On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations

  • EH Doha1,

    Affiliated with

    • AH Bhrawy2, 3Email author and

      Affiliated with

      • MA Saker4

        Affiliated with

        Boundary Value Problems20112011:829543

        DOI: 10.1155/2011/829543

        Received: 31 October 2010

        Accepted: 6 March 2011

        Published: 15 March 2011

        Abstract

        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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq1_HTML.gif . The Bernstein polynomial bases vanish except the first polynomial at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq2_HTML.gif , which is equal to 1 and the last polynomial at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq3_HTML.gif , which is also equal to 1 over the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq4_HTML.gif . This provides greater flexibility in imposing boundary conditions at the end points of the interval. The moments http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq5_HTML.gif is nothing but Bernstein polynomial itself. With the advent of computer graphics, Bernstein polynomial restricted to the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq6_HTML.gif becomes important in the form of Bezier curves [2]. 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., [3]).

        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 [4], between Bernstein polynomial basis and Chebyshev polynomial basis [5], and between Bernstein polynomial basis and Jacobi polynomial basis [6]. The second approach is the dual basis functions for Bernstein polynomials (see Jüttler [7]). Jüttler [7] 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 [10], Legendre [11], ultraspherical [12], Hermite [13], Jacobi [14], and Laguerre [15] polynomials. These polynomials have been used in both the solution of boundary value problems [1619] and in computational fluid dynamics [8]. 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., [1619]). 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

        The Bernstein polynomials of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq7_HTML.gif th degree form a complete basis over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq8_HTML.gif , and they are defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ1_HTML.gif
        (2.1)

        where the binomial coefficients are given by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq9_HTML.gif .

        The derivatives of the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq10_HTML.gif th degree Bernstein polynomials are polynomials of degree http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq11_HTML.gif and are given by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ2_HTML.gif
        (2.2)
        The multiplication of two Bernstein basis is
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ3_HTML.gif
        (2.3)
        and the moments of Bernstein basis are
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ4_HTML.gif
        (2.4)
        Like any basis of the space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq12_HTML.gif , the Bernstein polynomials have a unique dual basis http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq13_HTML.gif (also called the inverse or reciprocal basis) which consists of the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq14_HTML.gif dual basis functions
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ5_HTML.gif
        (2.5)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ6_HTML.gif
        (2.6)
        Jüttler [7] represented the dual basis function with respect to the Bernstein basis. The dual basis functions must satisfy the relation of duality
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ7_HTML.gif
        (2.7)
        Indefinite integral of Bernstein basis is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ8_HTML.gif
        (2.8)
        and all Bernstein basis function of the same order have the same definite integral over the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq15_HTML.gif , namely,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ9_HTML.gif
        (2.9)

        3. Derivatives of Bernstein Polynomials

        The main objective of this section is to prove the following two theorems for the derivatives of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq16_HTML.gif and Bernstein coefficients of the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq17_HTML.gif th derivative of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq18_HTML.gif .

        Theorem 3.1.

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ10_HTML.gif
        (3.1)

        Proof.

        For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq19_HTML.gif , (3.1) leads us to go back to (2.2).

        If we apply induction on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq20_HTML.gif , assuming that (3.1) holds, we want to show that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ11_HTML.gif
        (3.2)
        If we differentiate (3.1), then we have (with application of relation (2.2))
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ12_HTML.gif
        (3.3)
        which can be written as
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ13_HTML.gif
        (3.4)
        Set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq21_HTML.gif in the first term of the right-hand side of relation (3.4) to get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ14_HTML.gif
        (3.5)
        It can be easily shown that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ15_HTML.gif
        (3.6)

        which completes the induction and proves the theorem.

        We can express the Bernstein polynomial of any degree http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq22_HTML.gif in terms of any higher degree basis http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq23_HTML.gif using the following lemma.

        Lemma 3.2.

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ16_HTML.gif
        (3.7)

        For proof, see, Farouki and Rajan [20].

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq24_HTML.gif be a differentiable function of degree http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq25_HTML.gif defined on the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq26_HTML.gif , then we can write
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ17_HTML.gif
        (3.8)
        Further, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq27_HTML.gif denote the Bernstein coefficients of the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq28_HTML.gif th derivative of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq29_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ18_HTML.gif
        (3.9)

        Then, we can state and prove the following theorem.

