Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 5 (2010), 89 -- 98


Hossein Jafari and M. A. Firoozjaee

Abstract. A scheme is developed for the numerical study of the Korteweg-de Vries (KdV) and the Korteweg-de Vries Burgers (KdVB) equations with initial conditions by a homotopy approach. Numerical solutions obtained by homotopy analysis method are compared with exact solution. The comparison shows that the obtained solutions are in excellent agreement.

2010 Mathematics Subject Classification: 35A35; 65M99.
Keywords: KDVB equation; Homotopy analysis method; Exact solution.

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  1. M. A. Abdou and A. A. Soliman, Variational iteration method for solving Burgers and coupled Burgers equations. J Comput Appl Math,181(2) (2005), 245--251. MR2146836(2006a:65139). Zbl 1072.65127.

  2. A. A. Soliman, A numerical simulation and explicit solutions of KdV-Burgers' and Lax's seventh-order KdV equations, Solitons and Fractals, Chaos, 29 (2006), 294--302. MR2211466. Zbl 1099.35521.

  3. A. A. Soliman, New numerical technique for Burgers equation based on similarity reductions. In: International conference on computational fluid dynamics, Beijing, China, October 17--20, 2000, 559--566.

  4. M. J. Ablowitz and P.A. Clarkson, Solitons nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. xii+516 pp. MR1149378(93g:35108). Zbl 0762.35001.

  5. A. R. Bahadir, A fully implicit finite--difference scheme for two--dimensional Burgers equations, Appl Math Comput 137 (2003), 131-137. MR1949127(2004a:65097). Zbl 1027.65111.

  6. J. M. Burger, A mathematical model illustrating the theory of turbulence, Adv Appl Mech, I (1948), 171--99.

  7. J. D. Cole, On a quasilinear parabolic equations occurring in aerodynamics, Q Appl Math, 9 (1951), 225-236. MR0042889 (13,178c). Zbl 0043.09902.

  8. A. Coley, et al., editors, Backlund and Darboux transformations. Providence, RI: American Mathematical Society, 2001. MR1870397 (2002g:37001). Zbl 0974.00040.

  9. J. D. Fletcher, Generating exact solutions of the two-dimensional Burgers equations, Int J Numer Meth Fluids, 3(1983), 213--6. Zbl 0563.76082.

  10. E. G. Fan and H. Q. Zhang, A note on the homogeneous balance method, Phys Lett A, 246 (1998), 403--406. Zbl 1125.35308.

  11. E. Fan, Soliton solutions for a generalized Hirota--Satsuma coupled KdV equation and a coupled MKdV equation, Phys Lett A, 282 (2001), 18--22. MR1838205 (2002d:35177). Zbl 0984.37092.

  12. C.S. Gardner, J.M. Green , M.D. Kruska and R.M. Miura, Method for solving the Korteweg-de Vries equation, Phys Rev Lett, 19 (1967), 1095--1097. Zbl 1103.35360.

  13. J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int J Nonlinear Mech, 35 (2000), 37--43. MR1723761 (2000k:65103). Zbl 1091.74012.

  14. J. H. He, Homotopy perturbation method for solving boundary value problems, Phys Lett A, 350 (2006), 87--88. MR2199322.

  15. J. H. He, Some asymptotic methods for strongly nonlinear equations, Int J Mod Phys B, 20 (10) (2006), 1141--1199. MR2251264 (2007c:35151). Zbl 1102.34039.

  16. R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys Rev Lett, 27 (1971), 1192--1194. Zbl 1168.35423.

  17. R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys Lett A, 85 (1981), 407--408. MR0632382 (82j:35128).

  18. P.C. Jain and D.N. Holla, Numerical solution of coupled Burgers D equations, Int J Numer Meth Eng, 12 (1978), 213-222.

  19. D. Kaya, An application of the decomposition method for the KdVb equation, Appl Math Comput, 152 (2004), 279--288. MR2050064. Zbl 1053.65087.

  20. S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method, CRC Press, Boca Raton: Chapman & Hall (2004). MR2058313(2005h:65003). Zbl 1051.76001.

  21. S. J. Liao, On the homotopy anaylsis method for nonlinear problems, Appl Math Comput, 147 (2004),499--513.MR2012589. Zbl 1086.35005.

  22. S. J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Appl Math Comput, 169 (2005), 1186--1194. MR2174713. Zbl 1082.65534.

  23. S. J. Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J Heat Mass Transfer, 48 (2005), 2529--39.

  24. W. Malfeit, Solitary wave solutions of nonlinear wave equations, Am J Phys, 60 (1992),650--654.

  25. J. Satsuma and R. Hirota, A coupled KdV equation is one case of the four--reduction of the KP hierarchy. J Phys Soc Jpn, 51(1982), 3390--3397. MR0687745(84g:58057).

  26. A. A. Soliman, Collocation solution of the Korteweg-de Vries equation using septic splines, Int J Comput Math, 81 (2004), 325--331. MR2174994. Zbl 1058.65113.

  27. C. H. Su and C. S. Gardner, Derivation of the Korteweg de-Vries and Burgers equation, J Math Phys, 10 (1969), 536-539. MR0271526 (42 #6409). Zbl 0283.35020.

  28. M. Wadati, H. Sanuki and K. Konno, Relationships among inverse method, Backlund transformation and an infinite number of conservation laws, Prog Theor Phys, 53 (1975), 419--436. MR0371297 (51 #7516). Zbl 1079.35506.

  29. M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys Lett A, 213 (1996), 279--287. MR1390282 (96m:35289). Zbl 0972.35526.

  30. Y. T. Wu, X. G. Geng, X. B. Hu and S. M. Zhu, A generalized Hirota--Satsuma coupled Korteweg--de Vries equation and Miura transformations, Phys Lett A, 255 (1999), 259--264.MR1691458 (2000c:37109). Zbl 0935.37029.

  31. F. W. Wubs and E.D. de Goede, An explicit--implicit method for a class of time-dependent partial differential equations, Appl Numer Math, 9 (1992), 157--181. MR1147969(92j:65136). Zbl 0749.65068.

  32. C. T. Yan, A simple transformation for nonlinear waves, Phys Lett A, 224 (1996), 77--84. MR1427895 (97i:35161). Zbl 1037.35504.

  33. S. I. Zaki, A quintic B-spline finite elements scheme for the KdVB equation, Comput Meth Appl Mech Eng, 188 (2000), 121--134. Zbl 0957.65088.

Hossein Jafari M. A. Firoozjaee
Department of Mathematics and Computer Science, Department of Mathematics and Computer Science,
University of Mazandaran, University of Mazandaran,
Babolsar, Iran. Babolsar, Iran.
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