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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Unification of the Bernstein-type polynomials and their applications

Yilmaz Simsek

Author Affiliations

Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya, TR-07058, Turkey

Boundary Value Problems 2013, 2013:56  doi:10.1186/1687-2770-2013-56


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/56


Received:12 November 2012
Accepted:27 February 2013
Published:20 March 2013

© 2013 Simsek; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we investigate some new identities related to the unification of the Bernstein-type polynomials, Bernoulli polynomials, Euler numbers and Stirling numbers of the second kind. We also give some remarks and applications of the Bernstein-type polynomials related to solving high even-order differential equations by using the Bernstein-Galerkin method. We also give some applications on these polynomials and differential equations.

MSC: 11B68, 12D10, 14F10, 26C05, 26C10, 30B40, 30C15, 42A38, 44A10.

Keywords:
Bernstein polynomials; generating function; Bezier curves; Laplace transform; functional equation; high-order differential equations; Bernstein-Galerkin method; Bernoulli polynomials; Bernoulli numbers; Euler polynomials; Euler numbers; Genocchi polynomials; Genocchi numbers; Stirling numbers of the second kind

1 Introduction

Generating functions play an important role in the investigation of various useful properties of the sequences and differential equations. These functions are also used to find many properties and formulas for the sequences. In [1], the author constructed certain generating functions for the unification of the classical Bernstein polynomials. Using these generating functions, the author derived several interesting and useful identities for these polynomials. The Bernstein polynomials have been defined by many different ways, for example, by q-series, by complex function and by many algorithms. The Bernstein polynomials are used in approximations of functions as well as in other fields such as smoothing in statistics, in numerical analysis, constructing the Bezier curves. The Bernstein polynomials are also used to solve differential equations.

According to Farouki [2], the Bernstein polynomial basis was introduced 100 years ago (Bernstein, 1912) as a means to constructively prove the ability of polynomials to approximate any continuous function, to any desired accuracy, over a prescribed interval. Their slow convergence rate and the lack of digital computers to efficiently construct them caused the Bernstein polynomials to lie dormant in the theory rather than practice of approximation for the better part of a century. The Bernstein coefficients of a polynomial provide valuable insight into its behavior over a given finite interval, yielding many useful properties and elegant algorithms that are now being increasingly adopted in other application domains.

Recently, the author [1] introduced and investigated the following generating functions which use a unification of the classical Bernstein polynomials:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M4">View MathML</a>. The following function is a generating function of the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M6">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M7">View MathML</a>.

An explicit formula of the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a> is given by the following theorem [1].

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M9">View MathML</a>. Letb, nandsbe nonnegative integers. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M10">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M11">View MathML</a>

Remark 1.2 If we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M12">View MathML</a> in (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M13">View MathML</a>

which denotes the classical Bernstein basis function (cf. [1-7]). Consequently, the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a> are a unification of the Bernstein polynomials.

The remainder of this study is organized as follows.

Section 2: We give many properties of the unification of the Bernstein-type polynomials: partition of unity, alternating sum, subdivision property. We also give many functional equations and differential equations of this generating function. Using these equations, many properties of the unification of the Bernstein-type polynomials can be found. Section 3: Integral representations of the unification of the Bernstein-type polynomials are given. Using these representations, we give an identity. Section 4: By using the Laplace transform, we find some identities of the unification of the Bernstein-type polynomials. Section 5: By using a new generating function, we prove the Marsden identity for the unification of the Bernstein-type polynomials. Section 6: By using generating functions, we give relations between the unification of the Bernstein-type polynomial, the unification of the Bernoulli polynomial of higher order and the Stirling numbers of the second kind. Section 7: By using the unification of the Bernstein-type polynomials and the Bernstein-Galerkin methods, we solve high even-order differential equations. Section 8: We give some remarks on the unification of the Bernstein-type polynomials and Bezier-type curves.

2 Properties of the unification of the Bernstein-type polynomials

In this section, we investigate some properties of the unification of the Bernstein-type polynomials.

2.1 Partition of unity

The unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>does not have partition of unity. That is, by using (1), we derive the following functional equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M16">View MathML</a>

By using the same method as that in [7] and (2), we arrive at the formula for the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M17">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M18">View MathML</a>

Remark 2.1 The polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M19">View MathML</a> have partition of unity. That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M20">View MathML</a>

2.2 Alternating sum

By using (1), we derive the following functional equation which is used to find an alternating sum of the unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M22">View MathML</a>

By using the same method as that in [7] and (2), we arrive at a formula for the alternating sum of the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M17">View MathML</a>, which is given by the following theorem.

Theorem 2.2

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M24">View MathML</a>

2.3 Subdivision property

Here, we give partial differential equations and a functional equation of the generating function for the unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>. By using this functional equation, we derive the subdivision property unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>.

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M27">View MathML</a>

(3)

By using the above functional equation and (2), we derive the subdivision property for the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a> by the following theorem.

