In this article, we apply cubature formulas based on number-theoretic methods for approximate solution of the system of multidimensional integral equations φj(x1,x2,...,xn)=∫0x...∫0xΣr=1NKjr(x1,x2,...xs,y1,y2,...ys)××φj(y1,y2,...,ys)dy1dy2...dys+fj(x1,x2,...,xs)⁠, and we estimate the error of the approximate solution, where fj(x)∈Hs1(C1),Kjr∈H2s1(C2)

Topics
Integral equations
1.
N. S.
Bakhvalov
,
Numerical methods
, 1st ed. (M.: Science,
Boston
,
1975
) p.
631
.
Google Scholar
 
2.
N. P. Z. I. S.
Berezin
,
Calculation methods
(Science. T. 1.,
Moscow
,
1966
.) p.
466
p.
Google Scholar
 
3.
A. F.
Verlen
 and
A. S.
Sizikova
, “
Integral equations: Methods, algorithms, programs. reference manual
.” -
Kiev: Science thought.
12
,
543
p. (
1986
.), doi:
https://doi.org/10.1007/s10092-013-0080-x
.
Google Scholar
Crossref
Search ADS
 
4.
N. M.
Korobov
,
Approximate calculation of multiple integrals using number theory methods
(Nauka, DAN SSSR, -M.:
115
, No.
6
,
1957
,) pp. P.
1062
–
1065
.
Google Scholar
 
5.
N. M.
Korobov
, “
Application of number-theoretic grids in integral equations and interpolation formulas
,”
Tr. Matem. Inst. Academy of Sciences of the USSR
, - M.:
60
, S.
195
–
210
. (
1961
,).
Google Scholar
 
6.
N. M.
Korobov
,
On the calculation of optimal coefficients
(DAN SSSR. 1982. Volume
267
., Stockholm, No.
2
.
1959
).
Google Scholar
 
7.
N. M.
Korobov
, “
Trigometric sums and their applications.
” -M.:.
8.
M. I.
Israilov
 and
K. M.
Shadimetov
, “
Weighted optimal quadrature formulas for singular integrals of Cauchy type
,”
DAN Uz SSR.
8
.,
10
–
11
. (
1991
).
Google Scholar
 
9.
M. I.
Israilov
 and
K. M.
Shadimetov
, “
Optimal coefficients of weighted quadrature formulas for singular integrals of the cauchy type
,”
DAN Uz SSR.
11
,
7
–
9
. (
1991
.).
Google Scholar
 
10.
M. I.
Israilov
 and
I. M.
Abiraev
, “
Approximate solution of the system of Fredholm integral equations of the second kind using number-theoretic grids
,”
Uzbek Mathematical Journal.
2
,
43
–
51
(
1993
).
Google Scholar
 
11.
A. D.
Polyanina
 and
A. V.
Manzhirov
, “
Handbook of integral equations
,”
Physics and Mathematics
24
(
2003
).
Google Scholar
 
12.
E. V.
Sumin
,
V. B.
Sherstyukov
, and
O. V.
Sherstyukova
, “Fredholm and volterra integral equations, boundary value problems and methods for their solution.” Teaching aid.
Moscow
,
96
(
2016
.).
13.
A.
Avyt
, “
Approximate calculation of the Volterra-Stieltes linear integral equation of the second kind by the generalized trapezoid method
.”
Young scientist.
6
(
65
),
3
–
9
. (
2014
).
Google Scholar
 
14.
A. A.
Abdurashidov
, “
Aptoximate solution of linear and nonlinear Volterra integral equations by the method of variational iterations
,”
Young scientist
6
(
140
),
8
–
12
(
2017
).
Google Scholar
 
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