A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations Mohan K. Kadalbajoo a , Kapil K. Sharma b, * a Department of Mathematics, Indian Institute of Technology, Kanpur 208 016, India b Department of Mathematics, Punjab University, Chandigarh 160 014, India Abstract A boundary value problem for second order singularly perturbed delay differential equation is considered. When the delay argument is sufficiently small, to tackle the delay term, the researchers [M.K. Kadalbajoo, K.K. Sharma, Numerical analysis of singularly perturbed delay differential equations with layer behavior, Appl. Math. Comput. 157 (2004) 11–28, R.E. O’Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991] used Taylor’s series expansion and presented an asymptotic as well as numerical approach to solve such type boundary value problem. But the existing methods in the literature fail in the case when the delay argument is bigger one because in this case, the use of Taylor’s series expansion for the term containing delay may lead to a bad approximation. In this paper to short out this problem, we present a numerical scheme for solving such type of boundary value prob- lems, which works nicely in both the cases, i.e., when delay argument is bigger one as well as smaller one. To handle the delay argument, we construct a special type of mesh so that the term containing delay lies on nodal points after discret- ization. The proposed method is analyzed for stability and convergence. To demonstrate the efficiency of the method and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution several test examples are considered. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Delay differential equation; Singularly perturbed; Boundary layer; Oscillations; Finite difference 1. Introduction In this paper, we extend the numerical study of boundary value problems for singularly perturbed delay differential equations of the convection–diffusion type with delay in the convection term which was initiated in [4]. A singularly perturbed delay differential equation is an ordinary differential equation in which the high- est derivative is multiplied by a small parameter and involving at least one delay term. In the past, less atten- tion had been paid for the numerical solution of singularly perturbed delay differential equations. But in recent 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.08.089 * Corresponding author. Address: Department of Mathematics, Panjab University, Chandigarh 160 014, India. E-mail addresses: kadal@iitk.ac.in (M.K. Kadalbajoo), kapilks@pu.ac.in (K.K. Sharma). Available online at www.sciencedirect.com Applied Mathematics and Computation 197 (2008) 692–707 www.elsevier.com/locate/amc