Theorem 2.3

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M29">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M30">View MathML</a>

(4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M31">View MathML</a>denotes the classical Bernstein basis function.

Remark 2.4 Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M12">View MathML</a> into (4), we obtain the subdivision property for the classical Bernstein basis functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M33">View MathML</a>

Using (3), we give the following partial differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M34">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M35">View MathML</a>

By applying these partial differential equations, we obtain the following derivative relations which are related to the subdivision property unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>, respectively:

Theorem 2.5

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M37">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M38">View MathML</a>

3 Integral representations

In this section, we derive integral representations of the unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>. We also give an identity which connects the binomial coefficients, gamma and beta functions.

The beta function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M40">View MathML</a> is a function of two complex variables α and β, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M41">View MathML</a>

(5)

(cf. [[8], p.9, Eq. (60)]). The beta function is related to the gamma function; one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M42">View MathML</a>

Replacing α by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M43">View MathML</a> and β by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M44">View MathML</a> in the above equation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M45">View MathML</a>

(6)

(cf. [[8], p.9, Eq. (62)]).

Theorem 3.1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M46">View MathML</a>

(7)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M47">View MathML</a>

(8)

Proof

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M48">View MathML</a>

(9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M49">View MathML</a>

By using (5), we easily arrive at the desired result. □

Binomial coefficients play an important role in mathematics and mathematical physics, especially in statistics, probability and analytic number theory. Therefore, by using (7) and (8), we derive the following identity related to the binomial coefficients, gamma and beta functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M50">View MathML</a>

4 Identities

In this section, by using the Laplace transform, we give some identities of the unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>.

Using the generating function in (2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M52">View MathML</a>

(10)

Integrating equation (10) (by parts) with respect to t from zero to infinity, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M53">View MathML</a>

(11)

If we appropriately use the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M54">View MathML</a> of the following Laplace transform of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M55">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M56">View MathML</a>

(12)

by substituting (12) into (11), we arrive at the following theorem.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M57">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M58">View MathML</a>

(13)

Remark 4.2 If we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M12">View MathML</a> in (13), then we arrive at Theorem 15 in [7].

We modify (2) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M60">View MathML</a>

From the above equation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M61">View MathML</a>

Therefore, we arrive at the following theorem.

Theorem 4.3

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M62">View MathML</a>

5 Marsden identity

In this section, by using generating functions, we prove the Marsden identity for the unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>. This identity is associated with a formula for rational linear transformation of B-splines, which are of interest in computer-aided geometric design and approximation theory.

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M64">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M65">View MathML</a>

We derive the following functional equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M66">View MathML</a>

From the above functional equation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M67">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M68">View MathML</a>

(14)

From the above equation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M69">View MathML</a>

By comparing the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M70">View MathML</a> on the both sides of the above equation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M71">View MathML</a>

Therefore, we arrive at the Marsden identity which is given by the following theorem.

Theorem 5.1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M72">View MathML</a>

Remark 5.2 By using (14), we also obtain the Marsden identity for the classical Bernstein polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M73">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M74">View MathML</a>

6 Relations between the polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a>, unification of the Bernoulli polynomial of higher order and Stirling numbers of the second kind

The so-called unification of the Bernoulli, Euler and Genocchi polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M76">View MathML</a> were defined by Ozden [9]. The polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M76">View MathML</a> are defined by means of the following generating function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M78">View MathML</a>

(15)

where k is an integer parameter, a and b are real parameters and β is a complex parameter. Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M79">View MathML</a>

(cf. [9,10]).

The above generating function is related to some special polynomials as follows.

Remark 6.1 Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M80">View MathML</a> into (15), we have the Apostol-Bernoulli polynomials (cf. [11-13]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M81">View MathML</a>

substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M84">View MathML</a> into (15), we have the Apostol-Euler polynomials:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M85">View MathML</a>

substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M84">View MathML</a> into (15), we have the Apostol-Genocchi polynomials:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M89">View MathML</a>

substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M90">View MathML</a> into (15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M91">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M92">View MathML</a> denotes the classical Bernoulli polynomials and substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M84">View MathML</a> into (15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M95">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M96">View MathML</a> denotes the classical Euler polynomials.

Now, the modification of (15) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M97">View MathML</a>

(16)

The following definition provides a natural generalization and unification of λ-Stirling numbers of the second kind, which is defined by Srivastava [12,13].

Definition 6.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M99">View MathML</a>. The generalized λ-Stirling type numbers of the second kind <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M100">View MathML</a> are defined by means of the following generating function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M101">View MathML</a>

(17)

Remark 6.3 By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M102">View MathML</a> in (17), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M103">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M104">View MathML</a> denotes the Stirling numbers of the second kind. It is also well known that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M105">View MathML</a>

(18)

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M106">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M107">View MathML</a> being the Kronecker symbol (cf. [8,12,14,15]).

