On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations
© E. H. Doha et al. 2011
Received: 31 October 2010
Accepted: 6 March 2011
Published: 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.
Bernstein polynomials  have many useful properties, such as, the positivity, the continuity, and unity partition of the basis set over the interval . The Bernstein polynomial bases vanish except the first polynomial at , which is equal to 1 and the last polynomial at , which is also equal to 1 over the interval . This provides greater flexibility in imposing boundary conditions at the end points of the interval. The moments is nothing but Bernstein polynomial itself. With the advent of computer graphics, Bernstein polynomial restricted to the interval becomes important in the form of Bezier curves . Many properties of the Bézier curves and surfaces come from the properties of the Bernstein polynomials. Moreover, Bernstein polynomials have been recently used for the solution of differential equations, (see, e.g., ).
The Bernstein polynomials are not orthogonal; so their uses in the least square approximations are limited. To overcome this difficulty, two approaches are used. The first approach is the basis transformation, for the transformation matrix between Bernstein polynomial basis and Legendre polynomial basis , between Bernstein polynomial basis and Chebyshev polynomial basis , and between Bernstein polynomial basis and Jacobi polynomial basis . The second approach is the dual basis functions for Bernstein polynomials (see Jüttler ). Jüttler  derived an explicit formula for the dual basis function of Bernstein polynomials. The construction of the dual basis must be repeated at each time the approximation polynomial increased.
For spectral methods [8, 9], explicit formulae for the expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of those of the original expansion coefficients of the function itself are needed. Such formulae are available for expansions in Chebyshev , Legendre , ultraspherical , Hermite , Jacobi , and Laguerre  polynomials. These polynomials have been used in both the solution of boundary value problems [16–19] and in computational fluid dynamics . In most of these applications, use is made of formulae relating the expansion coefficients of derivatives appearing in the differential equation to those of the function itself, (see, e.g., [16–19]). This process results in an algebraic system or a system of differential equations for the expansion coefficients of the solution which then must be solved.
Due to the increasing interest on Bernstein polynomials, the question arises of how to describe their properties in terms of their coefficients when they are given in the Bernstein basis. Up to now, and to the best of our Knowledge, many formulae corresponding to those mentioned previously are unknown and are traceless in the literature for Bernstein polynomials. This partially motivates our interest in such polynomials.
Another motivation is concerned with the direct solution techniques for solving high even-order differential equations, using the Bernstein Galerkin approximation. Also, we use Bernstein Petrov-Galerkin approximation; we choose the trial functions to satisfy the underlying boundary conditions of the differential equations, and the test functions to be dual Bernstein polynomials which satisfy the orthogonality condition. The method leads to linear systems which are sparse for problems with constant coefficients. Numerical results are presented in which the usual exponential convergence behavior of spectral approximations is exhibited.
The remainder of this paper is organized as follows. In Section 2, we give an overview of Bernstein polynomials and the relevant properties needed in the sequel, and in Section 3, we prove the main results of the paper which are: (i) an explicit expression for the derivatives of Bernstein polynomials of any degree and for any order in terms of the Bernstein polynomials themselves and (ii) an explicit formula for the expansion coefficient of the derivatives of an infinitely differentiable function in terms of those of the original expansion coefficients of the functions itself. In Section 4, we discuss separately Bernstein Galerkin and Bernstein Petrov-Galerkin methods and describe how they are used to solve high even-order differential equations. Finally, Section 5 gives some numerical results exhibiting the accuracy and efficiency of our proposed numerical algorithms.
2. Relevant Properties of Bernstein Polynomials
where the binomial coefficients are given by .
3. Derivatives of Bernstein Polynomials
The main objective of this section is to prove the following two theorems for the derivatives of and Bernstein coefficients of the th derivative of .
For , (3.1) leads us to go back to (2.2).
which completes the induction and proves the theorem.
We can express the Bernstein polynomial of any degree in terms of any higher degree basis using the following lemma.
For proof, see, Farouki and Rajan .
Then, we can state and prove the following theorem.
and this completes the proof of Theorem 3.3.
The following two corollaries will be of fundamental importance in what follows.
4. An Application for the Solution of High Even-Order Differential Equations
4.1. Bernstein Galerkin Method
where is the inner product in , and its norm will be denoted by .
It is of fundamental importance to note here that the crucial task in applying the Galerkin-spectral Bernstein approximations is how to choose an appropriate basis for such that the linear system resulting from the Bernstein-Galerkin approximation to (4.4) is as simple as possible.
where for all . The boundary conditions lead to the first and the last expansion coefficients to be zero.
4.2. Bernstein Petrov-Galerkin Method
where and , for all . The boundary conditions lead to the first and the last expansion coefficients to be zero.
4.3. Using Coefficients of Differentiated Expansions
Thus, there are equations for the unknowns , in order to obtain a solution; it is only necessary to solve (4.23) with the help of (4.21) for the unknowns coefficients , .
5. Numerical Results
We solve in this section several numerical examples by using the algorithms presented in the previous section. Comparisons between Bernstein Galerkin method (BGM), Bernstein Petrov-Galerkin method (BPGM), and other methods proposed in [21–24] are made. We consider the following examples.
subject to the boundary conditions , with the exact solution .
and for .
with the analytical solution .
and for .
with the exact solution .
and for .
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