Boundary Value Problems
Volume 2011 (2011), Article ID 829543, 16 pages
doi:10.1155/2011/829543
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.
Abstract
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.