differential Eq 24 (2008), 239–248.
5. 5 A. K. Khalifa, K. R. Raslan, and H. M. Alzubaidi, A finite difference scheme for the MRLW and solitary wave interactions, Appl Math Comput 189 (2007), 346–354.
6. 6 T. Achouri, M. Ayadi, and K. Omrani, A fully Galerkin method for the Damped Generalized Regularized Long‐Wave (DGRLW) equation, Numer Methods Partial Differential Eq 25 (2009), 668–684.
7. 7 T. Kadri, N. Khiari, F. Abidi, and K. Omrani, Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation, Numer Methods Partial Differential Eq 24 (2008), 1501–1516.
8. 8 A. K. Khalifa, K. R. Raslan, and H. M. Alzubaidi, A collocation method with cubic B‐splines for solving the MRLW equation, J Comput Appl Math 212 (2008), 406–418.
9. 9 L. R. T. Gardner, G. A. Gardner, F. A. Ayoub, and N. K. Amein, Approximations of solitary waves of the MRLW equation by B‐spline finite element, Arab J Sci Eng 22 (1997), 183–193.
10. 10 T. Achouri and K. Omrani, Numerical solutions for the damped generalized regularized long‐wave equation with a variable coefficient by Adomian decomposition method, Commun Nonlinear Sci Numer Simul 14 (2009), 2025–2033.
11. 11 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.
12. 12 D. Kaya, A numerical simulation of solitary‐wave solutions of the generalized regularized long‐wave equation, Appl Math Comput 149 (2004), 833–841.
13. 13 A. K. Khalifa, K. R. Raslan, and H. M. Alzubaidi, Numerical study using ADM for the modified regularized long wave equation, Appl Math Mode 32 (2008), 2962–2972.
14. 14 J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl Math Comput 151 (2004), 287–292.
15. 15 J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals 26 (2005), 695–700.
16. 16 J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int J Nonlinear Sci Numerl Simul 6 (2005), 207–208.
17. 17 J. H. He, Homotopy perturbation method for solving boundary value problems, Phy Lett A 350 (2006), 87–88.
18. 18 J. H. He, Homotopy perturbation technique, Comput Meth Appl Mech Eng 178 (1999), 257–262.
19. 19 J. H. He, A coupling method of a homotopy technique and a perturbation technique for non‐linear problems, Int J Nonlinear Mech 35 (2000), 37–43.
20. 20 J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl Math Comput 135 (2003), 73–79.
21. 21 J. H. He, Some asymptotic methods for strongly nonlinear equations, Inte J Mod Phys B 20 (2006), 1141–1199.
22. 22 J. H. He, New interpretation of homotopy perturbation method, Int J Mod Phys B 20 (2006), 2561–2568.
23. 23 J. H. He, A modified Newton‐Raphson method, Commun Numer Meth Eng 20 (2004), 801–805.
24. 24 A. Yildirim, On the solution of the nonlinear Korteweg‐de Vries equation by the homotopy perturbation method, Commun Numer Meth Eng (2009), [Epub ahead of print (DOI: 10.1002/cnm.1146)].
25. 25 T. Ozis and A. Yildirim, Traveling wave solution of Korteweg‐de Vries equation using H's homotopy perturbation method, Int J Nonlinear Sci Numer Simul 8 (2007), 239–242.
26. 26 A. Yildirim, Exact solutions of nonlinear differential‐difference equations by H's homotopy perturbation method, Int J Nonlinear Sci Numer Simul 9 (2008), 111–114.
27. 27 J. Biazar and H. Ghazvini, Exact solutions for nonlinear Schrödinger equations by H's homotopy perturbation method, Phys Lett A 366 (2007), 79–84.
28. 28 D. D. Ganji, The application of H's homotopy perturbation method to nonlinear equations arising in heat transfer, Phys Lett A 355 (2006), 337–341.
29. 29 S. Abbasbandy, Numerical solutions of the integral equations: homotopy perturbation method and Adomia's decomposition method, Appl Math Comput 173 (2006), 493–500.
30. 30 Z. Odibat and S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals 36 (2008), 167–174.
31. 31 A. M. Siddiqui, R. Mahmood, and Q. K. Ghori, Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane, Chaos Solitons Fractals 35 (2008), 140–147.
32. 32 L. Cveticanin, Homotopy‐perturbation method for pure nonlinear differential equation, Chaos Solitons Fractals 30 (2006), 1221–1230.
33. 33 J. Biazar and H. Ghazvini, H's homotopy perturbation method for solving system of Volterra integral equations of the second kind, Chaos Solitons Fractals 39 (2009), 770–777.
34. 34 J. Biazar and H. Ghazvini, Numerical solution for special non‐linear Fredholm integral equation by HPM, Appl Math Comput 195 (2008), 681–687.
35. 35 M. Dehghan and F. Shakeri, Use of H's homotopy perturbation method for solving a partial differential equation arising in modeling of flow in porous media, J Porous Media 11 (2008), 765–778.
36. 36 F. Shakeri and M. Dehghan, Solution of the delay differential equations via homotopy perturbation method, Math Comput Modell 48 (2008), 486–498.
37. 37 A. Saadatmandi, M. Dehghan, and A. Eftekhari, Application of H's homotopy perturbation method for non‐linear system of second‐order boundary value problems, Nonlinear Anal: Real World Appl 10 (2009), 1912–1922.
38. 38 M. Dehghan and F. Shakeri, Solution of an integro‐differential equation arising in oscillating magnetic fields using H's homotopy perturbation method, Prog Electromagn Res (PIER) 78 (2008), 361–376.
39. 39 F. Shakeri and M. Dehghan, Inverse problem of diffusion equation by H's homotopy perturbation method, Phys Scr 75 (2007), 551–556.
40. 40 M. Dehghan and F. Shakeri, Solution of a partial differential equation subject to temperature overspecification by H's homotopy perturbation method, Phys Scr 75 (2007), 778–787.
41. 41 A. Yildirim and T. Özis, Solutions of Singular IVPs of Lane‐Emden type by Homotopy Perturbation Method, Phy Lett A 369 (2007), 70–76.
42. 42 A. Yildirim, Solution of BVPs for Fourth‐Order Integro‐Differential Equations by using Homotopy Perturbation Method, Comput Math Appl 56 (2008), 3175–3180.
43. 43 A. Yildirim, H's Homotopy Perturbation Method for nonlinear differential‐difference equations, Int J Comput Math (2009), in press.
44. 44 A. Yildirim, The Homotopy Perturbation Method for approximate solution of the modified KdV equation, Z Naturforsch A, J Phys Sci 63a (2008), 621–626.