A Wavelet-Galerkin method for a singularly perturbed convection-dominated diffusion equation Mohamed El-Gamel Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt Abstract There are few techniques available to numerically solve singularly perturbed parabolic problems. In this paper we show that the Wavelet-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on several examples and a comparison with the method of reduction order is made. It is shown that the Wavelet-Galerkin method yields better results. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Wavelet-Galerkin method; Singularly perturbed; Convection-dominated diffusion equation 1. Introduction Wavelet analysis is a new numerical concept which allows one to represent a function in terms of a set of basis functions, called wavelets, which are localized both in location and scale. As we noted earlier, spectral bases are infinitely differentiable, but have global support. On the other hand, basis functions used in finite difference or finite element methods have small compact support, but have poor continuity properties. As a result, spectral methods have good spectral localization, but poor spatial localiza- tion, while finite difference and finite element methods have good localization, but poor spectral localization. Wavelet bases seem to combine the advantages of both spectral and finite difference (or finite element) bases. One can expect that numerical methods based on wavelet bases are able to attain good spatial and spectral resolution. In wavelet applications to the solution of partial differential equations the most frequently used wavelets are those with compact support introduced by Daubechies [4]. Exploration of usage of Daubechies wavelets to solve partial differential equations has been undertaken by a number of investigators such as [1,2,9,11,13,18–20,25]. Additional references can be found in the recent review by El-Gamel [7,8]. The main objective of this paper is to present a new numerical approach for solving the singularly perturbed convection-dominated diffusion equation 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.03.017 E-mail address: gamel_eg@yahoo.com Applied Mathematics and Computation 181 (2006) 1635–1644 www.elsevier.com/locate/amc