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

Author

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  • M.K. Kadalbajoo

    (Indian Institute of Technology)

  • K.K. Sharma

    (Indian Institute of Technology)

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.

Suggested Citation

  • M.K. Kadalbajoo & K.K. Sharma, 2002. "Numerical Analysis of Boundary-Value Problems for Singularly-Perturbed Differential-Difference Equations with Small Shifts of Mixed Type," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 145-163, October.
    • Handle: RePEc:spr:joptap:v:115:y:2002:i:1:d:10.1023_a:1019681130824
    DOI: 10.1023/A:1019681130824
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    Cited by:

    1. Sabir, Zulqurnain & Wahab, Hafiz Abdul & Umar, Muhammad & Erdoğan, Fevzi, 2019. "Stochastic numerical approach for solving second order nonlinear singular functional differential equation," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    2. Ayşe Kurt & Salih Yalçınbaş & Mehmet Sezer, 2013. "Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2013, pages 1-9, August.
    3. Oğuz, Cem & Sezer, Mehmet, 2015. "Chelyshkov collocation method for a class of mixed functional integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 943-954.
    4. Sharma, Amit & Rai, Pratima, 2023. "Analysis of a higher order uniformly convergent method for singularly perturbed parabolic delay problems," Applied Mathematics and Computation, Elsevier, vol. 448(C).
    5. Gemechis File Duressa & Imiru Takele Daba & Chernet Tuge Deressa, 2023. "A Systematic Review on the Solution Methodology of Singularly Perturbed Differential Difference Equations," Mathematics, MDPI, vol. 11(5), pages 1-16, February.

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