Numerical Algorithms

, Volume 60, Issue 3, pp 463–481

An improved spectral homotopy analysis method for MHD flow in a semi-porous channel

  • Sandile Sydney Motsa
  • Stanford Shateyi
  • Gerald T. Marewo
  • Precious Sibanda
Original Paper


In this paper we report on a novel method for solving systems of highly nonlinear differential equations by blending two recent semi-numerical techniques; the spectral homotopy analysis method and the successive linearisation method. The hybrid method converges rapidly and is an enhancement of the utility of the original spectral homotopy analysis method (Motsa et al., Commun Nonlinear Sci Numer Simul 15:2293–2302, 2010; Computer & Fluids 39:1219–1225, 2010) and an improvement on other recent semi-analytical techniques. We illustrate the application of the method by solving a system of nonlinear differential equations that govern the problem of laminar viscous flow in a semi-porous channel subject to a transverse magnetic field. A comparison with the numerical solution confirms the validity and accuracy of the technique and shows that the method converges rapidly and gives very accurate results.


Improved spectral homotopy analysis method Coupled nonlinear equations Laminar two-dimensional flow Electrically conducting incompressible viscous fluid 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abbasbandy, S.: Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method. Appl. Math. Comput. 145, 887–893 (2003)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Abbasbandy, S.: The application of homotopy analysis method to solve a generalized HirotaSatsuma coupled KdV equation. Phys. Lett. A 361, 478–483 (2007)MATHCrossRefGoogle Scholar
  3. 3.
    Abbasbandy, S., Shivanian, E.: Multiple solutions of mixed convection in a porous medium on semi-infinte interval using pseudo-spectral collocation method. Commun. Nonlinear Sci. Numer. Simul. 16, 2745–2752 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Abbaoui, K., Cherruault, Y.: Convergence of Adomian’s method applied to non-linear equations. Math. Comput. Model. 20, 69–73 (1994)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Abdulaziz, O., Hashim, I., Saif, A.: Series solutions of time-fractional PDEs by homotopy analysis method, differential equations and nonlinear mechanics, vol. 2008, Article ID 686512. doi:10.1155/2008/686512 (2008)
  6. 6.
    Adomian, G.: Nonlinear stochastic differential equations. J Math. Anal. Appl. 55, 441–452 (1976)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Adomian, G.: A review of the decomposition method and some recent results for nonlinear equations. Comput. Math. Appl. 21, 101–127 (1991)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Rafiq, A., Javeria, A.: New iterative methods for solving nonlinear equations by using the modified homotopy perturbation method. Acta Univ. Apulensis 18, 129–137 (2009)MathSciNetMATHGoogle Scholar
  9. 9.
    Awad, F.G., Sibanda, P., Motsa, S.S., Makinde, O.D.: Convection from an inverted cone in a porous medium with cross-diffusion effects. Comput. Math. Appl. 61, 1431–1441 (2011)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Babolian, E., Biazar, J.: Solution of non-linear equations by modified Adomian decomposition method. Appl. Math. Comput. 132, 167–172 (2002)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Babolian, E., Biazar, J., Vahidi, A.R.: On the decomposition method for system of linear equations and system of linear Volterra integral equations. Appl. Math. Comput. 147, 19–27 (2004)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Modified homotopy analysis method for solving systems of second-order BVPs. Commun. Nonlinear Sci. Numer. Simul. 14, 430–442 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Bataineh, A.S., Noorani, M.S.M., Hashim, I.: On a new reliable modification of homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 409–423 (2009)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)MATHGoogle Scholar
  15. 15.
    Chen, X., L, Zheng, Zhang, X.: Convergence of the homotopy decomposition method for solving nonlinear equations. Adv. Dyn. Syst. Appl. 2(1), 59–64 (2007)MathSciNetMATHGoogle Scholar
  16. 16.
    Cherruault, Y.: Convergence of Adomian’s method. Kybernetes 18, 31–38 (1989)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Desseaux, A.: Influence of a magnetic field over a laminar viscous flow in a semi-porous channel. Int. J. Eng. Sci. 37, 1781–1794 (1999)MATHCrossRefGoogle Scholar
  18. 18.
    Don, W.S., Solomonoff, A.: Accuracy and speed in computing the Chebyshev Collocation Derivative. SIAM J. Sci. Comput. 16(6), 1253–1268 (1995)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Ganji, D.D., Nourollahi, M., Rostamian, M.: A comparison of variational iteration method with Adomian’s decompostion method in some highly nonlinear equations. Int. J. Sci. Technol. 2(2), 179–188 (2007)Google Scholar
  20. 20.
    Golbabaia, A., Javidi, M.: Application of homotopy perturbation method for solving eighth-order boundary value problems. Appl. Math. Comput. 191, 334–346 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    He, J.H.: Variational iteration method: a kind of non-linear analytical technique: some examples. Int. J. Non-Linear Mech. 34(4), 699–708 (1999)MATHCrossRefGoogle Scholar
  22. 22.
    He, J.H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114(2–3), 115–123 (2000)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)MATHCrossRefGoogle Scholar
  24. 24.
    He, J.H.: A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int. J. Non-linear Mech. 35, 37–43 (2000)MATHCrossRefGoogle Scholar
  25. 25.
    Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University (1992)Google Scholar
  26. 26.
