Numerical Algorithms

, Volume 66, Issue 3, pp 643–662 | Cite as

Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems

Original Paper


We develop a numerical algorithm for solving singularly perturbed one-dimensional parabolic convection-diffusion problems. The method comprises a standard finite difference to discretize in temporal direction and Sinc-Galerkin method in spatial direction. The convergence analysis and stability of proposed method are discussed in details, it is justifying that the approximate solution converges to the exact solution at an exponential rate. we know that the conventional methods for these problems suffer due to decreasing of perturbation parameter, but the Sinc method handel such difficulty as singularity. This scheme applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence.


Singularly perturbed convection-diffusion equation Sinc-Galerkin method Convergence analysis Stability 

Mathematics Subject Classifications (2010)

65M12 65M60 65M99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bialecki, B.: Sinc-collocation methods for two-point boundary value problems. IMA J. Numer. Anal. 11, 357–375 (1991)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Cai, X., Liu, F.: A Reynolds uniform scheme for singularly perturbed parabolic differential equation. ANZIAM J. 47(EMAC–2005), C633–C648 (2007)MathSciNetGoogle Scholar
  3. 3.
    Clavero, C., Jorge, J.C., Lisbona, F.: Uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems. J. Comput. Appl. Math. 154, 415–429 (2003)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Clavero, C., Gracia, J.L., Stynes, M.: A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems. J. Comput. Appl. Math. 235, 5240–5248 (2011)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Clavero, C., Gracia, J.L., Lisbona, F.: High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type. Numer. Algoritm. 22, 73–97 (1999)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Clavero, C., Gracia, J.L.: A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reactiondiffusion problems. Appl. Math. Comput. 218, 5067–5080 (2012)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Deb, R., Natesan, S.: Higher-order time accurate numerical methods for singularly perturbed parabolic partial differential equations. Int. J. Comput. Math. 86(7), 1204–1214 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    El-Gamel, M.: The sinc-galerkin method for solving singularly-perturbed reaction-diffusion problem. Electron. Trans. Numer. Anal. 23, 129–140 (2006)MATHMathSciNetGoogle Scholar
  9. 9.
    Kadalbajoo, M.K., Gupta, V., Awasthi, A.: A uniformly convergent B-spline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convection-diffusion problem. J. Comput. Appl. Math. 220, 271–289 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Kadalbajoo, M.K., Gupta, V.: Numerical solution of singularly perturbed convection-diffusion problem using parameter uniform B-spline collocation method. J. Math. Anal. Appl 355, 439–452 (2009)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia (1992)CrossRefMATHGoogle Scholar
  12. 12.
    Mukherjee, K., Natesan, S.: Richardson extrapolation technique for singularly perturbed parabolic convection-diffusion problems,Computing 92, 1–32 (2011)MATHMathSciNetGoogle Scholar
  13. 13.
    Mukherjee, K., Natesan, S.: Parameter-uniform hybrid numerical scheme for time-dependent convection-dominated initial-boundary-value problems. Computing 84(3–4), 209–230 (2009)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Nurmuhammada, A., Muhammada, M., Moria, M., Sugiharab, M.: Double exponential transformation in the Sinc-collocation method for a boundary value problem with fourth-order ordinary differential equation. J. Comput. Appl. Math. 182, 32–50 (2005)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Okayama, T., Matsuo, T., Masaaki Sugihara, M.: Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind. J. Comput. Appl. Math. 234, 1211–1227 (2010)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    O’Riordan, E., Pickett, M.L., Shishkin, G.I.: Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems. Math. Comput. 75(255), 1135–1154 (2006)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Rashidinia, J., Zarebnia, M.: Convergence of approximate solution of system of Fredholm integral equations. J. Math. Anal. Appl. 333, 1216–1227 (2007)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Ramos, J.I.: An exponentially fitted method for singularly perturbed one-dimentional parabolic problems. Appl. Math. Comput. 161, 513–523 (2005)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  20. 20.
    Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)CrossRefMATHGoogle Scholar
  21. 21.
    Saadatmandi, A., Dehghan, M.: The use of Sinc-collocation method for solving multi-point boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 17(2), 593–601 (2012)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Zarebnia, M., Rashidinia, J.: Approximate solution of systems of Volterra integral equations with error analysis. Int. J. Comput. Math. 87(13), 3052–3062 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of MathematicsIran University of Science and TechnologyNarmakIran

Personalised recommendations