Applied Mathematics E-Notes, 21(2021), 622-633 ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/ Robust Numerical Scheme For Solving Singularly Perturbed Differential Equations Involving Small Delays Mesfin Mekuria Woldaregay , Gemechis File Duressa Received 26 August 2020 Abstract In this article, we consider singularly perturbed differential equation containing delay parameter 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 convective term. The terms with the delay treated using Taylor’s approximation. The resulting singularly perturbed boundary value problem is solved using the technique of non-standard finite difference method. The stability of the scheme is analyzed and investigated using maximum principle and solution bound. The formulated scheme converges independent of the perturbation parameter with order of convergence O(N -1 ). The theoretical finding is validated using numerical examples. The obtained result in this article is accurate and parameter uniformly convergent. 1 Introduction Different mathematical models in science and engineering (such as in control theory, epidemiology, laser optics) take into account not only the present state of a physical system but also it includes the past history. Time delays are natural components of the dynamic processes of biology, ecology, physiology, economics, epidemiology and mechanics [5] and ‘to ignore them is to ignore reality’[2]. Some modelers ignore the lag effect and use differential equation model as substitute for delay differential equation model. Kuang ([10], pp. 11) comments on the dangers that researchers risk if they ignore lags (delays) which they think are small. Delay differential equations (DDEs) model problems where there is after effect affecting the variable of the problem as compared to differential equations which model the problem to current conditions. DDEs is said to be retarded type if the delay argument does not occur in the highest order derivative term, otherwise it is known as neutral DDEs. A singularly perturbed delay differential equations is differential equations in which its highest order derivative is multiplied by small perturbation parameter and having delay parameter(s) on the terms different from the highest order derivative. Singularly perturbed DDEs arise in the mathematical modeling of various physical phenomena, for example in micro scale heat transfer [17], fluid dynamics [7], diffusion in polymers [12], reaction-diffusion equations [3], a lot of model in diseases or physiological processes [13, 18] etc. Notations: In this paper, N is denoted for the number of mesh intervals. The symbol C is denoted for positive constant independent of ε and N . The norm ‖·‖ is used to denote maximum norm i.e. ‖f ‖ = max x ‖f (x)‖. Mathematics Subject Classifications: 65L06, 65L12, 65L15 Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia Department of Mathematics, Jimma University, Jimma, Ethiopia 622