Browsing by Author "Saker, M. A."

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Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations
(PERGAMON-ELSEVIER SCIENCE LTD, 2011) Doha, E. H.; Bhrawy, A. H.; Saker, M. A.
A new explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of high even-order boundary value problems (BVPs) with two point boundary conditions but by considering their integrated forms. The Bernstein-Petrov-Galerkin method (BPG) is applied to construct the numerical solution for such problems. The method is then tested on examples and compared with other methods. It is shown that the BPG yields better results. (C) 2010 Elsevier Ltd. All rights reserved.
Modified Jacobi-Bernstein basis transformation and its application to multi-degree reduction of Bezier curves
(ELSEVIER SCIENCE BV, 2016) Bhrawy, A. H.; Doha, E. H.; Saker, M. A.; Baleanu, D.
This paper reports new modified Jacobi polynomials (MJPs). We derive the basis transformation between MJPs and Bernstein polynomials and vice versa. This transformation is merging the perfect Least-square performance of the new polynomials together with the geometrical insight of Bernstein polynomials. The MJPs with indexes corresponding to the number of endpoints constraints are the natural basis functions for Least-square approximation of Bezier curves. Using MJPs leads us to deal with the constrained Jacobi polynomials and the unconstrained Jacobi polynomials as orthogonal polynomials. The MJPs are automatically satisfying the homogeneous boundary conditions. Thereby, the main advantage of using MJPs, in multi-degree reduction of Bezier curves on computer aided geometric design (CAGD), is that the constraints in CAGD are also satisfied and that decreases the steps of multi-degree reduction algorithm. Several numerical results for the multi-degree reduction of Bezier curves on CAGD are given. (C) 2016 Elsevier B.V. All rights reserved.
On Generalized Jacobi-Bernstein Basis Transformation: Application of Multidegree Reduction of Bezier Curves and Surfaces
(ASME, 2014) Doha, E. H.; Bhrawy, A. H.; Saker, M. A.
This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for least-square approximation of Bezier curves and surfaces. Application to multidegree reduction (MDR) of Bezier curves and surfaces in computer aided geometric design (CAGD) is given.
On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations
(SPRINGER, 2011) Doha, E. H.; Bhrawy, A. H.; Saker, M. A.
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.
A PSEUDOSPECTRAL METHOD FOR SOLVING THE TIME-FRACTIONAL GENERALIZED HIROTA-SATSUMA COUPLED KORTEWEG-DE VRIES SYSTEM
(EDITURA ACAD ROMANE, 2017) Saker, M. A.; Ezz-Eldien, S. S.; Bhrawy, A. H.
In this paper, a new space-time spectral algorithm is constructed to solve the generalized Hirota-Satsuma coupled Korteweg-de Vries (GHS-C-KdV) system of time-fractional order. The present algorithm consists of applying the collocation-spectral method in conjunction with the operational matrix of fractional derivative for the double Jacobi polynomials, which will be employed as a basis function for the spectral solution. The main characteristic behind this approach is that such problems will reduce to those of solving algebraic systems of equations that greatly simplifying the problem. For ensuring the accuracy and efficiency of the presented algorithm, we apply it to find the approximate solutions of two specific problems, namely, a homogeneous form of the GHS-C-KdV system and a inhomogeneous GHS-C-KdV system.