    Karmishin, A.M., Zhukov, A.I., Kolsov, V.G.: Methods of Dynamics Calculation and Testing for Thin-walled Structures. Mashinostroyenie, Moscow (1990)Google Scholar
  27. 27.
    Kierzenka, J., Shampine, L.: A BVP solver based on residual control and the Matlab PSE. ACM Trans. Math. Softw. 27, 299–316 (2001)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press (2003)Google Scholar
  29. 29.
    Liao, S.J.: Comparison between the homotopy analysis method and the homotopy perturbation method. Appl. Math. Comput. 169, 1186–1194 (2005)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297–354 (2007)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Liao, S.J.: Notes on the homotopy analysis method: Some definitions and theores. Commun. Nonlinear Sci. Numer. Simul. 14, 983–997 (2009)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2003–2016 (2010)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Li, J.L.: Adomian’s decomposition method and homotopy perturbation method in solving nonlinear equations. J. Comput. Appl. Math. 228, 168–173 (2009)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Liang, S., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation. Commun. Nonlinear Sci. Numer. Simul. 14, 4057–4064 (2009)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Liang, S., Jeffrey, D.J.: An efficient analytical approach for solving fourth order boundary value problems. Comput. Phys. Commun. 180, 2034–2040 (2009)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Makukula, Z., Sibanda, P., Motsa, S.S.: A note on the solution of the Von Kármán equations using series and Chebyshev spectral methods. BVP 2010, Article ID 471793, 1–17 (2010). doi:10.1155/2010/471793 Google Scholar
  37. 37.
    Makukula, Z., Sibanda, P., Motsa, S.S.: A novel numerical technique for two-dimensional Laminar flow between two moving porous walls. Math. Probl. Eng. 2010, Article ID 528956, 1–15 (2010). doi:10.1155/2010/528956 MathSciNetCrossRefGoogle Scholar
  38. 38.
    Makukula, Z.G., Sibanda, P., Motsa, S.S.: On new solutions for heat transfer in a visco-elastic fluid between parallel plates. M3AS 4(4), 221–230 (2010)MathSciNetGoogle Scholar
  39. 39.
    Makukula, Z., Motsa, S.S., Sibanda, P.: On a new solution for the viscoelastic squeezing flow between two parallel plates. JARAM 2, 31–38 (2010)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Chun-Mei, C., Gao, F.: A few numerical methods for solving nonlinear equations. Int. Math. Forum 3(29), 1437–1443 (2008)MathSciNetMATHGoogle Scholar
  41. 41.
    Motsa, S.S., Sibanda, P., Shateyi, S.: A new spectral-homotopy analysis method for solving a nonlinear second order BVP. Commun. Nonlinear Sci. Numer. Simul. 15, 2293–2302 (2010)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Motsa, S.S., Sibanda, P., Awad, F.G., Shateyi, S.: A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem. Comp. Fluid. 39, 1219–1225 (2010)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Motsa, S.S., Sibanda, P.: On the solution of MHD flow over a nonlinear stretching sheet by an efficient semi-analytical technique. Int. J. Numer. Methods Fluids doi:10.1002/fld.2541
  44. 44.
    Motsa, S.S., Shateyi, S.: A new approach for the solution of three-dimensional magnetohydrodynamic rotating flow over a shrinking sheet. Math. Probl. Eng. 2010, Article ID 586340, 1–15 (2010). doi:10.1155/2010/586340 Google Scholar
  45. 45.
    Motsa, S.S., Shateyi, S.: Successive linearisation solution of free convection non-Darcy flow with heat and mass transfer. In: El-Amin, M. (ed.) Advanced Topics in Mass Transfer, pp. 425–438. InTech Open Access Publishers (2011)Google Scholar
  46. 46.
    Mohyud-Din, S.T., Noor, M.A.: Homotopy perturbation method and pad approximants for solving Flierl-Petviashivili equation. Appl. Appl. Math. 3(2), 224–234 (2008)MathSciNetMATHGoogle Scholar
  47. 47.
    Osterle, J.F., J Young, F.: Natural convection between heated vertical plates in a horizontal magnetic field. J. Fluid Mech. 11, 512–518 (1961)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Sajida, M., Hayat, T.: Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations. Nonlinear Anal.: Real World Appl. 9, 2296–2301 (2008)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  50. 50.
    Shateyi, S., Motsa, S.S.: Variable viscosity on magnetohydrodynamic fluid flow and heat transfer over an unsteady stretching surface with Hall effect. BVP 2010, Article ID 257568, 1–20 (2010). doi:10.1155/2010/257568 Google Scholar
  51. 51.
    Trefethen, L.N.: Spectral Methods in MATLAB. SIAM (2000)Google Scholar
  52. 52.
    Umavathi, J.C.: A note on magnetoconvection in a vertical enclosure. Int. J. Non-Linear Mech. 31(3), 371–376 (1996)MATHCrossRefGoogle Scholar
  53. 53.
    Wu, T.M.: A new formula of solving nonlinear equations by Adomian and homotopy methods. Appl. Math. Comput. 172, 903–907 (2006)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Wu, T.M.: A study of convergence on the Newton-homotopy continuation method. Appl. Math. Comput. 168, 1169–1174 (2005)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Yildirim, A.: Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method. Comput. Math. Appl. 56, 3175–3180 (2008)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Ziabakhsh, Z., Domairry, G.: Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 1284–1294 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Sandile Sydney Motsa
    • 1
  • Stanford Shateyi
    • 2
  • Gerald T. Marewo
    • 3
  • Precious Sibanda
    • 1
  1. 1.School of Mathematical SciencesUniversity of KwaZulu-NatalScottsvilleRepublic of South Africa
  2. 2.Department of MathematicsUniversity of VendaThohoyandouRepublic of South Africa
  3. 3.Department of MathematicsUniversity of SwazilandKwaluseniSwaziland

Personalised recommendations