Skip to main content
Springer Nature Link
Log in

Numerical Analysis of Boundary-Value Problems for Singularly-Perturbed Differential-Difference Equations with Small Shifts of Mixed Type

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

Abstract

In this paper, we use a numerical method to solve boundary-value problems for a singularly-perturbed differential-difference equation of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift. Similar boundary-value problems are associated with expected first exit time problems of the membrane potential in models for the neuron. The stability and convergence analysis of the method is given. The effect of a small shift on the boundary-layer solution is shown via numerical experiments. The numerical results for several test examples 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

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. LANGE, C. G., and MIURA, R.M., Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations: (û) Small Shifts with Layer Behaûior, SIAM Journal on Applied Mathematics, Vol. 54, pp. 249-272, 1994.

    Google Scholar 

  2. LANGE, C. G., and MIURA, R. M., Singular-Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations, SIAM Journal on Applied Mathematics, Vol. 42, pp. 502-531, 1982.

    Google Scholar 

  3. LANGE, C. G., and MIURA, R. M., Singular-Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations: (ii) Rapid Oscillations and Resonances, SIAM Journal on Applied Mathematics, Vol. 45, pp. 687-707, 1985.

    Google Scholar 

  4. LANGE, C. G., and MIURA, R.M., Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations: (iii) Turning Point Problems, SIAM Journal on Applied Mathematics, Vol. 45, pp. 708-734, 1985.

    Google Scholar 

  5. LANGE, C. G., and MIURA, R.M., Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations: (ûi) Small Shifts with Rapid Oscillations, SIAM Journal on Applied Mathematics, Vol. 54, pp. 273-283, 1994.

    Google Scholar 

  6. STEIN, R. B., Some Models of Neuronal Variability, Biophysical Journal, Vol. 7, pp. 37-68, 1967.

    Google Scholar 

  7. TUCKWELL, H. C., and COPE, D. K., Accuracy of Neuronal Interspike Times Calculated from a Diffusion Approximation, Journal of Theoretical Biology, Vol. 83, pp. 377-387, 1980.

    Google Scholar 

  8. HUTCHINSON, G. E., Circular Casual Systems in Ecology, Annals of the New York Academy of Sciences, Vol. 50, pp. 221-246, 1948.

    Google Scholar 

  9. KADALBAJOO, M. K., and REDDY, Y. N., A Nonasymptotic Method for General Linear Singular-Perturbation Problems, Journal of Optimization Theory and Applications, Vol. 55, pp. 257-269, 1986.

    Google Scholar 

  10. NICHOLS, N. K., On the Numerical Integration of a Class of Singular-Perturbation Problems, Journal of Optimization Theory and Applications, Vol. 60, pp. 439-452, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    M.K. Kadalbajoo (Professor)

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

    K.K. Sharma (Research Scholar)

Authors
  1. M.K. Kadalbajoo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K.K. Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kadalbajoo, M., Sharma, K. Numerical Analysis of Boundary-Value Problems for Singularly-Perturbed Differential-Difference Equations with Small Shifts of Mixed Type. Journal of Optimization Theory and Applications 115, 145–163 (2002). https://doi.org/10.1023/A:1019681130824

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019681130824

Search

Navigation