Theorem 6.4Letb, nandsbe nonnegative integers with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M10">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M109">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M110">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M111">View MathML</a>denote the unification Bernoulli polynomial of higher order and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M112">View MathML</a>-Stirling numbers of the second kind, respectively.

Proof By (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M113">View MathML</a>

By using (2), (16) and (17) in the above equation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M114">View MathML</a>

From the above equation, after some calculation, we find the desired result. □

Theorem 6.5Letbandnbe nonnegative integers with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M115">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M116">View MathML</a>

(19)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M117">View MathML</a>denotes the Euler polynomials of higher order.

Proof By (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M118">View MathML</a>

From the above, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M119">View MathML</a>

By using the Cauchy product in the above, after some calculation, we find the desired result. □

We recall from the work of Gould [[14], Vol. 7, Eq. (2.45)] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M120">View MathML</a>

(20)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M121">View MathML</a>

By substituting (20) into (19), we arrive at the following result.

Corollary 6.6The following identity holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M122">View MathML</a>

7 Unification of the Bernstein-type polynomials for solving high even-order differential equations by the Bernstein-Galerkin methods

In [4], Doha et al. gave an application of the Bernstein polynomials for solving high even-order differential equations by using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods. The methods do not contain generating functions for proving explicitly the derivatives formula of the Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves. Here, we prove this formula for the unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a> by a higher-order partial differential equation and functional equations. We also give some remarks and applications related to these polynomials and the Bernstein-Galerkin method.

We modify (1) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M124">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M126">View MathML</a>. Let b, k, n and s be nonnegative integers and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M127">View MathML</a>, then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M128">View MathML</a>

(22)

so that, obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M129">View MathML</a>

By using the same method as in [15], we now give a higher-order partial differential equation for the generating function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M130">View MathML</a> as follows.

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M131">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M132">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M133">View MathML</a>

By using Leibnitz’s formula for the vth derivative, with respect to x, of the product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M134">View MathML</a> of the above two functions, we obtain the following higher-order partial differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M135">View MathML</a>

By using (1) in the above partial differential equation, we get the following higher order partial differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M136">View MathML</a>

By substituting (21) into the above equation, after some calculation, we arrive at the following theorem.

Theorem 7.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M137">View MathML</a>. Letb, sandvbe nonnegative integers with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M10">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M139">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M140">View MathML</a>

Integrating equation (22) (by parts) with respect to x from 0 to 1 and using Theorem 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M141">View MathML</a>

for all b and s.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M142">View MathML</a>

(23)

We recall from the work of Doha et al. [4] that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M143">View MathML</a> is a differentiable function of degree m and defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M144">View MathML</a>, then a linear combination of the Bernstein polynomials can be written. Therefore, we can easily have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M145">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M146">View MathML</a>, otherwise <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M147">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M148">View MathML</a>

By using the same method as in [4], we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M149">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M146">View MathML</a>, otherwise <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M147">View MathML</a>. We now give an application for the solution of high even-order differential equations. We also recall from the work of Doha et al. [4] that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M153">View MathML</a>

(24)

by the following boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M154">View MathML</a>

(cf. [4]). By using the same method as that of Doha et al. [4], we apply the unification of the Bernstein-type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M155">View MathML</a> to the Bernstein-Galerkin approximation for solving (24); that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M156">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M157">View MathML</a>

By applying the Bernstein-Galerkin approximation (24), we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M158">View MathML</a> as follows. For solving this equation, we need the following notations, which we recall from the work of Doha et al. [[4], p.9, Eq. (4.4)].

The inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M159">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M160">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M161">View MathML</a>

By using this inner product, we modify (24) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M162">View MathML</a>

(25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M164">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M165">View MathML</a>

The matrix representation of the above equation is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M166">View MathML</a>

where

By using (23), one can easily find A, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M168">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M169">View MathML</a>); that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M170">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M171">View MathML</a>

Remark 7.2 According to Doha et al.[4], it is important to apply the Galerkin-spectral Bernstein approximation for how to choose an appropriate basis for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M172">View MathML</a> such that the linear system resulting in the Bernstein-Galerkin approximation to (25) is possible. That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M173">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M174">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M163">View MathML</a>. The 2m boundary conditions lead to the first m, and the least m expansion coefficients are zero.

Remark 7.3 By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, Doha et al.[4] solved the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M176">View MathML</a>

subject to the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M177">View MathML</a>, with the exact solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M178">View MathML</a>

(cf. see for detail [4,16]).

8 Further remarks on Bezier curves

The unification of the Bernstein-type polynomials is used to construct Bezier-type curves which are used in computer-aided graphics design and related fields and also in the time domain, particularly in animation and interface design (cf. [2,6]).

The Bezier-type curve of degree n can be generalized by the author [1] as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M179">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M5">View MathML</a> denotes the unification of the Bernstein-type polynomials and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/56/mathml/M182">View MathML</a> are the control points.

The unification of the Bernstein-type polynomials might affect the shape of the curves.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author completed the paper himself. The author read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.

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