, Volume 54, Issue 2, pp 515–526 | Cite as

A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method

  • F. Z. GengEmail author
  • S. P. Qian


In this paper, a simple numerical method is proposed for solving singularly perturbed boundary layers problems exhibiting twin boundary layers. The method avoids the choice of fitted meshes. Firstly the original problem is transformed into a new boundary value problem whose solution does not change rapidly by a proper variable transformation; then the transformed problem is solved by using the reproducing kernel method. Two numerical examples are given to show the effectiveness of the present method.


Reproducing kernel method Two boundary layers Singularly perturbed problems 

Mathematics Subject Classification

65L11 65D15 



The work was supported by the National Natural Science Foundation of China (Nos. 11201041, 11026200), the Special Funds of the National Natural Science Foundation of China (No. 11141003) and Qing Lan Project of Jiangsu Province.


  1. 1.
    Natesan, S., Jayakumar, J., Vigo-Aguiar, J.: Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers. J. Comput. Appl. Math. 158, 121–134 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kadalbajoo, M.K., Arora, P., Gupta, V.: Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers. Comput. Math. Appl. 61, 1595–1607 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Rai, P., Sharma, K.K.: Nnmerical method for singularly perturbed differential-difference equation with turning point. Int. J. Pure Appl. Math. 73, 451–470 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Rai, P., Sharma, K.K.: Numerical study of singularly perturbed differential-difference equation arising in the modeling of neuronal variability. Comput. Math. Appl. 63, 118–132 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Rai, P., Sharma, K.K.: Numerical analysis of singularly perturbed delay differential turning point problem. Appl. Math. Comput. 218, 3483–3498 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Phaneendra, K., Rakmaiah, S., Chenna Krishna Reddy, M.: Numerical treatment of singular perturbation problems exhibiting dual boundary layers. Ain Shams Eng. J. (2015). doi: 10.1016/j.asej.2015.02.012
  7. 7.
    Becher, S., Roos, H.-G.: Richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers. J. Comput. Appl. Math. 290, 334–351 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Geng, F.Z., Qian, S.P.: Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers. Appl. Math. Lett. 26, 998–1004 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Geng, F.Z., Cui, M.G.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327, 1167–1181 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cui, M.G., Geng, F.Z.: Solving singular two-point boundary value problem in reproducing kernel space. J. Comput. Appl. Math. 205, 6–15 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cui, M.G., Lin, Y.Z., et al.: Nonlinear Numerical Analysis in Reproducing Kernel Space. Nova Science Publishers, Inc., Commack (2009)zbMATHGoogle Scholar
  13. 13.
    Li, X.Y., Wu, B.Y.: Error estimation for the reproducing kernel method to solve linear boundary value problems. J. Comput. Appl. Math. 243, 10–15 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li, X.Y., Wu, B.Y.: A continuous method for nonlocal functional differential equations with delayed or advanced arguments. J. Math. Anal. Appl. 409, 485–493 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang, W.Y., Yamamoto, M., Han, B.: Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation. Inverse Probl. 29, 1–15 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang, W.Y., Han, B., Yamamoto, M.: Inverse heat problem of determining time-dependent source parameter in reproducing kernel space. Nonlinear Anal. Real World Appl. 14, 875–887 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, Y., Su, L., Cao, X., Li, X.: Using reproducing kernel for solving a class of singularly perturbed problems. Comput. Math. Appl. 61, 421–430 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jiang, W., Lin, Y.Z.: Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space. Commun. Nonlinear Sci. Numer. Simulat. 16, 3639–3645 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Inc, M., Akgül, A.: Numerical solution of seventh-order boundary value problems by a novel method. Abstr. Appl. Anal. 2014, 1–9 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mohammadi, M., Mokhtari, R.: Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. J. Comput. Appl. Math. 235, 4003–4014 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Akram, G., Ur Rehman, H.: Numerical solution of eighth order boundary value problems in reproducing Kernel space. Numer. Algorithms 62, 527–540 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Arqub, O.A., Al-Smadi, M., Shawagfeh, N.: Solving Fredholm integro-differentialequations using reproducing kernel Hilbert space method. Appl. Math. Comput. 219, 8938–8948 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Abu, O. A.: Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput. Appl. (2015). doi: 10.1007/s00521-0152110-x
  24. 24.
    Arqub, O.A., Mohammed, A.S., Momani, S., Hayat, T.: Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Comput. (2015). doi: 10.1007/s00500-0151707-4
  25. 25.
    Momani, S., Arqub, O.A., Hayat, T., Al-Sulami, H.: A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Volterra type. Appl. Math. Comput. 240, 229–239 (2014)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Abbasbandy, S., Azarnavid, B., Alhuthali, M.S.: A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems. J. Comput. Appl. Math. 279, 293–305 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ketabchi, R., Mokhtari, R., Babolian, E.: Some error estimates for solving Volterra integral equations by using the reproducing kernel method. J. Comput. Appl. Math. 273, 245–250 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ghasemi, M., Fardi, M., Ghaziani, R.K.: Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space. Appl. Math. Comput. 268, 815–831 (2015)MathSciNetGoogle Scholar
  29. 29.
    Geng, F.Z., Qian, S.P.: Modified reproducing kernel method for singularly perturbed boundary value problems with a delay. Appl. Math. Model. 39, 5592–5597 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Department of MathematicsChangshu Institute of TechnologyChangshuPeople’s Republic of China

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