        Theorem 3.3.

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ19_HTML.gif
        (3.10)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ20_HTML.gif
        (3.11)

        Proof.

        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ21_HTML.gif
        (3.12)
        then making use of Theorem 3.1 (formula (3.1)) immediately yields
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ22_HTML.gif
        (3.13)
        If we change the degree of Bernstein polynomials using (3.7), then we can write
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ23_HTML.gif
        (3.14)
        Expanding the two summation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq30_HTML.gif and rearranging the coefficients of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq31_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq32_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ24_HTML.gif
        (3.15)

        and this completes the proof of Theorem 3.3.

        The following two corollaries will be of fundamental importance in what follows.

        Corollary 3.4.

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ25_HTML.gif
        (3.16)

        Proof.

        We can express explicitly the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq33_HTML.gif th derivatives of Bernstein polynomials from Theorem 3.1 to obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ26_HTML.gif
        (3.17)
        Now, (3.16) can be easily derived by using (2.3). Thanks to (2.9), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ27_HTML.gif
        (3.18)

        Corollary 3.5.

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ28_HTML.gif
        (3.19)

        Proof.

        Using Theorem 3.1, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ29_HTML.gif
        (3.20)
        It follows immediately from (3.7) and (2.7) that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ30_HTML.gif
        (3.21)

        4. An Application for the Solution of High Even-Order Differential Equations

        4.1. Bernstein Galerkin Method

        Consider the solution of the differential equation
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ31_HTML.gif
        (4.1)
        subject to the following boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ32_HTML.gif
        (4.2)
        Let us first introduce some basic notation which will be used in the sequel. We set
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ33_HTML.gif
        (4.3)
        then the Bernstein-Galerkin approximation to (4.1) is to find http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq34_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ34_HTML.gif
        (4.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq35_HTML.gif is the inner product in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq36_HTML.gif , and its norm will be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq37_HTML.gif .

        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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq38_HTML.gif such that the linear system resulting from the Bernstein-Galerkin approximation to (4.4) is as simple as possible.

        We can choose the basis functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq39_HTML.gif to be of the form
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ35_HTML.gif
        (4.5)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq40_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq41_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq42_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq43_HTML.gif . The http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq44_HTML.gif boundary conditions lead to the first http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq45_HTML.gif and the last http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq46_HTML.gif expansion coefficients to be zero.

        Therefore, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq47_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ36_HTML.gif
        (4.6)
        It is now clear that (4.4) is equivalent to
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ37_HTML.gif
        (4.7)
        Let us denote
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ38_HTML.gif
        (4.8)
        Then, (4.7) is equivalent to the following matrix equation
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ39_HTML.gif
        (4.9)
        where the elements of the matrices http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq49_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq51_HTML.gif are given explicitly using Corollary 3.4, as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ40_HTML.gif
        (4.10)

        4.2. Bernstein Petrov-Galerkin Method

        The Petrov-Galerkin method generates a sequence of approximate solutions that satisfy a weak form of the original differential equation as tested against polynomials in a dual space. To describe this method and the full discretization more precisely, we introduce some basic notation. We set
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ41_HTML.gif
        (4.11)
        Denoting by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq53_HTML.gif the spaces of Bernstein polynomials of degree ≤ http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq54_HTML.gif and dual Bernstein of degree ≤ http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq55_HTML.gif , then the Bernstein Petrov-Galerkin approximation to (4.1) is, to find http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq56_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ42_HTML.gif
        (4.12)
        We choose the trial Bernstein functions to satisfy the underlying boundary conditions of the differential equation, and we choose the test dual Bernstein functions to satisfy the orthogonality condition. Consider the test and trial functions of expansion http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq58_HTML.gif to be of the form
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ43_HTML.gif
        (4.13)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq59_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq60_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq62_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq63_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq64_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq65_HTML.gif . The http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq66_HTML.gif boundary conditions lead to the first http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq67_HTML.gif and the last http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq68_HTML.gif expansion coefficients to be zero.

