Encyclopedia of Complexity and Systems Science

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| Editors: Robert A. Meyers

Semi-analytical Methods for Solving the KdV and mKdV Equations

  • Doğan Kaya
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DOI: https://doi.org/10.1007/978-3-642-27737-5_305-3


Adomian decomposition method, homotopy analysis method, homotopy perturbation method, and variational perturbation method

These are some of the semi-analytical/numerical methods for solving Ordinary Differential Equation (in short ODE) or Partial Differential Equation (in short PDE) in literature. – Exact solution A solution to a problem that contains the entire physics and mathematics of a problem, as opposed to one that is approximate, perturbative, closed, etc. – Korteweg-de Vries equation The classical nonlinear equations of interest usually admit for the existence of a special type of the traveling wave solutions, which are either solitary waves or solitons. – Modified Korteweg-de Vries This equation is a modified form of the classical KdV equation in the nonlinear term. – Soliton This concept can be regarded as solutions of nonlinear partial differential equations.

Definition of the Subject

In this study, some semi-analytical/numerical methods are applied to solve the...

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Primary Literature

  1. Abbasbandy S (2006) The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 360:109–113ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. Abdou MA, Soliman AA (2005a) New applications of variational iteration method. Phys D 211:1–8MathSciNetCrossRefzbMATHGoogle Scholar
  3. Abdou MA, Soliman AA (2005b) Variational iteration method for solving Burger’s and coupled Burger’s equations. J Comput Appl Math 181:245–251ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. Adomian G (1988) A review of the decomposition method in applied mathematics. J Math Anal Appl 135:501–544MathSciNetCrossRefzbMATHGoogle Scholar
  5. Adomian G, Rach R (1992) Noise terms in decomposition solution series. Comput Math Appl 24:61–64MathSciNetCrossRefzbMATHGoogle Scholar
  6. Adomian G (1994) Solving frontier problems of physics: the decomposition method. Kluwer, BostonCrossRefzbMATHGoogle Scholar
  7. Chowdhury MSH, Hashim I, Abdulaziz O (2009) Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems. Commun Nonlinear Sci Numer Simul 14:371–378ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. Debtnath L (1997) Nonlinear partial differential equations for scientist and engineers. Birkhauser, BostonCrossRefGoogle Scholar
  9. Debtnath L (2007) A brief historical introduction to solitons and the inverse scattering transform – a vision of Scott Russell. Int J Math Edu Sci Technol 38:1003–1028CrossRefzbMATHGoogle Scholar
  10. Domairry G, Nadim N (2008) Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation. Int Commun Heat Mass Trans 35:93–102CrossRefGoogle Scholar
  11. Drazin PG, Johnson RS (1989) Solutions: an introduction. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  12. Edmundson DE, Enns RH (1992) Bistable light bullets. Opt Lett 17:586ADSCrossRefGoogle Scholar
  13. Fermi E, Pasta J, Ulam S (1974) Studies of nonlinear problems. American Mathematical Society, Providence, pp 143–156zbMATHGoogle Scholar
  14. Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg-de Vries equation. Phys Rev Lett 19:1095–1097ADSCrossRefzbMATHGoogle Scholar
  15. Gardner CS, Greene JM, Kruskal MD, Miura RM (1974) Korteweg-de Vries equation and generalizations, VI, methods for exact solution. Commun Pure Appl Math 27:97–133CrossRefzbMATHGoogle Scholar
  16. Geyikli T, Kaya D (2005a) An application for a modified KdV equation by the decomposition method and finite element method. Appl Math Comp 169:971–981MathSciNetCrossRefzbMATHGoogle Scholar
  17. Geyikli T, Kaya D (2005b) Comparison of the solutions obtained by B-spline FEM and ADM of KdV equation. Appl Math Comput 169:146–156MathSciNetzbMATHGoogle Scholar
  18. Hayat T, Abbas Z, Sajid M (2006) Series solution for the upper-convected Maxwell fluid over a porous stretching plate. Phys Lett A 358:396–403ADSCrossRefzbMATHGoogle Scholar
  19. Hayat T, Sajid M (2007) On analytic solution of thin film flow of a fourth grade fluid down a vertical cylinder. Phys Lett A 361:316–322ADSCrossRefzbMATHGoogle Scholar
  20. He JH (1997) A new approach to nonlinear partial differential equations. Commun Nonlinear Sci Numer Simul 2:203–205Google Scholar
  21. He JH (1999a) Homotopy perturbation technique. Comput Math Appl Mech Eng 178:257–262MathSciNetCrossRefzbMATHGoogle Scholar
  22. He JH (1999b) Variation iteration method - a kind of non-linear analytical technique: some examples. Int J Nonlinear Mech 34:699–708ADSCrossRefzbMATHGoogle Scholar
  23. He JH (2000) A coupling method of homotopy technique and a perturbation technique for nonlinear problems. Int J Nonlinear Mech 35:37–43CrossRefzbMATHGoogle Scholar
  24. He JH (2003) Homotopy perturbation method a new nonlinear analytical technique. Appl Math Comput 135:73–79MathSciNetzbMATHGoogle Scholar
  25. He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 20:1141–1199ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. He JH, Wu XH (2006) Construction of solitary solution and compacton-like solution by variational iteration method. Chaos Solitons Fractals 29:108–113ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. Helal MA, Mehanna MS (2007) A comparative study between two different methods for solving the general Korteweg-de Vries equation. Chaos Solitons Fractals 33:725–739ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. Hirota R (1971) Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192–1194ADSCrossRefzbMATHGoogle Scholar
  29. Hirota R (1973) Exact N-solutions of the wave equation of long waves in shallow water and in nonlinear lattices. J Math Phys 14:810–814ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. Hirota R (1976) Direct methods of finding exact solutions of nonlinear evolution equations. In: Miura RM (ed) Bäcklund transformations, vol 515, lecture notes in mathematics. Springer, Berlin, pp 40–86Google Scholar
  31. Inan IE, Kaya D (2006) Some exact solutions to the potential Kadomtsev-Petviashvili equation. Phys Lett A 355:314–318ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. Inan IE, Kaya D (2007) Exact solutions of the some nonlinear partial differential equations. Phys A 381:104–115MathSciNetCrossRefGoogle Scholar
