Boundary Value Problems
Volume 2011 (2011), Article ID 829543, 16 pages
Research Article

On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations

1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
4Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt

Received 31 October 2010; Accepted 6 March 2011

Academic Editor: S. Messaoudi

Copyright © 2011 E. H. Doha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A new formula expressing explicitly the derivatives of Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves is proved, and a formula expressing the Bernstein coefficients of the general-order derivative of a differentiable function in terms of its Bernstein coefficients is deduced. An application of how to use Bernstein polynomials for solving high even-order differential equations by Bernstein Galerkin and Bernstein Petrov-Galerkin methods is described. These two methods are then tested on examples and compared with other methods. It is shown that the presented methods yield better results.