        Therefore, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq69_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ44_HTML.gif
        (4.14)
        and, accordingly, (4.12) is equivalent to
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ45_HTML.gif
        (4.15)
        Let us denote
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ46_HTML.gif
        (4.16)
        Then, (4.15) is equivalent to the following matrix equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ47_HTML.gif
        (4.17)
        If we take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq71_HTML.gif as defined in (4.13) and if we denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq73_HTML.gif . Then, the elements http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq74_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq75_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq76_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq77_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq78_HTML.gif are given explicitly by using Corollary 3.5, as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ48_HTML.gif
        (4.18)

        4.3. Using Coefficients of Differentiated Expansions

        Here, we shall use Theorem 3.3 for the solution of the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq79_HTML.gif th-order differential (4.1)-(4.2). We approximate http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq80_HTML.gif by an expansion of Bernstein polynomials
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ49_HTML.gif
        (4.19)
        We seek to determine http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq81_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq82_HTML.gif , using Petrov-Galerkin method. Note here that we set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq84_HTML.gif to ensure that the boundary conditions (4.2) are satisfied. Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq86_HTML.gif are polynomials of degree at most http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq88_HTML.gif , respectively, we may write
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ50_HTML.gif
        (4.20)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ51_HTML.gif
        (4.21)
        It is to be noted here that (4.21) is obtained by making use of relation (3.11). The coefficients http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq89_HTML.gif are chosen so that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq90_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ52_HTML.gif
        (4.22)
        Substituting (4.19) and (4.20) into (4.22), multiplying by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq91_HTML.gif , and integrating over the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq92_HTML.gif yield
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ53_HTML.gif
        (4.23)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ54_HTML.gif
        (4.24)

        Thus, there are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq93_HTML.gif equations for the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq94_HTML.gif unknowns http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq95_HTML.gif , in order to obtain a solution; it is only necessary to solve (4.23) with the help of (4.21) for the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq96_HTML.gif unknowns coefficients http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq98_HTML.gif .

        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 [2124] are made. We consider the following examples.

        Example 5.1.

        Consider the boundary value problem (see, [22])
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ55_HTML.gif
        (5.1)

        subject to the boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq99_HTML.gif , with the exact solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq100_HTML.gif .

        Table 1 lists the maximum pointwise error http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq101_HTML.gif and maximum absolute relative error http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq102_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq103_HTML.gif using the BGM and BPGM with various choices of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq104_HTML.gif . Table 1 shows that our methods have better accuracy compared with the quintic nonpolynomial spline method developed in [22]; it is also shown that, in the case of solving linear system of order 14, we obtain a maximum absolute error of order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq105_HTML.gif . It is worthy noting here that the method of [22] gives the maximum absolute error http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq106_HTML.gif but by solving a linear system of order 64 instead of order 14 in our case.
        Table 1

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq108_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq109_HTML.gif .

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq110_HTML.gif

        BGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq111_HTML.gif

        BPGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq112_HTML.gif

        BGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq113_HTML.gif

        BPGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq114_HTML.gif

        2

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq115_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq116_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq117_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq118_HTML.gif

        4

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq119_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq120_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq121_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq122_HTML.gif

        6

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq123_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq124_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq125_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq126_HTML.gif

        8

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq127_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq128_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq129_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq130_HTML.gif

        10

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq131_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq132_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq133_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq134_HTML.gif

        12

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq135_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq136_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq137_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq138_HTML.gif

        14

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq139_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq140_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq141_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq142_HTML.gif

        16

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq143_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq144_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq145_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq146_HTML.gif

        18

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq147_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq148_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq149_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq150_HTML.gif

        Example 5.2.

        We consider the fourth-order two point boundary value problem (see, [21])
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ56_HTML.gif
        (5.2)

        with the analytical solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq151_HTML.gif .

        Table 2 lists the maximum pointwise error and maximum absolute relative error of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq152_HTML.gif using the BGM and BPGM with various choices of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq153_HTML.gif . In Table 3, a comparison between the error obtained by using BGM, BPGM, the sinc-Galerkin, and modified decomposition methods (see, [21]) is displayed. This definitely shows that our methods are more accurate.
        Table 2

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq155_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq156_HTML.gif .