  33. Kaya D (2003) A numerical solution of the sine-Gordon equation using the modified decomposition method. Appl Math Comp 143:309–317MathSciNetCrossRefzbMATHGoogle Scholar
  34. Kaya D (2006) The exact and numerical solitary-wave solutions for generalized modified Boussinesq equation. Phys Lett A 348:244–250ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. Kaya D (2007) Some methods for the exact and numerical solutions of nonlinear evolution equations. Paper presented at the 2nd international conference on mathematics: trends and developments (ICMTD), Cairo, pp 27–30 (in press)Google Scholar
  36. Kaya D, Al-Khaled K (2007) A numerical comparison of a Kawahara equation. Phys Lett A 363:433–439ADSCrossRefzbMATHGoogle Scholar
  37. Kaya D, El-Sayed SM (2003a) An application of the decomposition method for the generalized KdV and RLW equations. Chaos Solitons Fractals 17:869–877ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. Kaya D, El-Sayed SM (2003b) Numerical soliton-like solutions of the potential Kadomtsev-Petviashvili equation by the decomposition method. Phys Lett A 320:192–199ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. Kaya D, El-Sayed SM (2003c) On a generalized fifth order KdV equations. Phys Lett A 310:44–51ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. Kaya D, Inan IE (2005) A convergence analysis of the ADM and an application. Appl Math Comp 161:1015–1025MathSciNetCrossRefzbMATHGoogle Scholar
  41. Khater AH, El-Kalaawy OH, Helal MA (1997) Two new classes of exact solutions for the KdV equation via Bäcklund transformations. Chaos Solitons Fractals 8:1901–1909ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. Korteweg DJ, de Vries H (1895) On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos Mag 39:422–443MathSciNetCrossRefzbMATHGoogle Scholar
  43. Liao SJ (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. Shanghai Jiao Tong University, Ph.D. thesisGoogle Scholar
  44. Liao SJ (1995) An approximate solution technique not depending on small parameters: a special example. Int J Nonlinear Mech 30:371MathSciNetCrossRefzbMATHGoogle Scholar
  45. Liao SJ (2003) Beyond perturbation: introduction to homotopy analysis method. Chapman & Hall/CRC Press, Boca RatonCrossRefGoogle Scholar
  46. Liao SJ (2004a) On the homotopy analysis method for nonlinear problems. Appl Math Comp 147:499MathSciNetCrossRefzbMATHGoogle Scholar
  47. Liao SJ (2004b) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147:499–513MathSciNetzbMATHGoogle Scholar
  48. Liao SJ (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comp 169:1186–1194MathSciNetCrossRefzbMATHGoogle Scholar
  49. Miura RM (1968) Korteweg-de Vries equations and generalizations, I; a remarkable explicit nonlinear transformations. J Math Phys 9:1202–1204ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. Miura RM, Gardner CS, Kruskal MD (1968) Korteweg-de Vries equations and generalizations, II; existence of conservation laws and constants of motion. J Math Phys 9:1204–1209ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. Momani S, Abuasad S (2006) Application of He’s variational iteration method to Helmholtz equation. Chaos Solitons Fractals 27:1119–1123ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. Polat N, Kaya D, Tutalar HI (2006) A analytic and numerical solution to a modified Kawahara equation and a convergence analysis of the method. Appl Math Comp 179:466–472MathSciNetCrossRefzbMATHGoogle Scholar
  53. Ramos JI (2008) On the variational iteration method and other iterative techniques for nonlinear differential equations. Appl Math Comput 199:39–69MathSciNetzbMATHGoogle Scholar
  54. Rayleigh L (1876) On waves. Lond Edinb Dublin. Philos Mag 5:257zbMATHGoogle Scholar
  55. Russell JS (1844) Report on waves. In: Proceedings of the 14th meeting of the British Association for the Advancement of science, BAAS, LondonGoogle Scholar
  56. Sajid M, Hayat T (2008a) Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations. Nonlinear Analy Real World Appl 9:2296–2301MathSciNetCrossRefzbMATHGoogle Scholar
  57. Sajid M, Hayat T (2008b) The application of homotopy analysis method for thin film flow of a third order fluid. Chaos Solitons Fractal 38:506–515ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. Sajid M, Hayat T, Asghar S (2007) Comparison between HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn 50:27–35MathSciNetCrossRefzbMATHGoogle Scholar
  59. Shawagfeh N, Kaya D (2004) Series solution to the Pochhammer-Chree equation and comparison with exact solutions. Comp Math Appl 47:1915–1920MathSciNetCrossRefzbMATHGoogle Scholar
  60. Ugurlu Y, Kaya D (2008) Exact and numerical solutions for the Cahn-Hilliard equation. Comp Math Appl 56:2987–3274CrossRefzbMATHGoogle Scholar
  61. Wazwaz AM (1997) Necessary conditions for the appearance of noise terms in decomposition solution series. J Math Anal Appl 5:265–274zbMATHGoogle Scholar
  62. Wazwaz AM (2002) Partial differential equations: methods and applications. Balkema, RotterdamzbMATHGoogle Scholar
  63. Wazwaz AM (2007a) Analytic study for fifth-order KdV-type equations with arbitrary power nonlinearities. Commun Nonlinear Sci Numer Simul 12:904–909ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. Wazwaz AM (2007b) A variable separated ODE method for solving the triple sine-Gordon and the triple sine-Gordon equations. Chaos Solitons Fractals 33:703–710ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. Wazwaz AM (2007c) The extended tan h method for abundant solitary wave solutions of nonlinear wave equations. App Math Comp 187:1131–1142CrossRefzbMATHGoogle Scholar
  66. Wazwaz AM (2007d) The variational iteration method for solving linear and nonlinear systems of PDEs. Comput Math Appl 54:895–902MathSciNetCrossRefzbMATHGoogle Scholar
  67. Wazwaz AM, Helal MA (2004) Variants of the generalized fifth-order KdV equation with compact and noncompact structures. Chaos Solitons Fractals 21:579–589ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. Whitham GB (1974) Linear and nonlinear waves. Wiley, New YorkzbMATHGoogle Scholar
  69. Zabusky NJ, Kruskal MD (1965) Interactions of solitons in collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243ADSCrossRefzbMATHGoogle Scholar

