e-Uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs Mohan K. Kadalbajoo a , Kailash C. Patidar b, * , Kapil K. Sharma c a Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur 208016, India b Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa c MAB Universite ´ Bordeaux 1, 351, cours de la Libr ´ation, F-33405 Talence cedex, France Abstract We consider some problems arising from singularly perturbed general differential difference equations. First we con- struct (in a new way) and analyze a ‘‘fitted operator finite difference method (FOFDM)’’ which is first order e-uniformly convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the ‘‘fitted mesh finite difference method (FMFDM)’’. This FMFDM is a direct method and e-uniformly convergent with the nodal error as Oðn 2 ln 2 nÞ which is an improvement over the exist- ing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson’s extrapola- tion or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as Oðn 1 ln nÞ where n denotes the total number of sub-intervals of [0, 1]. Comparative numerical results are presented in support of the theory. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Differential difference equations; Singular perturbations; Boundary value problems; Fitted operator methods; Fitted mesh methods; Shishkin mesh 1. Introduction Consider the singularly perturbed differential difference equation (SPDDE): ey 00 ðxÞþ aðxÞy 0 ðxÞþ aðxÞy ðx  dÞþ fðxÞy ðxÞþ bðxÞy ðx þ gÞ¼ f ðxÞ on X ¼ð0; 1Þ; ð1:1Þ under the interval and boundary conditions y ðxÞ¼ /ðxÞ on  d 6 x 6 0; y ðxÞ¼ cðxÞ on 1 6 x 6 1 þ g; ð1:2Þ 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.03.043 * Corresponding author. E-mail addresses: kadal@iitk.ac.in (M.K. Kadalbajoo), kpatidar@uwc.ac.za (K.C. Patidar), sharma@math.u-bordeaux1.fr (K.K. Sharma). Applied Mathematics and Computation xxx (2006) xxx–xxx www.elsevier.com/locate/amc ARTICLE IN PRESS