Acta Applicandae Mathematicae

, 108:395 | Cite as

Solution of Delay Differential Equation by Means of Homotopy Analysis Method

  • A. K. Alomari
  • M. S. M. Noorani
  • R. Nazar


The algorithm of approximate analytical solution for delay differential equations (DDE) is obtained via homotopy analysis method (HAM) and modified homotopy analysis method (MHAM). Various examples of linear, nonlinear and system of initial value problems of DDE are solved and the results obtained show that these algorithms are accurate and efficient for the DDE. The convergence of this algorithm is also proved.


Homotopy analysis method Modified homotopy analysis method Convergent solution Delay differential equation 


  1. 1.
    Wikipedia A: Delay differential equation,, 10 April 2008
  2. 2.
    Ulsoy, A.G.: Analytical solution of a system of homogeneous delay differential equations via the Lambert function. In: Proceedings of the American Control Conference, Chicago, IL, June 2000 Google Scholar
  3. 3.
    Evans, D.J., Raslan, K.R.: The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math. 82, 49–54 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Adomian, G., Rach, R.: Nonlinear Stochastic differential delay equation. J. Math. Anal. Appl. 91, 94–101 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Adomian, G., Rach, R.: A nonlinear delay differential equation. J. Math. Anal. Appl. 91, 301–304 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Shakeri, F., Dehghan, M.: Solution of delay differential equations via a homotopy perturbation method. Math. Comput. Model. 48, 486–498 (2008). doi: 10.1016/j.mcm.2007.09.016 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Liao, S.J.: Homotopy analysis method and its application. Ph.D. dissertation, Shanghai Jiao Tong University, Shanghai (1992) Google Scholar
  8. 8.
    Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman and Hall/CRC. Boca Raton (2003) Google Scholar
  9. 9.
    Liao, S.J.: Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput. 169, 1186–1194 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Liao, S.J.: An explicit totally analytic approximation of Blasius viscous flow problems. Int. J. Nonlinear Mech. 34, 759–778 (1999) zbMATHCrossRefGoogle Scholar
  11. 11.
    Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Studies Appl. Math. 117, 239–263 (2006) zbMATHCrossRefGoogle Scholar
  13. 13.
    Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007) zbMATHCrossRefGoogle Scholar
  14. 14.
    Hayat, T., Khan, M.: Homotopy solutions for a generalized second-grade fluid past a porous plate. Nonlinear Dyn. 42, 395–405 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hayat, T., Abbas, Z., Sajid, M.: Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction. Phys. Lett. A 372, 2400–2408 (2008) Google Scholar
  16. 16.
    Hayat, T., Sajid, M., Ayub, M.: On explicit analytic solution for MHD pipe flow of a fourth grade fluid. Commun. Nonlinear Sci. Numer. Simul. 13, 745–751 (2008) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Xu, H., Liao, S.J.: Dual solutions of boundary layer flow over an upstream moving plate. Commun. Nonlinear Sci. Numer. Simul. 13, 350–358 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Abbasbandy, S.: Solitary wave solutions to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method. Nonlinear Dyn. 52, 35–40 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    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). doi: 10.1016/j.cnsns.2007.10.007 CrossRefMathSciNetGoogle Scholar
  20. 20.
    Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Tan, Y., Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul. 13, 539–546 (2008). doi: 10.1016/j.cnsns.2006.03.008 zbMATHCrossRefGoogle Scholar
  22. 22.
    Alomari, A.K., Noorani, M.S.M., Nazar, R.: Solutions of heat-like and wave-like equations with variable coefficients by means of the homotopy analysis method. Chin. Phys. Lett. 25, 589–592 (2008) CrossRefGoogle Scholar
  23. 23.
    Alomari, A.K., Noorani, M.S.M., Nazar, R.: Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2008.01.008 (in press)
  24. 24.
    Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation. Phys. Lett. A 361, 478–483 (2007) zbMATHCrossRefGoogle Scholar
  25. 25.
    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). doi: 10.1016/j.cnsns.2007.09.012 CrossRefMathSciNetGoogle Scholar
  26. 26.
    Bataineh, A.S., Noorani, M.S.M., Hashim, I.: The homotopy analysis method for Cauchy reaction-diffusion problems. Phys. Lett. A 372, 613–618 (2008) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Wu, Y., Cheung, K.: Homotopy solution for nonlinear differential equations in wave propagation problems. Wave Motion. doi: 10.1016/j.wavemoti.2008.07.002 (in press)
  28. 28.
    Johnson, W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Monthly 109(3), 217–234 (2002) zbMATHCrossRefGoogle Scholar
  29. 29.
    Wille, D.R., Baker, C.T.H.: DELSOL-a numerical code for the solution of systems of delay–differential equations. Appl. Numer. Math. 9, 223–234 (1992) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Faculty of Science and TechnologyUniversiti Kebangsaan MalaysiaBangiMalaysia

Personalised recommendations