Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2010, Article ID 495184, 17 pages doi:10.1155/2010/495184 Research Article A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem Musa C ¸ akır 1 and Gabil M. Amiraliyev 2 1 Department of Mathematics, Faculty of Sciences, 100. Y. University, 65080 Van, Turkey 2 Department of Mathematics, Faculty of Sciences, Sinop University, 57000 Sinop, Turkey Correspondence should be addressed to Musa C ¸ akır, cakirmusa@hotmail.com Received 30 October 2009; Accepted 13 April 2010 Academic Editor: Michela Redivo-Zaglia Copyright q 2010 M. C ¸ akır and G. M. Amiraliyev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameter ε, of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically. 1. Introduction This paper is concerned with ε-uniform numerical method for the singularly perturbed semilinear boundary-value problem BVP: Lu : ε 2 u ′′  εaxu ′ - f x, u 0, 0 < x < ℓ, 1.1 L 0 u : -εu ′ 0 ψu0  0, 1.2 uℓ  - ϕuℓ 1   0, 0 <ℓ 1 < ℓ, 1.3 where ε is a small positive parameter, the functions ax ≥ 0,f x, u, and ψu,ϕu are sufficiently smooth on 0,ℓ , 0,ℓ  × R, and R, respectively, and furthermore 0 <β ≤ ∂f ∂u ≤ β ∗ < ∞, dψ du ≥ δ> 0,     dϕ du     ≤ κ< 1. 1.4 The solution u generally has boundary layers near x  0 and x  ℓ .