Abstract
Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a linear second-order delay differential equation is examined. It is proved that it gives essentially a first-order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.
Similar content being viewed by others
References
Amiraliyev, G.M., Cimen, E.: Numerical method for a singularly perturbed convection-diffusion problem with delay. Appl. Math. Comput. 216(8), 2351–2359 (2010)
Amiraliyev, G.M., Mamedov, Y.D.: Difference schemes on the uniform mesh for a singularly perturbed pseudo-parabolic equations. Turk. J. Math. 19, 207–222 (1995)
Andreev, V.B.: The Green function and a priori estimates of solutions of monotone three-point singularly perturbed finite-difference schemes. Differ. Equ. 37, 923–933 (2001)
Cimen, E.: A priori estimates for solution of singularly perturbed boundary value problem with delay in convection term. J. Math. Anal. 8(1), 202–211 (2017)
Derstein, M.W., Gibbs, H.M., Hopf, F.A., Kaplan, D.L.: Bifurcation gap in a hybrid optical system. Phys. Rev. A 26(6), 3720–3722 (1982)
Doolan, E.R., Miller, J.J.H., Schilders, W.H.A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin (1980)
Du, B.: Stability analysis of periodic solution for a complex-valued neural networks with bounded and unbounded delays. Asian J. Control 20(2), 881–892 (2018)
Du, B., Lian, X., Cheng, X.: Partial differential equation modeling with Dirichlet boundary conditions on social networks. Bound. Value Probl. 2018(50), 1–11 (2018)
Du, B., Zhang, W., Yang, Q.: Robust state estimation for neutral-type neural networks with mixed time delays. J. Nonlinear Sci. Appl. 10(5), 2565–2578 (2017)
Els’gol’ts, E.L.: Qualitative methods in mathematical analysis. In: Translations of Mathematical Monographs. vol. 12. AMS, Providence (1964)
Erneux, E.: Applied Delay Differential Equations. Springer, Berlin (2009)
Farrel, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman Hall/CRC, Boca Raton (2000)
Geng, F.Z., Qian, S.P.: Modified reproducing kernel method for singularly perturbed boundary value problems with a delay. Appl. Math. Model. 39(18), 5592–5597 (2015)
Glizer, V.Y.: Controllability conditions of linear singularly perturbed systems with small state and input delays. Math. Control Signals Syst. 28(1), 1–29 (2016)
Kadalbajoo, M.K., Ramesh, V.P.: Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy. Appl. Math. Comput. 188, 1816–1831 (2007)
Keane, A., Krauskopf, B., Postlethwaite, C.M.: Climate models with delay differential equations. Chaos 27(114309), 1–15 (2017)
Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary-value problems for differential-difference equations. SIAM J. Appl. Math. 42, 502–531 (1982)
Liu, L.B., Chen, Y.: Maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay. Appl. Math. Comput. 227(15), 801–810 (2014)
Longtin, A., Milton, J.: Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Math. Biosci. 90, 183–199 (1988)
Mackey, M.C., Glass, L.: Oscillations and chaos in physiological control systems. Science 197, 287–289 (1977)
Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems, revised edition. World Scientific, Singapore (2012)
Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008)
Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001)
Selvi, P.A., Ramanujam, N.: An iterative numerical method for singularly perturbed reaction-diffusion equations with negative shift. J. Comput. Appl. Math. 296, 10–23 (2016)
Stein, R.B.: Some models of neuronal variability. Biophys. J. 7, 37–68 (1967)
Subburayan, V.: A parameter uniform numerical method for singularly perturbed delay problems with discontinuous convection coefficient. Arab J. Math. Sci. 22(2), 191–206 (2016)
Subburayan, V., Ramanujam, N.: Asymptotic initial value technique for singularly perturbed convection–diffusion delay problems with boundary and weak interior layers. Appl. Math. Lett. 25(12), 2272–2278 (2012)
Zarin, H.: On discontinuous Galerkin finite element method for singularly perturbed delay differential equations. Appl. Math. Lett. 38(1), 27–32 (2014)
Acknowledgements
We thank the editor(s) and the referee(s) for their favourable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cimen, E., Amiraliyev, G.M. Uniform Convergence Method for a Delay Differential Problem with Layer Behaviour. Mediterr. J. Math. 16, 57 (2019). https://doi.org/10.1007/s00009-019-1335-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-019-1335-9