الفهرس 
FundamentalsOnly 14 pages are availabe for public view
 |  | from | 154 |  |  |
| | | from | 154 | | |
Abstract
The Bernstein polynomials began to enjoy widespread use as a versatile means of intuitively constructing and manipulating geometric shapes, simple and efficient recursive algorithms, recognition of its excellent numerical stability properties, and an increasing diversification of its repertoire of applications. With the advent of computer graphics, Bernstein polynomials restricted to the interval [0, 1], becomes important in the form of B´ezier curves. Indeed, many properties of the B´ezier curves and surfaces come from the properties of Bernstein polynomials. Moreover, Bernstein polynomials have been frequently used in both the solution of differential equations and approximation theory.
In this work, we are concerned with how to describe derivatives and integrals of Bern- stein polynomial basis in terms of Bernstein polynomials themselves. One of the main objectives of this thesis is to introduce and develop new efficient algorithms based on the Bernstein-Galerkin and Bernstein-dual-Petrov-Galerkin approximations for solving high even-order differential equations. These algorithms are based on appropriate construction of trial and test functions for the Petrov-Galerkin formulation that lead to discrete systems with specially structured matrices that can be efficiently inverted, while the convergence rates of the algorithms are exponential for problems with smooth solutions. Moreover, we introduce and develop new efficient algorithms based on the Bernstein-Jacobi basis trans- formations for multi-degree reduction of B´ezier curves and surfaces. We demonstrate the advantage of using different generalizations of Jacobi polynomials.
We also aim to introduce and develop new family of orthogonal polynomials that gen- eralize the constrained Jacobi polynomials. The expansion for the function (solution), in terms of Jacobi polynomials (1 − x)a(1 + x)bP (α,β)(x) (n ≥ 0, α > −1, β > −1, a, b ≥ 0), enables one to get the sought for general constraints generalized Jacobi-Galerkin approximation for any possible values of the real parameters α and β and constraints a and b. That is, instead of developing approximation results for each particular pairs of indexes (α, β) and a, b, it would be very useful to carry out a study on polynomials (1 − x)a(1 + x)bP (α,β)(x) with general indexes which can then be directly applied to otherapplications.
In Chapter 1, we present a general introduction to Bernstein polynomials, B´ezier curves and B´ezier surfaces. A brief account of orthogonal polynomials, their properties and expansion of functions in terms of them are given. Some general properties and
important relations concerned with the Jacobi polynomials P (α,β)(x) are considered. We also clarify the differences between the three most commonly used spectral methods, namely, the Galerkin, Collocation and tau methods.
In Chapter 2, we consider in detail how to construct efficient algorithms for solving high even-order differential equations with constant coefficients subject to homogeneous and nonhomogeneous boundary conditions using Bernstein-Galerkin and Bernstein-dual- Petrov-Galerkin method based on both derivatives and integrals of Bernstein polynomials. Several numerical examples are considered to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.
In Chapter 3, a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis is given. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. The GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for Least-Squares approximation of B´ezier curves and surfaces. Some applications to multi- degree reduction of B´ezier curves and surfaces are given. An efficient algorithms for solving high even-order differential equations is also given.
In Chapter 4, we establish and prove the basis transformation between the modified Jacobi polynomials (MJPs) and Bernstein polynomials and vice versa. This transforma- tion is merging the perfect Least-Squares performance of the MJPs with the geometrical insight of the Bernstein form. Using MJPs leads us to deal with the constrained Ja- cobi polynomials and the unconstrained Jacobi polynomials as orthogonal polynomials. Moreover, we apply this transformation to multi-degree reduction of B´ezier curves and sur- faces on computer aided geometric design (CAGD). We also present and develop spectral method based on this transformation for solving high even-order differential equations. Finally several illustrative examples are implemented to demonstrate the efficiency and applicability of the proposed algorithms.
Finally in Chapter 5, we propose and define a new orthogonal basis function based
on generalized Jacobi polynomials and modified Jacobi polynomials, and then derive the transformations between this new basis and Bernstein basis and vice versa. The so- called general constraints generalized Jacobi polynomials (GCGJPs) satisfy some selected properties that are essentially relevant to spectral approximations and computer aided geometric design. The main advantage of proposed GCGJPs is that the classical Jacobi polynomials, the constrained Jacobi polynomials and GJPs are directly special cases of GCGJPs. We also present the transformation between GCGJPs and Bernstein basis. Using this transformation, we present two applications, the first application is multi- degree reduction of B´ezier curves and surfaces. The second application is a simple and efficient method for solving high even-order differential equations.
The obtained numerical results are tabulated and displayed graphically whenever pos- sible. These results show that the proposed algorithms of solutions are reliable accurate. Comparisons with previously obtained results by other researchers or exact known solu- tions are made throughout the context whenever available.
To the best of our knowledge, the formulae and algorithms stated and proved in Chapters 2 up to 5 are completely new. The Programs used in this thesis are performed using the PC machine, with Intel(R) Core(TM) 2 Duo CPU 2.00 GHz, 4.00 GB of RAM,
and the symbolic computation software Mathematica 8.0 has been also used.