Numerical Algorithms

, Volume 64, Issue 4, pp 593–605 | Cite as

Determination of optimal convergence-control parameter value in homotopy analysis method

Original Paper


In the framework of the Homotopy Analysis Method (HAM) the so-called convergence-control parameter \(c_{0}\) (Liao (Int J Non-Linear Mech 32:815–822, 1997) originally used the symbol \(\hbar \) to denote the auxiliary parameter. But, \(\hbar \) is well-known as Planck’s constant in quantum mechanics. To avoid misunderstanding, Liao (Commun Nonlinear Sci Numer Simulat 15:2003–2016, 2010) suggest to use the symbol \(c_0\) to denote the basic convergence-control parameter.) has a key role in convergence of obtained series solution of solving non-linear equations. In this paper a modified approach in the determining of the convergence-control parameter value \(c_{0}\) is proposed. The purpose of this paper is to find a proper convergence-control parameter \(c_0\) to get a convergent series solution, or a faster convergent one. This modified approach minimizes the norm of a discrete residual function, systematically, in which seeks to find an optimal value of the convergence-control parameter \(c_0\) at each order of HAM approximation, instead of the so-called \(c_0\)-curve process. The proved theorems and numerical results demonstrate the validity, efficiency, and performance of the proposed approach.


Two-point boundary value problem Homotopy Analysis Method Optimal value Convergence-control parameter 

Mathematics Subject Classifications (2010)

65H20 65L10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Liao, S.J.: A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics. Int. J. Non-Linear Mech. 32, 815–822 (1997)CrossRefMATHGoogle Scholar
  2. 2.
    Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat. 15, 2003–2016 (2010)CrossRefMATHGoogle Scholar
  3. 3.
    Liao, S.J.: On the proposed homotopy analysis technique for nonlinear problems and its applications. Ph. D. dissertation, Shanghai Jiao Tong University (1992)Google Scholar
  4. 4.
    Liao, S.J.: Numerically solving nonlinear problems by the homotopy analysis method. Comput. Mech. 20, 530–540 (1997)CrossRefMATHGoogle Scholar
  5. 5.
    Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman and Hall/CRC Press, Boca Raton (2003)CrossRefGoogle Scholar
  6. 6.
    Liao, S.J.: Notes on the homotopy analysis method: Some definitions and theorems. Commun. Nonlinear Sci. Numer. Simulat. 14, 983–997 (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Liao, S.J.: The Homotopy Analysis Method in Nonlinear Differential Equations. Higher Education Press and Springer, Beijing and Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simulat. 13, 539–546 (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Abbasbandy, S.: Soliton solutions for the 5th-order KdV equation with the homotopy analysis method. Nonlinear Dyn. 51, 83–87 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Wang, Z., Zou, L., Zhang, H.: Applying homotopy analysis method for solving differential-difference equation. Phys. Lett. A 369, 77–84 (2007)CrossRefMATHGoogle Scholar
  11. 11.
    Song, L., Zhang, H.Q.: Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation. Phys. Lett. A 367, 88–94 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Abbasbandy, S.: The application of the homotopy analysis method to solve a generalized Hirota Satsuma coupled KdV equation. Phys. Lett. A 361, 478–483 (2007)CrossRefMATHGoogle Scholar
  13. 13.
    Van Gorder, R.A.: Analytical method for the construction of solutions to the Föppl-von Kármán equations governing deflections of a thin flat plate. Int. J. Non-linear Mech. 47, 1–6 (2012)CrossRefGoogle Scholar
  14. 14.
    Abbasbandy, S., Tan, Y., Liao, S.J.: Newton-homotopy analysis method for nonlinear equations. Appl. Math. Comput. 188, 1794–1800 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Abbasbandy, S., Magyari, E., Shivanian, E.: The homotopy analysis method for multiple solutions of nonlinear boundary value problems. Commun. Nonlinear Sci. Numer. Simulat. 14, 3530–3536 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Izadian, J., Mohammadzade Attar, M., Jalili, M.: Numerical solution of deformation equations in homotopy analysis method. Appl. Math. Sci. 6, 357–367 (2012)MATHGoogle Scholar
  17. 17.
    Motsa, S.S., Sibanda, P., Shateyi, S.: A new spectral homotopy analysis method for solving a nonlinear second order BVP. Commun. Nonlinear Sci. Numer. Simulat. 15, 2293–2302 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liao, S.J.: On the relationship between the homotopy analysis method and Euler transform. Commun. Nonlinear Sci. Numer. Simulat. 15, 1421–1431 (2010)CrossRefMATHGoogle Scholar
  19. 19.
    Hayat, T., Ellahi, R., Ariel, P.D., Asghar, S.: Homotopy solutions for the channel flow of a third grade fluid. Nonlinear Dyn. 45, 55–64 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hayat, T., Naz, R., Sajid, M.: On the homotopy solution for Poiseuille flow of a fourth grade fluid. Commun. Nonlinear Sci. Numer. Simulat. 15, 581–589 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Liang, S., Jeffrey, D.J.: An efficient analytical approach for solving fourth order boundary value problems. Comput. Phys. Commun. 180, 2034–2040 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Liang, S., Jeffrey, D.J.: Approximate solutions to a parameterized sixth order boundary value problem. Comput. Math. Appl. 59, 247–253 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Yabushita, K., Yamashita, M., Tsuboi, K.: An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys. A: Math. Theoretical 40, 8403–8416 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wu, Y.Y., Chueng, K.F.: Homotopy solution for nonlinear differential equations in wave propagation problems. Wave Motion 46, 1–14 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wu, Y.Y., Chueng, K.F.: Two-parameter homotopy method for nonlinear equations. Numer. Algorithms 53, 555–572 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Marinca, V., Herisanu, N.: Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int. Commun. Heat Mass Trans. 35, 710–715 (2008)CrossRefGoogle Scholar
  27. 27.
    Marinca, V., Hersanu, N., Bota, C., Marinca, B.: An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Appl. Math. Lett. 22, 245–251 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Niu, Z.,Wang, C.: A one-step optimal homotopy analysis method for nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat. 15, 2026–2036 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Van Gorder, R.A.: Control of error in the homotopy analysis of semi-linear elliptic boundary value problems. Numer. Algorithms (2012). doi: 10.1007/s11075-012-9554-1 MathSciNetGoogle Scholar
  30. 30.
    Greenspan, D., Casulli, V.: Numerical Analysis for Applied Mathematics, Science, and Engineering. Addison-Wesley (1988)Google Scholar
  31. 31.
    Ha, S.N.: A nonlinear shooting method for two-point boundary value problems. Comput. Math. Appl 42, 1411–1420 (2001)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Doedel, E.: Finite difference methods for nonlinear two-point boundary-value problems. SIAM J. Numer. Anal 16, 173–185 (1979)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Deacon, A.G., Osher, S.: Finite-element method for a boundary-value problem of mixed type. SIAM J. Numer. Anal 16, 756–778 (1979)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Turkyilmazoglu, M.: Some issues on HPM and HAM methods: a convergence scheme. Math. Comput. Model. 53, 1929–1936 (2011)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Turkyilmazoglu, M.: Numerical and analytical solutions for the flow and heat transfer near the equator of an MHD boundary layer over a porous rotating sphere. Int. J. Therm. Sci. 50, 831–842 (2011)CrossRefGoogle Scholar
  36. 36.
    Turkyilmazoglu, M.: Solution of the Thomas-Fermi equation with a convergent approach. Commun. Nonlinear Sci. Num. Simulat. 17, 4097–4103 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran

Personalised recommendations