Skip to main content
SpringerLink
Log in

On using the modified variational iteration method for solving the nonlinear coupled equations in the mathematical physics

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

This paper applied the modified variational iteration method to the nonlinear coupled partial differential equations via the generalized nonlinear Hirota Satsuma coupled KdV equations, the nonlinear coupled Kortewge–de Vries KdV equations and the nonlinear shallow water equations together with the initial conditions. The proposed modification is made by introducing Adomian’s polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discritization, liberalization, perturbation, or restrictive assumptions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abbaoui K., Cherruault Y.: Convergence of Adomian’s method applied to differential equations. Appl. Math. Comput. 28, 103–109 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abbaoui K., Cherruault Y.: New ideas for proving convergence of decomposition methods. Appl. Math. Comput. 29, 103–108 (1995)

    MATH  MathSciNet  Google Scholar 

  3. Abbaoui K., Cherruault Y.: The decomposition method applied to Cauchy problem. Kybernetes 28, 68–74 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Abbaoui K., Cherruault Y.: Convergence of Adomian’s method applied to nonlinear equations. Math. Comput. Model. 20, 69–73 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Abbasbandy S.: A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials. J. Comput. Appl. Math. 20, 759–763 (2007)

    Google Scholar 

  6. Abbasbandy S.: Numerical solution of nonlinear Klein–Gordon equations by variational iteration method. Int. J. Num. Math. Eng. 70, 876–881 (2007)

    Article  MathSciNet  Google Scholar 

  7. Ablowitz M.J., Clarkson P.A.: Solitons, Nonlinear Evolution Equation and Inverse Scattering. Cambridge University Press, New York (1991)

    Book  Google Scholar 

  8. Alabdullatic M., Abdusalam H.A.: Adomian decomposition method for nonlinear reaction diffusion system of Lotka-Volterra type. Int. J. Math. Forum 2, 87–96 (2007)

    Google Scholar 

  9. Al-Khaled K., Al-Refai M., Alawneh A.: Traveling wave solutions using the variational method and tanh method for nonlinear coupled equations. Appl. Math. Comput. 202, 233–242 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Basto M., Semiao V.: Numerical study of modified Adomian’s method applied to Burgers equation. Comput. Appl. Math. 206, 927–949 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cherruault Y., Saccomandi G., Some B.: New results for convergence of Adomian’s method applied to integral equations. Math. Comput. Model. 16, 85–93 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Deng S.F.: Bäcklund transformations and soliton solutions for K P equation. Chaos Solitons Fractals 25, 475–480 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fan E.G.: Extended tanh function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ganji D.D., Rafei M.: Solitary wave solutions for a generalized Hirota- Satsuma coupled KdV equation by homotopy perturbation method. Phys. Lett. A 356, 131–137 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ganji D.D., Nourollahi M., Rostamian M.: A comparison of variational iteration method with Adomian’s decomposition method in some highly nonlinear equations. Int. J. Sci. Tech. 2, 179–188 (2007)

    Google Scholar 

  16. Guellal S., Cherruault Y.: Application of the decomposition method to identify the disturbed parameters of the elliptical equation. Math. Comput. Model. 4, 51–55 (1995)

    Article  MathSciNet  Google Scholar 

  17. He J.H.: Variational iteration method-a kind of nonlinear analytical technique: some examples. Int. J. Nonlinear Mech. 34, 699–708 (1999)

    Article  MATH  Google Scholar 

  18. He J.H.: A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Nonlinear Mech. 35, 37–43 (2000)

    Article  MATH  Google Scholar 

  19. He J.H.: New interpretation of homotopy method. Int. J. Modern Phys. B 20, 2561–2568 (2006)

    Article  Google Scholar 

  20. He J.H.: Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 6, 207–208 (2005)

    Google Scholar 

  21. He J.H.: Modified Lindstedt poincare methods for some strongly nonlinear oscillations part I: expansion of a constant. Int. J. Nonlinear Mech. 37, 309–314 (2002)

    Article  MATH  Google Scholar 

  22. He J.H.: Modified Lindstedt poincare methods for some strongly nonlinear oscillations part II: a new transformation. Int. J. Nonlinear Mech. 37, 315–320 (2002)

