tailieunhanh - Báo cáo hóa học: "Research Article On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 829543 16 pages doi 2011 829543 Research Article On the Derivatives of Bernstein Polynomials An Application for the Solution of High Even-Order Differential Equations E. H. Doha 1 A. H. Bhrawy 2 3 and M. A. Saker4 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 Correspondence should be addressed to A. H. Bhrawy alibhrawy@ 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. 1. Introduction Bernstein polynomials 1 have many useful properties such as the positivity the continuity and unity partition of the basis set over the interval 0 1 . The Bernstein polynomial bases vanish except the first polynomial at x 0 which is equal to 1 and the last polynomial at x 1 which is also