Nonlinear Analysis 71 (2009) e1579–e1588 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Parameter-uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction–diffusion problems S. Chandra Sekhara Rao ∗ , Mukesh Kumar Department of Mathematics, Indian Institute of Technology Delhi, Hauz khas, New Delhi-110 016, India article info MSC: 65L10 65L12 Keywords: Singular perturbation problem Semilinear reaction–diffusion problem Exponential splines Piecewise-uniform Shishkin mesh ε-uniform convergence abstract We consider a Dirichlet boundary value problem for singularly perturbed semilinear reaction–diffusion equation. The problem is discretized using an exponential spline difference scheme derived on the basis of splines in tension on piecewise-uniform Shishkin type mesh. The convergence analysis is given and the method is shown to be almost second order accurate in the discrete maximum norm, uniformly in the perturbation parameter ε. Numerical experiments are conducted to demonstrate the theoretical results. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Consider a class of singularly perturbed semilinear reaction–diffusion problem F ε u(x) ≡−ε 2 u ′′ (x) + f (x, u(x)) = 0, for x ∈ (0, 1) (1) u(0) = 0, u(1) = 0, (2) where ε is a small positive perturbation parameter. Let I =[0, 1]. We assume that f (x, u(x)) ∈ C ∞ (I × R) for convenience. In general, as ε tends to zero, the solution u(x) of (1)–(2) may exhibit boundary or internal layers of various types. The location of the layers and the behavior of the solution u(x) depends on the character of f (x, u(x)) (see [1,2]). We assume that f u (x, u(x)) ≥ β 2 > 0, for all (x, u(x)) ∈ I × R. (3) This is a standard stability condition which implies that the problem (1)–(2) and the reduced problem f (x, u 0 (x)) = 0, for x ∈ I , has a unique solution in C ∞ (I ). Note that in general, the reduced solution u 0 (x) does not satisfy either of the boundary conditions in (2). Thus the solution u(x) of (1)–(2) usually exhibits sharp boundary layers at the endpoints of the interval I when the parameter ε is near to zero. Singular perturbation problems often arise in many areas of science and engineering such as heat transfer problem with high Peclet numbers [3], drift-diffusion equations of semiconductor device physics [4], Navier–Stokes equations of fluid flow with high Renoylds number [5], Michaelis-Menten theory for enzyme reactions [6] and the mathematical models of liquid crystal materials and chemical reactions [7]. When polynomial based numerical methods are applied to (1)–(2) on uniform mesh, one does not obtain accurate results on all I , even in the linear case. This leads to the development of the numerical ∗ Corresponding author. Tel.: +91 11 26591489; fax: +91 11 26581005. E-mail address: scsr@maths.iitd.ernet.in (S. Chandra Sekhara Rao). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.210