Abstract
In this paper, we consider singularly perturbed differential equations having delay on the convection and reaction terms. The considered problem exhibits boundary layer on the left or right side of the domain depending on the sign of the coefficient of the convection term. We approximated the terms containing the delay using Taylor series approximation. The resulting singularly perturbed boundary value problem is treated using exponentially fitted operator mid-point upwind finite difference method. The stability of the scheme is analysed and investigated using maximum principle and by constructing barrier functions for solution bound. We formulated the uniform converges of the scheme. The scheme converges uniformly with linear order of convergence. To validate the theoretical analysis and finding, we consider three examples exhibiting boundary layer on the left and right side of the domain. The obtained numerical result in this paper is accurate and parameter uniformly convergent.
Similar content being viewed by others
References
Bestehorn M, Grigorieva EV (2004) Formation and propagation of localized states in extended systems. Ann Phys 13(7–8):423–431
Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61(1):41
Kadalbajoo MK, Ramesh V (2007) Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy. Appl Math Comput 188(2):1816–1831
Kellogg RB, Tsan A (1978) Analysis of some difference approximations for a singular perturbation problem without turning points. Math Comput 32(144):1025–1039
Kumar D, Kadalbajoo M (2012) Numerical treatment of singularly perturbed delay differential equations using b-spline collocation method on Shishkin mesh. JNAIAM 7(3–4):73–90
Liu Q, Wang X, De Kee D (2005) Mass transport through swelling membranes. Int J Eng Sci 43(19–20):1464–1470
Mackey MC, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197(4300):287–289
Miller JJ, Oriordan E, Shishkin GI (1996) Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific, Singapore
Omalley RE (1991) Singular perturbation methods for ordinary differential equations, vol 89. Springer, Berlin
Roos HG, Stynes M, Tobiska L (2008) Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, vol 24. Springer Science and Business Media, Berlin
Tian H (2002) The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag. J Math Anal Appl 270(1):143–149
Tzou D (1997) Micro- to macroscale heat transfer. Taylor & Francis, Washington, DC, p 1997
Wazewska-Czyzewska M, Lasota A (1976) Mathematical models of the red cell system. Matematyta Stosowana 6(1):25–40
Woldaregay MM, Duressa GF (2020) Higher-order uniformly convergent numerical scheme for singularly perturbed differential difference equations with mixed small shifts. Int J Differ Equ 2020:1–15
Woldaregay MM, Duressa GF (2021a) Fitted numerical scheme for singularly perturbed differential equations having small delays. Caspian J Math Sci 10 (in press)
Woldaregay MM, Duressa GF (2021b) obust numerical scheme for solving singularly perturbed differential equations involving small delays. Appl Math E-Notes 21 (in press)
Acknowledgements
The authors thanks associate editor and the reviewers for their constructive comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no any potential conflict of interest.
Additional information
Communicated by Abdellah Hadjadj.
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
Woldaregay, M.M., Duressa, G.F. Robust mid-point upwind scheme for singularly perturbed delay differential equations. Comp. Appl. Math. 40, 178 (2021). https://doi.org/10.1007/s40314-021-01569-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-021-01569-5