Research Article
On the Birkhoff Quadrature Formulas Using
Even and Odd Order of Derivatives
S. Hatami,
1
S. M. Hashemiparast,
1
and S. Shateyi
2
1
Department of Applied Mathematics, K. N. Toosi University of Technology, Jolfa Avenue, Seyed Khandan,
P.O. Box 15875-4416, Tehran, Iran
2
Department of Mathematics and Applied Mathematics, University of Venda, Private Bag X5050, Tohoyandou 0950, South Africa
Correspondence should be addressed to S. Shateyi; stanford.shateyi@univen.ac.za
Received 2 December 2014; Accepted 13 February 2015
Academic Editor: Bing-Jean Lee
Copyright © 2015 S. Hatami et al. Tis is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce some New Quadrature Formulas by using Jacoby polynomials and Laguerre polynomials. Tese formulas can be
obtained for a fnite and infnite interval and also separately for the even or odd order of derivatives. By using the properties of
error functions of the above orthogonal polynomials we can obtain the error functions for these formulas. Application of the new
approaches increases their precision degrees. Finally, some examples are given to illuminate the details.
1. Introduction
Let be a positive integer,
()
() the derivative of order
for the function (),
() the set of polynomials of degree
at most , and () a positive and integrable function on the
interval [,] throughout this paper.
We consider the quadrature formula as follows:
(;):=∫
()().
(1)
If
1
,...,
are chosen as the distinct zeros of orthog-
onal polynomial of degree in the family of orthogonal
polynomials [1, 2] associated with (), then the formula
(;)=
0
(;) :=
∑
=1
(
),
(2)
is exact for
2−1
. Tat is, the positive weights {
}
=1
are
usually determined in a way that formula (2) is exact for
the polynomials of degree as high as possible. Attempts to
obtain similar quadrature formulas have not been restricted
just to (); many researchers began to obtain some New
Quadrature Formulas based on () and its derivatives. For
example, Tur´ an was among the frst who considered in his
interesting paper in 1950 [3] the following quadrature rules:
(;) :=
2
∑
=0
∑
=1
()
(
,
). (3)
He showed that these rules have maximum degree of
precision as 2( + 1) − 1. Tur´ an’s attempt on quadrature
formulae attracted other researchers to expand this feld. Tey
began to follow his works and obtained several formulas [4–
8]. Te other case of quadrature formulae based on ()
and its derivatives is called Gaussian Birkhof quadrature.
For instance, Jetter [9] obtained new Gaussian quadrature
formulas based on Birkhof-type data. He used special cases
of data as pyramidal type data in incidence matrices. He
and Dyn [10] also worked on existence condition for these
quadrature formulas that are the generalized form of the
Gaussian quadrature. In another paper, he showed in [11]
that the formulas introduced in [9] are unique. Bojanov
and Nikolov [12] showed that the error of the quadrature
formulas depends monotonically on the data. Wang and
Guo [13] obtained the asymptotic estimate of nodes and
weights of Gaussian-Lobatto-Legendre-Birkhof quadrature
formulas. Tey presented a user-oriented implementation of
pesudospectral methods based on these quadrature nodes
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2015, Article ID 468629, 8 pages
http://dx.doi.org/10.1155/2015/468629