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq157_HTML.gif

        BGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq158_HTML.gif

        BPGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq159_HTML.gif

        BGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq160_HTML.gif

        BPGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq161_HTML.gif

        4

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq162_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq163_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq164_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq165_HTML.gif

        6

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq166_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq167_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq168_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq169_HTML.gif

        8

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq170_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq171_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq172_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq173_HTML.gif

        10

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq174_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq175_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq176_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq177_HTML.gif

        12

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq178_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq179_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq180_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq181_HTML.gif

        14

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq182_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq183_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq184_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq185_HTML.gif

        16

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq186_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq187_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq188_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq189_HTML.gif

        18

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq190_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq191_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq192_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq193_HTML.gif

        Table 3

        Comparison between different methods for Example 5.2.

        Error

        BGM

        BPGM

        Sinc-Galerkin in [21]

        Decomposition in [21]

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq194_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq195_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq196_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq197_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq198_HTML.gif

        Example 5.3.

        Consider the sixth-order BVP (see, [21, 23, 24])
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ57_HTML.gif
        (5.3)

        with the exact solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq199_HTML.gif .

        Table 4 lists the maximum pointwise error and maximum absolute relative error of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq200_HTML.gif using BGM and BPG with various choices of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq201_HTML.gif . Table 5 exhibits a comparison between the error obtained by using BGM, BPGM, and Sinc-Galerkin in [21], septic splines in [23] and modified decomposition in [24]. From this Table, one can check that our methods are more accurate.
        Table 4

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq203_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq204_HTML.gif .

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq205_HTML.gif

        BGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq206_HTML.gif

        BPGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq207_HTML.gif

        BGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq208_HTML.gif

        BPGM http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq209_HTML.gif

        6

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq210_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq211_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq212_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq213_HTML.gif

        8

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq214_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq215_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq216_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq217_HTML.gif

        10

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq218_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq219_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq220_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq221_HTML.gif

        12

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq222_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq223_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq224_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq225_HTML.gif

        14

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq226_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq227_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq228_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq229_HTML.gif

        16

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq230_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq231_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq232_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq233_HTML.gif

        18

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq234_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq235_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq236_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq237_HTML.gif

        Table 5

        Comparison between the errors of different methods in Example 5.3.

        Error

        BGM

        BPGM

        Sinc-Galerkin [21]

        Septic spline [23]

        Decomposition [24]

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq238_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq239_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq240_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq241_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq242_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq243_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq244_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq245_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq246_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq247_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq248_HTML.gif

        Authors’ Affiliations

        (1)
        Department of Mathematics, Faculty of Science, Cairo University
        (2)
        Department of Mathematics, Faculty of Science, King Abdulaziz University
        (3)
        Department of Mathematics, Faculty of Science, Beni-Suef University
        (4)
        Department of Basic Science, Institute of Information Technology, Modern Academy

