American Journal of Numerical Analysis, 2013, Vol. 1, No. 1, 22-31
Available online at http://pubs.sciepub.com/ajna/1/1/4
© Science and Education Publishing
DOI:10.12691/ajna-1-1-4
Optimal Quadrature Formulas for the Cauchy Type
Singular Integral in the Sobolev Space
(2)
2
( 1,1) L −
Kholmat M. Shadimetov, Abdullo R. Hayotov
*
, Dilshod M. Akhmedov
Department of Computational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan
*Corresponding author: hayotov@mail.ru
Received October 12, 2013; Revised November 20, 2013; December 03, 2013
Abstract This paper studies the problem of construction of the optimal quadrature formula in the sense of Sard in
(2)
2
( 1,1) L − S.L.Sobolev space for approximate calculation of the Cauchy type singular integral. Using the discrete
analogue of the operator
4 4
/ d dx we obtain new optimal quadrature formulas. Furthermore, explicit formulas of the
optimal coefficients are obtained. Finally, in numerical examples, we give the error bounds obtained for the case
0.02 h = by our optimal quadrature formula and compared with the corresponding error bounds of the quadrature
formula (15) of the work [26] at different values of singular point t . The numerical results show that our quadrature
formula is more accurate than the quadrature formula constructed in the work [26].
Keywords: optimal quadrature formula, singular integral of Cauchy type, Sobolev space
Cite This Article: Kh.M. Shadimetov, A.R. Hayotov, and D.M. Akhmedov, “Optimal Quadrature Formulas
for the Cauchy Type Singular Integral in the Sobolev Space
(2)
2
( 1,1) L − .” American Journal of Numerical Analysis
1, no. 1 (2013): 22-31. doi: 10.12691/ajna-1-1-4.
1. Introduction
Many problems of sciences and technics is naturally
reduced to singular integral equations. Moreover (see. [1])
plane problems are reduced to one dimensional singular
equations. The theory of one domensional singular
integral equations is given in [2,3]. It is known that the
solutions of such integral equations are expressed by
singular integrals. Therefore approximate calculation of
the singular integrals with high exactness is actual
problem of numerical analysis.
For the singular integral of Cauchy type
1
1
() x
dx
x t
ϕ
−
−
∫
we
consider the following quadrature formula
1
=0
1
()
[ ]( )
N
x
dx C x
x t
β
β
ϕ
βϕ
−
≅
−
∑
∫
(1.1)
with the error functional
[ 1,1]
=0
()
() [ ]( ),
N
x
x C x x
x t
β
β
ε
βδ
−
= − −
−
∑
(1.2)
where 1< < 1, t − [ ] C β are the coefficients, x
β
are the
nodes, = 2, 3, 4... N ,
[ 1,1]
() x ε
−
is the characteristic
function of the interval [ 1,1] − , δ is the Dirac delta
function, ϕ is a function of the space
( )
2
(0,1)
m
L . Here
( )
2
(0,1)
m
L is the Sobolev space of functions with a square
integrable m − th generalized derivative.
The difference
( )
1
=0
1
, ()()
()
[ ]( )
N
x x dx
x
dx C x
x t
β
β
ϕ ϕ
ϕ
βϕ
∞
−∞
−
=
= −
−
∫
∑
∫
(1.3)
is called the error of the formula (1.1).
By the Cauchy-Schwarz inequality
( )
( ) ( )*
2 2
, | ( 1,1) | ( 1,1)
m m
L L ϕ ϕ ≤ − ⋅ −
the error (1.3) of the formula (1.1) on functions of the
space
( )
2
( 1,1)
m
L − is reduced to finding the norm of the
error functional in the conjugate space
( )*
2
( 1,1)
m
L − .
Obviously the norm of the error functional depends
on the coefficients and the nodes of the quadrature
formula (1.1). The problem of finding the minimum of the
norm of the error functional by coefficients and by
nodes is called the S.M. Nikol’skii problem, and the
obtained formula is called the optimal quadrature formula
in the sense of Nikol’skii. This problem was first
considered by S.M. Nikol’skii [4], and continued by many
authors, see e.g. [5-10] and references therein.
Minimization of the norm of the error functional by
coefficients when the nodes are fixed is called Sard’s
problem and the obtained formula is called the optimal
quadrature formula in the sense of Sard. First this
problem was investigated by A. Sard [11].
There are several methods of construction of optimal
quadrature formulas in the sense of Sard such as the spline