Skip to main content
SpringerLink
Log in

Cubic spline method for a class of nonlinear singularly-perturbed boundary-value problems

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we present a numerical method for solving a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. The original second-order problem is reduced to an asymptotically equivalent first-order problem and is solved by a numerical method using a fourth-order cubic spline in the inner region. The method has been analyzed for convergence and is shown to yield anO(h 4) approximation to the solution. Some test examples have been solved to demonstrate the efficiency of the method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bender, C. M., andOrszag, S. A.,Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, New York, 1978.

    Google Scholar 

  2. Kevorkian, J. A., andCole, J. D.,Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, New York, 1981.

    Google Scholar 

  3. O'Malley, R. E.,Introduction to Singular Perturbations, Academic Press, New York, New York, 1974.

    Google Scholar 

  4. Roberts, S. M.,A Boundary-Value Technique for Singular Perturbation Problems, Journal of Mathematical Analysis and Applications, Vol. 87, pp. 489–508, 1982.

    Google Scholar 

  5. Kadalbajoo, M. K., andReddy, Y. N.,An Initial-Value Technique for a Class of Nonlinear Singular Perturbation Problems, Journal of Optimization Theory and Applications, Vol. 53, pp. 395–406, 1987.

    Google Scholar 

  6. Ahlberg, J. H., Nilson, E. N., andWalsh, J. L.,The Theory of Splines and Their Applications, Academic Press, New York, New York, 1967.

    Google Scholar 

  7. Hilderbrand, F. B.,Introduction to Numerical Analysis, McGraw-Hill, New York, New York, 1956.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Kanpur, India

    M. K. Kadalbajoo (Professor) & R. K. Bawa (Research Scholar)

Authors
  1. M. K. Kadalbajoo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. K. Bawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by I. Galligani

The authors thank the referee for his helpful comments.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kadalbajoo, M.K., Bawa, R.K. Cubic spline method for a class of nonlinear singularly-perturbed boundary-value problems. J Optim Theory Appl 76, 415–428 (1993). https://doi.org/10.1007/BF00939375

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00939375

Key Words

Search

Navigation