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A Parameter Uniform Numerical Method for an Initial Value Problem for a System of Singularly Perturbed Delay Differential Equations with Discontinuous Source Terms

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Differential Equations and Numerical Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 172))

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Abstract

In this paper an initial value problem for a system of singularly perturbed first order delay differential equations with discontinuous source terms is considered on the interval (0, 2]. The source terms are assumed to have simple discontinuities at the point \(d \in (0,2)\). The components of the solution exhibit initial layers and interior layers. The interior layers occuring in the solution are of two types-interior layers due to delay and interior layers due to the discontinuity of the source terms. A numerical method composed of the standard backward difference operator and a piecewise-uniform Shishkin mesh which resolves the initial and interior layers is suggested. This method is proved to be essentially first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical illustrations are provided to support the theory.

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Acknowledgments

The first author wishes to acknowledge the financial assistance extended through INSPIRE fellowship by the Department of Science and Technology, Government of India. The authors thank the unknown referee for the valuable suggestions which resulted in this improved version of the paper.

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Authors and Affiliations

  1. Bishop Heber College, Tiruchirappalli, Tamil Nadu, India

    Nagarajan Shivaranjani & John J. H. Miller

  2. Trinity College, Dublin, Ireland

    Valarmathi Sigamani

Authors
  1. Nagarajan Shivaranjani
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  2. John J. H. Miller
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  3. Valarmathi Sigamani
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Corresponding author

Correspondence to John J. H. Miller .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Bishop Heber College, Tiruchirappalli, Tamil Nadu, India

    Valarmathi Sigamani

  2. Institute for Numerical Computation and Analysis, Dublin, Ireland

    John J. H. Miller

  3. Department of Mathematics, Bharathidasan University, Tiruchirappalli, India

    Ramanujam Narasimhan

  4. Department of Mathematics, Bishop Heber College, Tiruchirappalli, India

    Paramasivam Mathiazhagan

  5. Department of Mathematics, Bishop Heber College, Tiruchirappalli, Tamil Nadu, India

    Franklin Victor

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Shivaranjani, N., Miller, J.J.H., Sigamani, V. (2016). A Parameter Uniform Numerical Method for an Initial Value Problem for a System of Singularly Perturbed Delay Differential Equations with Discontinuous Source Terms. In: Sigamani, V., Miller, J., Narasimhan, R., Mathiazhagan, P., Victor, F. (eds) Differential Equations and Numerical Analysis. Springer Proceedings in Mathematics & Statistics, vol 172. Springer, New Delhi. https://doi.org/10.1007/978-81-322-3598-9_8

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