A second order uniformly convergent numerical scheme for parameterized singularly perturbed delay differential problems

In this work, we propose a hybrid difference scheme for solving parameterized singularly perturbed delay differential problems. A unified error analysis framework for the proposed hybrid scheme is given that allows to conclude uniform convergence of 𝓞(N−2ln2N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}(N^{-2}\ln ^{2} N)$\end{document} on Shishkin meshes and 𝓞(N−2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}(N^{-2})$\end{document} on Bakhvalov meshes, where N is the number of mesh intervals in the domain. Numerical results are included to confirm the theoretical estimates.