Abstract
In this paper, a boundary value problem for a second-order singularly perturbed delay differential equation is considered. The solution of this problem exhibits boundary layers at x = 0 and x = 2 and interior layers at x = 1. A numerical method composed of a classical finite difference scheme applied on a piecewise-uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first-order convergent in the maximum norm uniformly in the perturbation parameter. Numerical illustrations support the theory.
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References
Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary-value problems for differential - difference equations. SIAM J. Appl. Math. 42(3), 502–530 (1982)
Cen, Z.: A hybrid finite difference scheme for a class of singularly perturbed delay differential equations. Neural, Parallel Sci. Comput. 16, 303–308 (2008)
Hongjiong, T.: Numerical methods for singularly perturbed delay differential equations. In: Proceedings of the International Conference on Boundary and Interior Layers, Computational and Asymptotic Methods - BAIL 2004. ONERA, Toulouse (2004)
Nicaise, S., Xenophontos, C.: Robust approximation of singularly perturbed delay differential equations by the hp finite element method. Comput. Meth. Appl. Math. 13(1), 21–37 (2013)
Subburayan, V., Ramanujam, N.: An initial value technique for singularly perturbed reaction diffusion problems with a negative shift. Novi Sad J. Math. 43(2), 67–80 (2013)
Subburayan, V., Ramanujam, N.: An initial value technique for singularly perturbed system of reaction-diffusion type delay differential equations. J. KSIAM. 17(4), 221–237 (2013)
Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific Publishing Co., Singapore, New Jersey, London, Hong Kong (1996)
Acknowledgements
The authors express their sincere thanks to Dr. N. Ramanujam, UGC Emeritus Fellow, Bharathidasan University, Tiruchirappalli, for the useful discussions on delay differential equations.
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Manikandan, M., Shivaranjani, N., Miller, J.J.H., Valarmathi, S. (2014). A Parameter-Uniform Numerical Method for a Boundary Value Problem for a Singularly Perturbed Delay Differential Equation. In: Ansari, A. (eds) Advances in Applied Mathematics. Springer Proceedings in Mathematics & Statistics, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-06923-4_7
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DOI: https://doi.org/10.1007/978-3-319-06923-4_7
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