    Article  MATH  Google Scholar 

  23. He J.H.: The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. Math. Comput. 151, 287–292 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Inc M.: On numerical soliton solution of the Kaup-Kupershmidt equation and convergence analysis of the decomposition method. Appl. Math. Comput. 172, 72–85 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Inokuti M., Sekine H., Mura T.: Gerenral use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nasser, S. (eds) Variational Method in the Mechanics of Solids, pp. 156–162. Pergamon, New York (1978)

    Google Scholar 

  26. Ismail H.N.A., Raslan K.R.: Solitary wave solutions for the general KDV equation by the Adomian decomposition method. Appl. Math. Comput. 154, 17–29 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mavoungou T., Cherruault Y.: Convergence of Adomian’s method and application to nonlinear partial differential equations. Kybernetes 21, 13–25 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  28. Mehdi D., Asgar H.: The solution of coupled Burger’s equation using Adomian–Pade technique. Appl. Math. Comput. 189, 1034–1047 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  29. Noor M.A., Mohyud-Din S.T.: Solution of singular and nonsingular initial and boundary value problems by modified variational iteration method. Math. Prob. Eng. 10, 1–23 (2008)

    MATH  Google Scholar 

  30. Ramos J.I.: On the variational iteration method and other iterative techniques for nonlinear differential equations. Appl. Math. Comput. 199, 39–69 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Rogers C., Schief W.K.: Bäcklund and Darboux transformations, Geometry and Modern Applications in Soliton Theory. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  32. Tsigaridas G., Fragos A., Polyzos I., Fakis M., Ioannou A., Giannetas V., Persephonis P.: Evolution of near-soliton initial conditions in nonlinear wave equations through their Bäcklund transformation. Chaos Solitons Fractals 23, 1841–1854 (2005)

    MATH  MathSciNet  Google Scholar 

  33. Vakhnenko V.O., Parkes E.J., Morrison A.J.: A Bäcklund transformation and the inverse scattering transformation for the generalized Vakhnenko equation. Chaos Solitons Fractals 17, 683–693 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wang M.L., Li X.Z.: Applications of F-expansion to periodic wave solutions for a new hamiltonian amplitude equation. Chaos Solitons Fractals 24, 1257–1268 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wu X.H., He J.H.: Parameter expanding methods for strongly nonlinear oscillators. Comput. Math. Appl. 54, 966–986 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zayed E.M.E., Zedan H.A., Gepreel K.A.: Group analysis and modified extended tanh-function to find the inveriant solutions and soliton solutions for nonlinear Euler equation. Int. J. Nonlinear Sci. Numer. Simul. 5, 221–234 (2004)

    Google Scholar 

  37. Zayed E.M.E., Zedan H.A., Gepreel K.A.: On the soltiary wave solutions for nonlinear Hirota-Satsuma coupled KdV of equations. Chaos Solitons Fractals 22, 285–303 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Zayed E.M.E., Abourabia A., Gepreel K.A., Horbaty M.M.: On the rational solitary wave solutions for the nonlinear Hirota-Satsuma coupled KdV system. Appl. Anal. 85, 751–768 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zayed E.M.E., Gepreel K.A., Horbaty M.M.: Exact solutions for some nonlinear differential equations using complex hyperbolic function methods. Appl. Anal. 87, 509–522 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zayed E.M.E., Nofal T.A., Gepreel K.A.: Homotopy perturbation and Adomian decomposition methods for solving nonlinear Boussinesq equations. Commun. Appl. Nonlinear Anal. 15, 57–70 (2008)

    MATH  MathSciNet  Google Scholar 

  41. Zayed E.M.E., Nofal T.A., Gepreel K.A.: The homotopy perturbation method for solving nonlinear Burgers and new coupled modified Korteweg-de Vries equations. Z. Naturforsch A 63a, 627–633 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

    Elsayed M. E. Zayed

  2. Department of Basic Sciences, Higher Technological Institute, Tenth of Ramadan City, Egypt

    Hanan M. Abdel Rahman

Authors
  1. Elsayed M. E. Zayed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hanan M. Abdel Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elsayed M. E. Zayed.

Additional information

Communicated by Editor in Chief.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zayed, E.M.E., Rahman, H.M.A. On using the modified variational iteration method for solving the nonlinear coupled equations in the mathematical physics. Ricerche mat. 59, 137–159 (2010). https://doi.org/10.1007/s11587-010-0075-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-010-0075-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation