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SOLVING DEFINITE INTEGRALS USING NUMERICAL METHOD
Shaxobiddin Sobirovich Kuziev
Kokand University, Kokand, Uzbekistan
shaxobiddin.qoziyev.89@gmail.com
Abstract.
In this work, an algorithm for constructing the effective formula for
numerical computation of definite integrals in
( )
2
(0,1)
m
L
space is given. Using this
algorithm, a derivative quadrature formula is constructed using the first and second
order derivatives of the function for equally distributed nodal points on the section
[0,1]. For this, the form of the norm of the optimal error functional is found.
Finding the conditional extremum of the function was used to minimize the norm
of the error function. A system of linear algebraic equations for optimal coefficients
is obtained. It is proved that the solution of the system of equations exists and is
unique using the Wandermonde determinant. The formula constructed in this work
gives good results if the values of the derivatives of the function at the nodes are
given.
Keywords.
Sobolev space, optimal error function, Lagrange multipliers,
Wandermonde determinant, derivative optimal quadrature formula.
INTRODUCTION
As a result of numerous scientific and practical researches carried out on a
global scale, modeling of synthesized holograms, mechanics of liquids and gases
leads to the construction of optimal quadrature formulas. Usually, the use of simple
interpolation quadrature formulas in solving such problems requires large-scale
computing. Therefore, creating optimal numerical solution algorithms that allow
calculating the solutions of typical problems in mathematics with sufficient
Yangi O'zbekiston taraqqiyotida tadqiqotlarni o'rni va rivojlanish omillari
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accuracy and developing ways to use modern computing tools for this purpose, as
well as constructing optimal quadrature formulas derived in certain Gilbert and
Banach spaces and evaluating their errors are important tasks of computational
mathematics. is one of the directions. Nowadays, the construction of derivative
optimal quadrature formulas in the approximate calculation of exact integrals is of
great importance. In particular, using the values of the functions up to the second-
order derivative at the nodes in certain Gilbert and Banach spaces, the construction
of derivative optimal quadrature formulas and the estimation of their errors are
widely used. Professor Kh.M. Shadimetov and his students S.L. Sobolev
’
s research
continues, and new results are obtained on the construction of formulas in
°
( )
2
(0,1)
m
L
,
( )
2
( )
m
L
R
,
( )
2
( )
m
K
P
spaces [1]. We consider this formula
1
2
0
1
0
0
0
( )
(
)
( '(0)
'(1))
''(
)
12
N
N
h
x dx
C
h
C
h
=
=
+
−
+
(1)
this difference (1) is called the error of the quadrature formula
(
)
1
2
0
1
0
0
0
( )
(
)
( '(0)
'(1))
''(
)
1
,
2
N
N
N
h
x dx
C
h
C
h
=
=
−
−
=
−
−
l
the error functional corresponding to this difference is
2
0
1
0,1
0
0
( )
( )
(
)
( '( )
'(
1))
''(
),
12
N
N
N
h
x
x
C
x h
x
x
C
x h
=
=
=
−
−
+
−
−
−
−
l
(2)
where,
( )
x
is the Dirac
’s
delta-function,
[0,1]
( )
x
is the characteristic function of
the interval
[0,1]
,
1
[ ]
C
are the unknown coefficients of the quadrature formula (1),
( )
2
(0,1)
m
f
L
,
( )
2
(0,1)
m
L
is the space of functions which are square integrable with
m
-th
generalized derivative.
The problem of constructing a derived quadrature formula of the form (1) in
the
( )
2
(0,1)
m
L
space is given [2].
ANALYTICAL EXPRESSIONS FOR THE COEFFICIENTS OF THE
QUADRATIC FORMULA
To find the coefficients of the derivative optimal quadrature formula, we have
Yangi O'zbekiston taraqqiyotida tadqiqotlarni o'rni va rivojlanish omillari
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the following system of equations [22]
2
5
1
3
0
[ ]
(
)
(
),
0, ,
2(2
5)!
m
N
m
m
h
h
C
P
h
f h
N
m
−
−
=
−
+
=
=
−
(3)
3
2
5
3
2
2
2
5
3
3
2
2
0
1
( 1)
(
)
(
)
(2
5
)!
(
3)!
2 !(
3
)!
(2
2)!
i
i
j
m
i
i
m
m
i
i
j
i
m
m
i
j
B
h
B h
B
h
h
f h
m
i
i
j i
j
m
+ −
−
+
−
− −
+ −
+
−
=
=
−
=
−
+
+
− −
+
+ −
−
(4)
3
3
1
0
1
!
[ ](
)
,
0,
3.
!(
3
)!
j
N
j
j
B
h
C
h
m
j
j
+ −
+ −
=
=
= −
=
−
+ −
(5)
To solve a system of linear algebraic equations (3) - (5) with
4
m
in
( )
2
(0,1)
m
L
space, we use the approach based on
2
[ ]
m
D
−
discrete analog of
2
2
2
2
/
m
m
d
dx
−
−
differential operator. For this,
< 0
and
>
N
are
1
[ ] = 0
C
, and we rewrite equation
(3) in the convolutional form
2
1
3
(
)* [ ]
(
) =
(
),
= 0,1,...,
m
m
m
G
h
C
P
h
F h
N
−
−
+
(6)
and enter the following definitions:
2
1
(
) =
(
)* [ ],
m
v h
G
h
C
−
(7)
3
(
) = (
)
(
)
m
u h
v h
P
h
−
+
(8)
2
[ ]
m
D
−
discrete analogue of
2
4
2
4
/
m
m
d
dx
−
−
differential operator satisfying
2
2
(
)*
(
) = (
)
m
m
hD
h
G
h
h
−
−
equality was constructed and its properties were studied.
1
2
[ ] =
(
)* (
).
m
C
hD
h
u h
−
(9)
To calculate the convolution of (7), we need to find the representation of the
function
(
)
u h
for all integer values of
. If
[0,1]
h
,
(
) =
(
)
m
u h
f h
. For
< 0
and
>
N
we find the representation of
(
)
u h
.
For
< 0
2
5
2
5
1
2
1
1
=0
|
|
| (
) ( ) |
(
) =
[ ]*
(
)
[ ]*
=
[ ]
2(2
5)!
2(2
5)!
N
m
m
m
h
h
h
v h
C
G
h
C
C
m
m
−
−
−
−
=
=
−
−
3
3
2
5
2
5
2
5
3
1
=0
1
=
2
=0
!
(
)
( 1)
(
)
( 1)
=
[ ]( )
2(2
5
)! !
!(
3
)!
2(2
5
)! !
i
j
m
i
m
N
m
i
i
m
i
i
i
j
i
i
j
i m
i B
h
h
h
C
h
m
i i
j i
j
m
i i
+ −
−
−
− −
− −
+ −
=
−
−
−
−
− −
+ −
− −
.
for
>
N
3
3
2
5
2
5
2
5
3
1
=0
1
=
2
=0
!
(
)
( 1)
(
)
( 1)
(
) =
[ ]( )
2(2
5
)! !
!(
3
)!
2(2
5
)! !
i
j
m
i
m
N
m
i
i
m
i
i
i
j
i
i
j
i m
i B
h
h
h
v h
C
h
m
i i
j i
j
m
i i
+ −
−
−
− −
− −
+ −
=
−
−
−
−
+
− −
+ −
− −
.
Yangi O'zbekiston taraqqiyotida tadqiqotlarni o'rni va rivojlanish omillari
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We introduce the following definitions
3
3
2
5
3
2
5
=0
1
!
(
)
( 1)
(
) =
,
2(2
5
)! !
!(
3
)!
i
j
m
i
m
i
i
i
j
m
i
j
i B
h
h
R
h
m
i i
j i
j
+ −
−
− −
+ −
−
=
−
− −
+ −
2
5
2
5
3
1
=
2
=0
(
)
( 1)
(
) =
[ ]( )
2(2
5
)! !
m
N
m
i
i
i
m
i m
h
Q
h
C
h
m
i i
−
− −
−
−
−
− −
(10)
after these designations
2
5
3
2
5
3
(
)
(
),
0,
(
)
(
)
(
),
.
m
m
m
m
R
h
Q
h
v h
R
h
Q
h
N
−
−
−
−
−
=
−
+
(11)
So,
2
5
3
2
5
3
(
)
(
),
< 0,
(
)
(
) ,
0
,
(
)
(
),
< 0,
m
m
m
m
m
R
h
Q
h
u h
F h
N
R
h
Q
h
−
−
−
+
−
−
+
=
−
+
(12)
where
3
3
3
3
3
3
(
) =
(
)
(
),
(
) =
(
)
(
).
m
m
m
m
m
m
Q
h
P
h
Q
h
Q
h
P
h
Q
h
−
−
−
−
+
−
−
−
−
+
(13)
3
(
)
m
Q
h
−
−
va
3
(
)
m
Q
h
+
−
are unknown polynomials of degree
(
3)
m
−
.
Theorem.
1
[ ]
C
,
= 1, 2,...,
1
N
−
coefficients of derivative optimal quadrature
formula of form (1) for
4
m
in
( )
2
(0,1)
m
L
space have the following form
[3,4,5]
(
)
3
3
1
=1
[ ] =
,
= 1, 2,...,
1
m
N
k k
k k
k
C
h
a q
b q
N
−
−
+
−
(14)
So,
1
[ ]
C
optimal coefficients depend on
2
6
m
−
unknowns, and
2
6
m
−
systems
of equations are needed to determine them. First we find from equation (5) the
expression of the coefficients
1
[0]
C
,
1
[ ]
C N
for
0
=
and
1
=
1
1
1
1
1
=1
=1
[0] =
[ ](
)
[ ],
N
N
C
C
h
C
−
−
−
−
(15)
1
1
1
=1
[ ] =
[ ](
).
N
C N
C
h
−
−
(16)
It can be seen from equations (15) and (16) that coefficients
1
[0]
C
and
1
[ ]
C N
are expressed using coefficients
1
[ ] (
1,
1)
C
N
=
−
. Therefore, we find the
coefficients
1
[0]
C
and
1
[ ]
C N
. For this, we calculate the following sum in the system
of equations (3)
Yangi O'zbekiston taraqqiyotida tadqiqotlarni o'rni va rivojlanish omillari
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(
)
2
5
1
2
1
=0
=0
(
) ( )
=
[ ]
(
) ( ) =
[ ]
2(2
5)!
m
N
N
m
h
h
S
C
G
h
h
C
m
−
−
−
−
=
−
(
)
(
)
2
5
2
5
1
1
=0
=0
(
) ( )
(
) ( )
=
[ ]
[ ]
(2
5)!
2(2
5)!
m
m
N
h
h
h
h
C
C
m
m
−
−
−
−
−
=
−
−
(
)
(
)
2
5
2
5
1
2
5
2
5
1
1
1
1
1
2
=1
=0
(
) ( )
(
) ( )
(
)
(
)
=
[0]
[ ]
[ ]
[0]
(2
5)!
(2
5)!
2(2
5)!
(2
5)!
m
m
N
m
m
h
h
h
h
h
h
C
C
C
C
S
S
m
m
m
m
−
−
−
−
−
−
−
+
−
=
+ −
−
−
−
−
(17)
We simplify the sums
1
S
and
2
S
in the last expression
2
5
3
2
5
3
2
5
3
2
5
1
0
1
0
1
0
(
)
1
0
0
!(2
5
)!
