Open Access

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

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 https://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 https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq8_HTML.gif , and they are defined by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq9_HTML.gif .

The derivatives of the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq11_HTML.gif and are given by
https://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
https://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
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq14_HTML.gif dual basis functions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ5_HTML.gif
(2.5)
where
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq15_HTML.gif , namely,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq16_HTML.gif and Bernstein coefficients of the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq17_HTML.gif th derivative of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq18_HTML.gif .

Theorem 3.1.

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

Proof.

For https://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 https://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
https://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))
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ13_HTML.gif
(3.4)
Set https://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
https://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
https://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 https://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 https://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.

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

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

Proof.

Since
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ23_HTML.gif
(3.14)
Expanding the two summation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq30_HTML.gif and rearranging the coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq31_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq32_HTML.gif , we get
https://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.

https://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 https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ27_HTML.gif
(3.18)

Corollary 3.5.

https://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
https://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
https://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
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq34_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ34_HTML.gif
(4.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq35_HTML.gif is the inner product in https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq39_HTML.gif to be of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ35_HTML.gif
(4.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq40_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq41_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq42_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq43_HTML.gif . The https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq44_HTML.gif boundary conditions lead to the first https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq45_HTML.gif and the last https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq47_HTML.gif , we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ37_HTML.gif
(4.7)
Let us denote
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq49_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq50_HTML.gif , https://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:
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ41_HTML.gif
(4.11)
Denoting by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq52_HTML.gif and https://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 ≤ https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq54_HTML.gif and dual Bernstein of degree ≤ https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq56_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq58_HTML.gif to be of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ43_HTML.gif
(4.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq59_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq60_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq61_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq62_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq63_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq64_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq65_HTML.gif . The https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq66_HTML.gif boundary conditions lead to the first https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq67_HTML.gif and the last https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq69_HTML.gif , we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ45_HTML.gif
(4.15)
Let us denote
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ47_HTML.gif
(4.17)
If we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq70_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq73_HTML.gif . Then, the elements https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq74_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq75_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq76_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq77_HTML.gif , https://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:
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq80_HTML.gif by an expansion of Bernstein polynomials
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ49_HTML.gif
(4.19)
We seek to determine https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq81_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq83_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq86_HTML.gif are polynomials of degree at most https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq88_HTML.gif , respectively, we may write
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ50_HTML.gif
(4.20)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq89_HTML.gif are chosen so that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq90_HTML.gif satisfies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq91_HTML.gif , and integrating over the interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq92_HTML.gif yield
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ53_HTML.gif
(4.23)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ54_HTML.gif
(4.24)

Thus, there are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq93_HTML.gif equations for the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq94_HTML.gif unknowns https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq96_HTML.gif unknowns coefficients https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq97_HTML.gif , https://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])
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq99_HTML.gif , with the exact solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq101_HTML.gif and maximum absolute relative error https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_IEq102_HTML.gif of https://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 https://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 https://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 https://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

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

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

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

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

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

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

2

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

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

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

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

4

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

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

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

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

6

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

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

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

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

8

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

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

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

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

10

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

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

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

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

12

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

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

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

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

14

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

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

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

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

16

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

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

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

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

18

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

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

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

https://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])
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ56_HTML.gif
(5.2)

with the analytical solution https://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 https://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 https://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

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

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

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

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

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

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

4

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

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

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

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

6

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

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

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

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

8

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

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

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

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

10

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

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

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

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

12

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

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

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

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

14

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

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

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

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

16

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

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

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

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

18

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

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

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

https://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]

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

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

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

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

https://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])
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F829543/MediaObjects/13661_2010_Article_60_Equ57_HTML.gif
(5.3)

with the exact solution https://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 https://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 https://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

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

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

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

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

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

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

6

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

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

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

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

8

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

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

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

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

10

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

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

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

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

12

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

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

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

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

14

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

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

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

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

16

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

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

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

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

18

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

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

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

https://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]

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

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

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

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

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

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

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

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

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

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

https://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

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© E. H. Doha et al. 2011

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