A numerical solution of the Burgers’ equation using cubic B-splines _ Idris Da g a , Dursun Irk b, * ,B€ ulent Saka b a Computer Engineering Department, Osmangazi University, 26480, Eski s ßehir, Turkey b Mathematics Department, Osmangazi University, 26480, Eski s ßehir, Turkey Abstract In the present paper numerical solutions of the one-dimensional Burgers’ equation are obtained by a method based on collocation of cubic B-splines over finite elements. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are found in good agreement with exact solutions. Time-space inte- gration of the Burgers’ equation yields a system of difference equation which is shown to be unconditionally stable. Ó 2004 Elsevier Inc. All rights reserved. 1. Introduction Consider the one-dimensional Burgers’ equation U t þ UU x ¼ kU xx ; a 6 x 6 b; t P 0 ð1Þ with the initial condition U ðx; 0Þ¼ f ðxÞ a 6 x 6 b ð2Þ and the boundary conditions U ða; tÞ¼ b 1 ; U ðb; tÞ¼ b 2 ; ð3Þ * Corresponding author. E-mail addresses: idag@ogu.edu.tr ( _ I. Da g), dirk@ogu.edu.tr (D. Irk), bsaka@ogu.edu.tr (B. Saka). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.01.028 Applied Mathematics and Computation 163 (2005) 199–211 www.elsevier.com/locate/amc