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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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Volume 19, Issue 3-4


Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline

A. S. V. Ravi Kanth
  • Corresponding author
  • Department of Mathematics, National Institute of Technology Kurukshetra, Haryana, 136 119, India
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/ P. Murali Mohan Kumar
Published Online: 2018-04-03 | DOI: https://doi.org/10.1515/ijnsns-2017-0126


In this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.

Keywords: nonlinear singular perturbation problems; parametric cubic spline; differential difference equations; quasilinearization; oscillations

PACS: 65L10; 65L11


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About the article

Received: 2017-06-11

Accepted: 2018-03-15

Published Online: 2018-04-03

Published in Print: 2018-06-26

The authors would like to thank National Board for Higher Mathematics(NBHM), Government of India, for providing financial support under the grant number 2/48(12)/2013/NBHM(R.P.)/R&D II/1084.

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 357–365, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0126.

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