INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. 2011; 27:1427–1445 Published online 10 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.1370 COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Uniformly convergent numerical method for singularly perturbed differential-difference equation using grid equidistribution Jugal Mohapatra and Srinivasan Natesan ∗, † Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati-781 039, India SUMMARY In this paper, a class of singularly perturbed differential-difference equations with small delay and shift terms is considered. A numerical method comprising of upwind finite difference operator on an adaptive grid, which is formed by equidistributing the arc-length monitor function, is constructed for approximating the solution. The method is proved to be robust, in the sense that the discrete solution obtained converges in the maximum norm to the exact solution uniformly with respect to the perturbation parameter. Parameter- uniform error bounds for the numerical approximations are established. Numerical examples support the theoretical results. Copyright 2010 John Wiley & Sons, Ltd. Received 18 March 2009; Revised 27 October 2009; Accepted 2 December 2009 KEY WORDS: singular perturbation problems; differential-difference equations; boundary layer; upwind scheme; adaptive mesh; uniform convergence 1. INTRODUCTION In this paper, we consider the following singularly perturbed differential–difference equation (DDE) in the domain  = (0, 1): L ε u ε (x ) ≡-εu ′′ ε (x ) - p(x )u ′ ε (x ) - (x )u ε (x - ) + q (x )u ε (x ) - (x )u ε (x + ) = F (x ), u ε (x ) = (x ), -x 0, u ε (x ) = (x ), 1x 1 + , (1) where 0<ε ≪ 1 is the perturbation parameter, the functions p(x ), (x ), q (x ), (x ), F (x ), (x ) and (x ) are sufficiently smooth functions and the delay parameter  and the shift parameter  are such that 0<,  ≪ 1, both are of o(ε). These equations are also known as DDE and are widespread in many branches of sciences and have been used for many years in control theory, biophysics, mechanics, etc. [1]. The DDEs provide more realistic models than the conventional singularly perturbed differential equations. For example, in population dynamics, these small parameters display time-lag or after-effect and hence play an important role in modelling real-life phenomena. The solution of (1) exhibits a single boundary layer at the left or right end of the domain depending upon whether p(x ) - (x ) + (x )>0 (or <0). Lange and Miura [2–4] provided an ∗ Correspondence to: Srinivasan Natesan, Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati-781 039, India. † E-mail: natesan@iitg.ernet.in Contract/grant sponsor: Council of Scientific and Industrial Research (CSIR), Government of India; contract/grant number: 9/731(0071)/08-EMRI Copyright 2010 John Wiley & Sons, Ltd.