1
1
1
1
i
i
m
m
m
m
m
N
j
m
j
i
j
i
j
k k
k k
k
k
k
k
k
j
k
i
k
i
a q
b q
q
h
h
S
j
m
j
q
q
q
q
−
−
−
−
−
+
− −
=
=
=
=
=
= −
+
− −
−
−
−
−
(18)
we will simplify the sum of
2
S
. For this, using the orthogonality condition (5),
we obtain the following
2
5
2
5
2
1
0
0
(
)
(
)
[ ]
2 !(2
5
)!
N
m
m
j
j
j
h
h
S
C
j
m
j
−
− −
=
=
−
=
=
− −
3
2
5
3
2
5
2
5
3
1
2
0
0
0
!
(
)
( 1)
(
)
( 1)
[ ]( )
2 !(2
5
)!
2 !(2
5
)!
!(
3
)!
j
i
j
m
N
m
m
j
j
m
j
j
j
i
j
j m
j
i
j B
h
h
h
C
h
j
m
j
j
m
j
i j
i
+ −
−
−
− −
− −
+ −
= −
=
=
=
−
−
=
−
− −
− −
+ −
.
(19)
putting equations (18), (19) into (17), we get the following sum
2
5
3
2
5
2
5
3
2
5
1
0
1
0
(
)
(
)
1
=
[0]
0
(2
5)!
!(2
5
)!
1
1
i
m
m
m
m
j
m
j
i
j
k k
k
k
j
k
i
a q
h
h
h
S
C
m
j
m
j
q
q
−
−
−
−
+
− −
=
=
=
−
+
−
− −
−
−
3
3
2
5
2
5
3
2
5
2
5
3
1
1
0
2
0
0
0
!
(
)
( 1)
(
)
( 1)
0
[ ]( )
1
1
2 !(2
5
)!
2 !(2
5
)!
!(
3
)!
i
j
i
j
m
m
m
N
m
N
m
j
j
m
j
j
j
i
i
j
j
k k
k
k
k
k
i
j m
j
i
j B
h
b q
q
h
h
C
h
q
q
j
m
j
j
m
j
i j
i
+ −
−
−
−
−
− −
− −
+ −
=
=
= −
=
=
=
−
−
+
−
+
−
−
− −
− −
+ −
(20)
CONCLUSION
It is known that numerical analytical solutions of differential equations and
integral equations formed in mathematical modeling of natural processes are
expressed by optimal quadrature and cubature formulas and are the research object
of numerical algebra, numerical integration theory and other similar issues. In this
regard, it is important to calculate the approximation of exact integrals, as well as
to estimate their errors in the Gilbert and Banach spaces of differentiable functions,
to construct derivative optimal quadrature formulas. In this work, using the
Yangi O'zbekiston taraqqiyotida tadqiqotlarni o'rni va rivojlanish omillari
20-
to’plam
2-son Iyun 2025
174
algorithm proposed by Sobolev, a new derived optimal quadrature formula was
constructed.
REFERENCES
1.
Sobolev S.L. Introduction to the theory of cubature formulas. - M.:
Nauka, 1974. - 808 p.
2.
Shadimetov, K., Nuraliev, F., Kuziev, S. Optimal quadrature formula
of hermite type in the space of differentiable functions // International Journal
of
Analysis
and
Applications,
2024,
22,
25,
pp.
1-19.
https://doi.org/10.28924/2291-8639-22-2024-25
3.
Shadimetov, K., Nuraliev, F., Kuziev, S. Coefficients and Errors of
the Optimal Quadrature Formula of the Hermite Type // AIP Conference
Proceedings,
2024,
3147(1),
030030,
pp.
1-12.
https://doi.org/10.1063/5.0210357
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Qo
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ziyev, S. (2021, April). Methods, tools and forms of distance
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Конференции
.
5.
Nuraliev, F. A., & Kuziev, S. S. (2024). Optimal Quadrature Formulas
with Derivative in the Space: Optimal Quadrature Formulas with Derivative in
the Space.
MODERN PROBLEMS AND PROSPECTS OF APPLIED
MATHEMATICS
,
1
(01).
6.
