A numerical treatment for singularly perturbed differential equations with integral boundary condition G.M. Amiraliyev a, * , I.G. Amiraliyeva a , Mustafa Kudu b a Department of Mathematics, Faculty of Art and Sciences, Yuzuncu Yil University, Van 65080, Turkey b Department of Mathematics, Faculty of Art and Sciences, Ataturk University, Erzincan 24100, Turkey Abstract We consider a uniform finite difference method on Shishkin mesh for a quasilinear first order singularly perturbed boundary value problem (BVP) with integral boundary condition. We prove that the method is first order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. The param- eter uniform convergence is confirmed by numerical computations. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Finite difference; Singular perturbation; Shishkin mesh; Integral boundary condition; Error estimates 1. Introduction In this paper we consider the following singular perturbation problem (BVP) with integral boundary condition eu 0 þ f ðt; uÞ¼ 0; t 2 I ¼ð0; T ; T > 0; ð1:1Þ uð0Þ¼ luðT Þþ Z T 0 bðsÞuðsÞ ds þ d ; ð1:2Þ where 0 < e 6 1 is the perturbation parameter, l and d are given constants. b(t) and f(t, u) are assumed to be sufficiently continuously differentiable functions in I ¼ I [ft ¼ 0g and I  R respectively and moreover of ou P a > 0: Note that the boundary condition (1.2) includes periodic and initial conditions as special cases. For e  1 the function u(t) has a boundary layer of thickness O(e) near t = 0 (see Section 2). 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.07.060 * Corresponding author. E-mail address: gamirali2000@yahoo.com (G.M. Amiraliyev). Applied Mathematics and Computation 185 (2007) 574–582 www.elsevier.com/locate/amc