Comparison of the solutions obtained by Adomian decomposition and wavelet-Galerkin methods of boundary-value problems Mohamed El-Gamel Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt Abstract In this paper, we will compare the performance of Adomian decomposition method and the wavelet-Galerkin method applied to the solution of boundary-value problems involving non-homogeneous heat and wave equations. It is shown that the Adomian decomposition method in many instances gives better results. In the wavelet-Galerkin solutions, Daubechies six wavelets are used because they give better results than those of lower degree wavelets. The results are then compared with those obtained using the Adomian decomposition method. Although the Adomian decomposition solution required slightly more computational effort than the wavelet-Galerkin solution, it resulted in more accurate results than the wavelet-Galerkin method. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Wavelet-Galerkin; Connection coefficients; Daubechies wavelets; Adomian’s decomposition method; Numerical solutions 1. Introduction The Adomian decomposition method (ADM) has been applied to a wide class of stochastic and determin- istic problems in many interesting mathematics and physics areas [1–5,9,27]. This method has some significant advantages over numerical methods. It provides analytic, verifiable, rapidly convergent approximation which yields insight into the character and the behavior of the solution just as in the closed form solution. Adomian gave a review of the decomposition method in [4]. Several authors have compared the ADM with some existing techniques in solving different types of prob- lems. Bellomo and Monaco [8] have compared the ADM and the perturbation technique are used in solving random non-linear differential equations. Edwards et al. [11] have introduced their comparison of ADM and Runge–Kutta methods for approximate solutions of some predator prey model equations. Wazwaz [24] pro- posed a new approach to develop a non-perturbative approximate solution for the Thomas–Fermi equation. This approach is based upon a modification of the ADM . Recently, he introduced a comparison between the 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.010 E-mail address: gamel_eg@yahoo.com Applied Mathematics and Computation 186 (2007) 652–664 www.elsevier.com/locate/amc