Journal of Optimization Theory and Applications

, Volume 146, Issue 3, pp 795–809 | Cite as

A Parameter-Uniform B-Spline Collocation Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems



We consider a Dirichlet boundary value problem for a class of singularly perturbed semilinear reaction-diffusion equations. A  B-spline collocation method on a piecewise-uniform Shishkin mesh is developed to solve such problems numerically. The convergence analysis is given and the method is shown to be almost second-order convergent, uniformly with respect to the perturbation parameter ε in the maximum norm. Numerical results are presented to validate the theoretical results.


Singular perturbation Reaction-diffusion problems B-spline collocation method Shishkin mesh 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chang, K.W., Howes, F.A.: Nonlinear Singular Perturbation Phenomena. Springer, New York (1984) MATHGoogle Scholar
  2. 2.
    O’Malley, R.E. Jr.: Singular Perturbation Methods for Ordinary Differential Equations. Springer, New York (1991) MATHGoogle Scholar
  3. 3.
    Jacob, M.: Heat Transfer. Wiley, New York (1959) Google Scholar
  4. 4.
    Van Roosbroeck, W.V.: Theory of flows of electrons and holes in germanium and other semiconductors. Bell Syst. Tech. J. 29, 560–607 (1950) Google Scholar
  5. 5.
    Hirsch, C.: Numerical Computation of Internal and External Flows. Wiley, New York (1988) MATHGoogle Scholar
  6. 6.
    Murray, J.D.: Lectures on Nonlinear Differential Equation Models in Biology. Clarendon Press, Oxford (1977) Google Scholar
  7. 7.
    Weekman, V.W. Jr., Gorring, R.L.: Influence of volume change on gas-phase reactions in porous catalysts. J. Catal. 4, 260–270 (1965) CrossRefGoogle Scholar
  8. 8.
    Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. Z. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969) MATHGoogle Scholar
  9. 9.
    Vulanović, R.: Mesh construction for discretization of singularly perturbed boundary value problems. Ph.D. thesis, University of Novi Sad (1986) Google Scholar
  10. 10.
    Shishkin, G.I.: A difference scheme for a singularly perturbed parabolic equation with a discontinuous boundary condition. Z. Vychisl. Mat. Mat. Fiz. 28, 1679–1692 (1988) MathSciNetGoogle Scholar
  11. 11.
    Roos, H.-G.: Layer-adapted grids for singular perturbation problems. Z. Angew. Math. Mech. 78, 291–301 (1998) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996) MATHGoogle Scholar
  13. 13.
    Vulanović, R.: On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh. Univ. Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 13, 187–201 (1983) MATHGoogle Scholar
  14. 14.
    Surla, K., Uzelac, Z.: A uniformly accurate spline collocation method for a normalized flux. J. Comput. Appl. Math. 166, 291–305 (2004) CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Surla, K.: On modelling of semilinear singularly perturbed reaction-diffusion problem. Nonlinear Anal. 30, 61–66 (1997) CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Rao, S.C.S., Kumar, M.: Parameter-uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction-diffusion problems. Nonlinear Anal. 71, e1579–e1588 (2009) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kadalbajoo, M.K., Reddy, Y.N.: Initial-value technique for a class of nonlinear singular perturbation problems. J. Optim. Theory Appl. 53, 395–406 (1987) CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Kadalbajoo, M.K., Patidar, K.C.: Spline techniques for solving singularly-perturbed nonlinear problems on nonuniform grids. J. Optim. Theory Appl. 114, 573–591 (2002) CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Rao, S.C.S., Kumar, M.: B-spline collocation method for nonlinear singularly-perturbed two-point boundary-value problem. J. Optim. Theory Appl. 134, 91–105 (2007) CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    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) CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    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) CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Varah, J.M.: A lower bound for the smallest singular value of a matrix. Linear Algebra Appl. 11, 3–5 (1975) CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Carlson, R.E., Hall, C.A.: Error bounds for bicubic spline interpolation. J. Approx. Theory 7, 41–47 (1973) CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Hall, C.A.: Natural cubic and bicubic spline interpolation. SIAM J. Numer. Anal. 7, 41–47 (1973) MATHGoogle Scholar
  25. 25.
    Gracia, J.L., Lisbona, F., Clavero, C.: High order ε-uniform methods for singularly perturbed reaction-diffusion problems. In: Vulkov, L., Waśniewski, J., Yalamov, P. (eds.) Lecture Notes in Computer Science, vol. 1988, pp. 350–358. Springer, Berlin (2001) Google Scholar
  26. 26.
    Bohl, E.: Finite Modelle gewööhnlicher Randwertaufgaben. Teubner, Stuttgart (1981) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiHauz KhasIndia

Personalised recommendations