On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations
1 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
4 Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt
Boundary Value Problems 2011, 2011:829543 doi:10.1155/2011/829543Published: 15 March 2011
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.