        References

        1. Lorentz GG: Bernstein Polynomials, Mathematical Expositions, no. 8. University of Toronto Press, Toronto, Canada; 1953:x+130.
        2. Farin G: Curves and Surfaces for Computer Aided Geometric Design. Academic Press, Boston, Mass, USA; 1996.
        3. Bhatti MI, Bracken P: Solutions of differential equations in a Bernstein polynomial basis. Journal of Computational and Applied Mathematics 2007, 205(1):272–280. 10.1016/j.cam.2006.05.002MATHMathSciNetView Article
        4. Boyd JP: Exploiting parity in converting to and from Bernstein polynomials and orthogonal polynomials. Applied Mathematics and Computation 2008, 198(2):925–929.MATHMathSciNetView Article
        5. Rababah A: Transformation of Chebyshev-Bernstein polynomial basis. Computational Methods in Applied Mathematics 2003, 3(4):608–622.MATHMathSciNetView Article
        6. Rababah A: Jacobi-Bernstein basis transformation. Computational Methods in Applied Mathematics 2004, 4(2):206–214.MATHMathSciNetView Article
        7. Jüttler B: The dual basis functions for the Bernstein polynomials. Advances in Computational Mathematics 1998, 8(4):345–352. 10.1023/A:1018912801267MATHMathSciNetView Article
        8. Canuto C, Hussaini MY, Quarteroni A, Zang TA: Spectral Methods in Fluid Mechanics, Scientific Computation. Springer, Berlin, Germany; 1988:xxii+563.
        9. Livermore PW: Orthogonal Galerkin polynomials. Journal of Computational Physics 2010, 229(6):2046–2060. 10.1016/j.jcp.2009.11.022MATHMathSciNetView Article
        10. Karageorghis A: A note on the Chebyshev coefficients of the general order derivative of an infinitely differentiable function. Journal of Computational and Applied Mathematics 1988, 21(1):129–132. 10.1016/0377-0427(88)90396-2MATHMathSciNetView Article
        11. Phillips TN: On the Legendre coefficients of a general-order derivative of an infinitely differentiable function. IMA Journal of Numerical Analysis 1988, 8(4):455–459. 10.1093/imanum/8.4.455MATHMathSciNetView Article
        12. Karageorghis A, Phillips TN: On the coefficients of differentiated expansions of ultraspherical polynomials. Applied Numerical Mathematics 1992, 9(2):133–141. 10.1016/0168-9274(92)90010-BMATHMathSciNetView Article
        13. Doha EH: On the connection coefficients and recurrence relations arising from expansions in series of Hermite polynomials. Integral Transforms and Special Functions 2004, 15(1):13–29. 10.1080/10652460310001600618MATHMathSciNetView Article
        14. Doha EH: On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. Journal of Physics. A 2004, 37(3):657–675. 10.1088/0305-4470/37/3/010MATHMathSciNetView Article
        15. Doha EH: On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials. Journal of Physics. A 2003, 36(20):5449–5462. 10.1088/0305-4470/36/20/307MATHMathSciNetView Article
        16. Doha EH, Bhrawy AH: Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials. Numerical Algorithms 2006, 42(2):137–164. 10.1007/s11075-006-9034-6MATHMathSciNetView Article
        17. Doha EH, Bhrawy AH: Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials. Applied Numerical Mathematics 2008, 58(8):1224–1244. 10.1016/j.apnum.2007.07.001MATHMathSciNetView Article
        18. Doha EH, Bhrawy AH: A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. Numerical Methods for Partial Differential Equations 2009, 25(3):712–739. 10.1002/num.20369MATHMathSciNetView Article
        19. Doha EH, Bhrawy AH, Abd-Elhameed WM: Jacobi spectral Galerkin method for elliptic Neumann problems. Numerical Algorithms 2009, 50(1):67–91. 10.1007/s11075-008-9216-5MATHMathSciNetView Article
        20. Farouki RT, Rajan VT: Algorithms for polynomials in Bernstein form. Computer Aided Geometric Design 1988, 5(1):1–26. 10.1016/0167-8396(88)90016-7MATHMathSciNetView Article
        21. El-gamel M: A comparison between the sinc-Galerkin and the modified decomposition methods for solving two-point boundary-value problems. Journal of Computational Physics 2007, 223(1):369–383. 10.1016/j.jcp.2006.09.025MATHMathSciNetView Article
        22. Ramadan MA, Lashien IF, Zahra WK: High order accuracy nonpolynomial spline solutions for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq249_HTML.gif order two point boundary value problems. Applied Mathematics and Computation 2008, 204(2):920–927. 10.1016/j.amc.2008.07.038MATHMathSciNetView Article
        23. Siddiqi SS, Akram G: Septic spline solutions of sixth-order boundary value problems. Journal of Computational and Applied Mathematics 2008, 215(1):288–301. 10.1016/j.cam.2007.04.013MATHMathSciNetView Article
        24. Wazwaz A: The numerical solution of sixth-order boundary value problems by the modified decomposition method. Applied Mathematics and Computation 2001, 118(2–3):311–325. 10.1016/S0096-3003(99)00224-6MATHMathSciNetView Article

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        © E. H. Doha et al. 2011

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