Skip to main content
SpringerLink
Log in

Solution of coupled system of nonlinear differential equations using homotopy analysis method

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this article, the homotopy analysis method has been applied to solve a coupled nonlinear diffusion–reaction equations. The validity of this method has been successful by applying it for these nonlinear equations. The results obtained by this method have a good agreement with one obtained by other methods. This work illustrates the validity of the homotopy analysis method for the nonlinear differential equations. The basic ideas of this approach can be widely employed to solve other strongly nonlinear problems.

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. Khater, A.H., Helal, M.A., El-Kalaawy, O.H.: Backlund transformations: exact solutions for the KdV and Calogero–Degasperis–Fokas mKdV equations. Math. Models Methods Appl. Sci. 21(8), 719–731 (1998)

    MATH  MathSciNet  Google Scholar 

  2. Wahlquist, H.D., Stabrook, F.B.: Backlund transformation for solutions of the KdV equation. Phys. Rev. Lett. 31, 1386–1390 (1973)

    Article  MathSciNet  Google Scholar 

  3. Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)

    Google Scholar 

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

    Article  Google Scholar 

  5. Baldwin, D., Goktas, U., Hereman, W., Hong, L., Martino, R.S., Miller, J.C.: Symbolic computation of exact solutions in hyperbolic and elliptic functions for nonlinear PDEs. J. Symb. Comput. 37, 669–705 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Wazwaz, A.M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 154(3), 713–723 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Wazwaz, A.M.: Variants of the generalized KdV equation with compact and noncompact structures. Comput. Math. Appl. 47, 583–591 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Wazwaz, A.M.: Distinct variants of the KdV equation with compact and noncompact structures. Appl. Math. Comput. 150, 365–377 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Wazwaz, A.M.: Compactons in a class of nonlinear dispersive equations. Math. Comput. Model. 37(3/4), 333–341 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. El-Wakil, S.A., Labany, S.K., Zahran, M.A., Sabry, R.: New exact solutions for a generalized variable coefficients 2D KdV equation. Chaos Solitons Fractals 19(5), 1083–1086 (2004)

    Article  MathSciNet  Google Scholar 

  11. El-Wakil, S.A., Abulwafa, E.M., Elhanbaly, A., Abdou, M.A.: The extended homogeneous balance method and its applications for a class of nonlinear evolution equations. Chaos Solitons Fractals 33(5), 1512–1522 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wazwaz, A.M.: Solitary wave solutions and periodic solutions for higher-order nonlinear evolution equations. Appl. Math. Comput. 181, 1683–1692 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ding, S., Zhao, X.: Exact traveling wave solutions of the Boussinesq equation. Chaos Solitons Fractals 29, 1032–1036 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. He, J.H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114, 115–123 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic, Boston (1994)

    MATH  Google Scholar 

  16. Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University (1992)

  17. Liao, S.J.: An approximate solution technique which does not depend upon small parameters: a special example. Int. J. Non-Linear Mech. 30, 371–380 (1995)

    Article  MATH  Google Scholar 

  18. Liao, S.J.: An approximate solution technique which does not depend upon small parameters (Part 2): an application in fluid mechanics. Int. J. Non-Linear Mech. 32(5), 815–822 (1997)

    Article  MATH  Google Scholar 

  19. Liao, S.J.: An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int. J. Non-Linear Mech. 34(4), 759–778 (1999)

    Article  MATH  Google Scholar 

  20. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Transf. 48, 2529–2539 (2005)

    Article  Google Scholar 

  22. He, J.H.: Homotopy perturbation method. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)

    Article  MATH  Google Scholar 

  23. El-Wakil, S.A., Abdou, M.A., Elhanbaly, A.: Adomian decomposition method for solving the diffusion–convection–reaction equations. Appl. Math. Comput. 177, 729–736 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. El-Wakil, S.A., Abdou, M.A.: New applications of variational iteration method using Adomian polynomials. Nonlinear Dyn. 52, 41–49 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Tehran, Iran

    Mehdi Ganjiani

  2. Department of Nuclear Engineering, Shahid Beheshti University, Evin, Tehran, 1983963113, Iran

    Hossein Ganjiani

Authors
  1. Mehdi Ganjiani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hossein Ganjiani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehdi Ganjiani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ganjiani, M., Ganjiani, H. Solution of coupled system of nonlinear differential equations using homotopy analysis method. Nonlinear Dyn 56, 159–167 (2009). https://doi.org/10.1007/s11071-008-9386-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-008-9386-7

Keywords

Search

Navigation