Solution of the Blasius Equation by Using Adomian Kamal Transform

  • Rachana KhandelwalEmail author
  • Padama Kumawat
  • Yogesh Khandelwal
Original Paper


In this article, we present solution of Blasius differential equation with condition at infinity and convert the series solution into rational function by using Pad\( \breve{\hbox{e}} \)s approximation. A new method is introduced, called Adomian Kamal transform method, which is a combination of Adomian decomposition method and Kamal transform, for handling a differential equation of mixing layer that arises in viscous incompressible fluid. It offered not only the numerical values, but also the power series close-form solutions.


Blasuis equation Pad\( \breve{\hbox{e}} \)s approximation Adomian decomposition Kamal transform method 



I thank the reviewers for helpful comments.


  1. 1.
    Baleanu, D., Inc, M.: Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation. De Gruyter 16, 364–370 (2018)Google Scholar
  2. 2.
    Baleanu, D., Inc, M.: Space-time fractional Rosenou–Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws. Adv. Differ. Equ. 46, 1–14 (2018). MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baleanu, D., Inc, M.: Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation. De Gruyter 16, 302–310 (2018)Google Scholar
  4. 4.
    Tchier, F., Inc, M.: Time fractional third-order variant Boussinesq system: symmetry analysis, explicit solutions, conservation laws and numerical approximations. Eur. Phys. J. Plus. 133, 1–15 (2018)CrossRefGoogle Scholar
  5. 5.
    Inc, M., Yusuf, A.: Investigation of the logarithmic-KdV equation involving Mittag-Leffler type kernel with Atangana-Baleanu derivative. J. Phys. A (2018). MathSciNetCrossRefGoogle Scholar
  6. 6.
    Inc, M., Yusuf, A.: Fractional optical solitons for the conformable space–time nonlinear Schrodinger equation with Kerr law nonlinearity. Opt. Quant. Electron. 50(3), 139 (2018)CrossRefGoogle Scholar
  7. 7.
    Yusuf, A., Inc, M., Aliyu, A.I., Baleanu D.: Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equation. Adv. Differ. Equ. 2018, 319 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Aliyu, I.A., Inc, M.: A fractional model of vertical transmission and cure of vector-borne diseases pertaining to the Atangana-Baleanu fractional derivatives. Chaos Solitons Fractals 116, 268–277 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Inc, M., Aliyu, I.A.: Gray optical soliton, linear stability analysis and conservation laws via multipliers to the cubic nonlinear Schrödinger equation. Optik 164, 472–478 (2018)CrossRefGoogle Scholar
  10. 10.
    Inc, M., Aliyu, I.A.: Optical solitons, explicit solutions and modulation instability analysis with second-order spatio-temporal dispersion. Eur. Phys. J. Plus. 132, 528 (2017)CrossRefGoogle Scholar
  11. 11.
    Inc, M., Aliyu, I.A.: Optical and singular solitary waves to the PNLSE with third order dispersion in Kerr media via two integration approaches. Optik 163, 142–151 (2018)CrossRefGoogle Scholar
  12. 12.
    Inc, M., Hashemi, S.M.: Exact solutions and conservation laws of the Bogoyavlenskii equation. Acta Phys. Pol. A 133, 1133–1137 (2018)CrossRefGoogle Scholar
  13. 13.
    Wazwaz, A.M.: The variation iteration method for solving two forms of Blasius equation on a Half-infinite domain. Appl. Math. Comput. 188, 485–491 (2007)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ebaid, A., Al-Armani, N.: A new approach for a class of the Blasius problem via a transformation and Adomians method. Abst. Appl. Anal. (2013). MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ogunlaran, M., Sagay-Yusuf, H.: Adomain Sumudu transform method for the Blasius equation. Br. J. Math. Comput. Sci. 14(3), 1–8 (2016)CrossRefGoogle Scholar
  16. 16.
    Abdelilah, K., Hassan, S.: The new integral Kamal transform. Adv. Theor. Appl. Math. 11(4), 451–458 (2016)Google Scholar
  17. 17.
    Khandelwal, R., Kumawat, P.: Kamal decomposition method and its application in solving coupled system of nonlinear PDE’s. Malaya J. Mat. 6(3), 619–625 (2018). CrossRefGoogle Scholar
  18. 18.
    Khandelwal, R., Choudhary, P.: Solution of fractional ordinary differential equation by Kamal transform. Int. J. Stat. Appl. Math. 3(2), 279–284 (2018)Google Scholar
  19. 19.
    Hassan, Y.Q., Zhu, L.M.: A note on the use of modified Adomian decomposition method for solving singular boundary value problems of higher-order ordinary deferential equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3261–3265 (2009)CrossRefGoogle Scholar
  20. 20.
    Wazwaz, A.M.: The modified decomposition method and Padĕ approximants for solving the Thomas-Fermi equation. Appl. Math. Comput. 105(1), 11–19 (1999)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Baker, G.A.: Essentials of Padĕ Approximants. Academic Press, London (1975)zbMATHGoogle Scholar
  22. 22.
    Khandelwal, Y., Umar, A.B.: Solution of the Blasius Equation by using Adomain Mahgoub transform. Int. J. Math. Trends Technol. 56(5), 303–306 (2018)CrossRefGoogle Scholar
  23. 23.
    Motsa, S., Marewo, G.T.: An improved spectral homotopy analysis method for solving boundary layer problems. Bound. Value Probl. (2011)Google Scholar
  24. 24.
    Belgacem, F.B.M., Karaballi, A.A.: Sumudu transform fundamental properties investigations and applications. J. Appl. Math. Stoch. Anal. 1–23 (2006).
  25. 25.
    Belgacem, F.B.M., Karaballi, A.A., Kalla S.L.: Analytical investigations of the Sumudu transform and applications to integral production equations. Math. Probl. Eng. 2003(3), 103–118 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method, p. 60. Springer, Berlin (2013)Google Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsMaharishi Arvind UniversityJaipurIndia

Personalised recommendations