An analytical–numerical method for solving the Korteweg–de Vries equation S. O ¨ zer, S. Kutluay * Department of Mathematics, Faculty of Arts and Science, _ I no¨ nu¨ University, 44280 Malatya, Turkey Abstract In this paper, an analytical–numerical method is applied to the one-dimensional Korteweg–de Vries equation with a variant of boundary and initial conditions to obtain its numerical solutions at small times. Two test problem with known exact solutions are studied to demonstrate the accuracy of the present method. The obtained results are compared with the exact solution of each problem and are found to be in good agree- ment with each other. The numerical scheme is also compared with earlier work and shown to be accurate and efficient. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Korteweg–de Vries equation; Analytical–numerical method 1. Introduction The Korteweg–de Vries equation (KdV) is one of the most important non- linear dispersive waves which was first introduced in [1] to describe the behav- iour of one-dimensional shallow-water waves with small but finite amplitude. 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.011 * Corresponding author. E-mail address: skutluay@inonu.edu.tr (S. Kutluay). Applied Mathematics and Computation 164 (2005) 789–797 www.elsevier.com/locate/amc