Books and Reviews

  1. The following, referenced by the end of the paper, is intended to give some useful for further readingGoogle Scholar
  2. For another obtaining of the KdV equation for water waves, see Kevorkian and Cole (1981); one can see the work of the Johnson (1972) for a different water-wave application with variable depth, for waves on arbitrary shears in the work of Freeman and Johnson (1970) and Johnson (1980) for a review of one and two-dimensional KdV equations. In addition to these; one can see the book of Drazin and Johnson (1989) for some numerical solutions of nonlinear evolution equations. In the work of the Zabusky, Kruskal and Deam (F1965) and Eilbeck (F1981), one can see the motion pictures of soliton interactions. See a comparison of the KdV equation with water wave experiments in Hammack and Segur (1974)Google Scholar
  3. For further reading of the classical exact solutions of the nonlinear equations can be seen in the works: the Lax approach is described in Lax (1968); Calogero and Degasperis (1982, A.20), the Hirota’s bilinear approach is developed in Matsuno (1984), the Bäckland transformations are described in Rogers and Shadwick (1982); Lamb (1980, Chap. 8), the Painleve properties is discussed by Ablowitz and Segur (1981, Sect. 3.8), In the book of Dodd, Eilbeck, Gibbon and Morris (1982, Chap. 10) can found review of the many numerical methods to solve nonlinear evolution equations and shown many of their solutionsGoogle Scholar

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Authors and Affiliations

  1. 1.Istanbul Commerce University, Faculty of EngineeringIstanbulTurkey