International scientific journal
“Interpretation and researches”
Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2
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NOCHIZIQLI INTEGRO-DIFFERENSIAL TENGLAMALAR YECHIMLARI
HAQIDA
Xolboyev Nurjon Abdijabbor o‘g‘li
JDPU o‘qituvchisi
Saydaliyeva Dilshoda Rustam qizi
JDPU magistranti
Annotatsiya.
Impulsiv,
maksimallarga
ega
oddiy
integro-differensial
tenglamalar sistemasi uchun nolokal chegaraviy masala o‘rganildi. Chegaraviy
masala integral shart bilan berilgan. Chegaraviy masala yechimining mavjudligi
isbotlangan.
Kalit so'zlar:
impulsiv integro-differensial tenglamalar, nolokal chegaraviy
shart, ketma-ket yaqinlashishlar, yechimning mavjudligi va yagonaligi, yechimning
uzluksiz bog‘liqligi.
Quyidagi integro-differensial tenglamalar sistemasini ko‘rib chiqamiz
[0, ],
,
1,2,...,
i
t
T
t
t i
p
=
lar uchun
(
)
1
2
0
( )
( , ) ( )
, ( ), max
( ) |
;
T
x t
H t s x s ds
f t x t
x
h h
=
+
(1)
tenglama nolokal chegaraviy shart bilan
0
(0)
( ) ( )
T
Ax
K t x t dt
B
+
=
(2)
va impuls
( ) ( )
( )
(
)
,
1,2,..., ,
i
i
i
i
x t
x t
I x t
i
p
+
−
−
=
=
(3)
bilan berilgan bo‘lsin, bu yerda
( )
0
1
1
0
...
,
,
n n
n n
p
p
t
t
t
t
T A R
K t
R
+
=
=
berilgan matritsa va
( )
0
det
0,
,
T
Q
Q
A
K t dt
= +
:[0, ]
,
n
n
n
f
T
R
R
R
→
:
n
n
i
I
R
R
→
berilgan funksiyalar;
1
2
0
,
h
h
t
( , ( )),
1, 2,
j
j
h
h
t x t
j
=
=
noldan farqli haqiqiy parametr,
( )
(
)
0
lim
,
i
i
h
x t
x t
h
+
+
→
=
+
( )
(
)
0
lim
i
i
h
x t
x t
h
−
−
→
=
−
( )
x t
funksiyaning mos ravishda
i
t
t
=
nuqtadagi o'ng va chap limitlari.
International scientific journal
“Interpretation and researches”
Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2
49
(
)
[0, ],
n
C
T R
fazoni aniqlanish sohasi
[0, ]
T
, qiymatlar sohasi
n
R
bo‘lgan
( )
x t
uzluksiz vektor funksiyalardan tashkil topgan Banax fazosi sifatida kiritamiz. Bu
fazoda norma quyidagicha kiritilgan:
[0, ]
1
max
( ) .
n
j
t
T
j
x
x t
=
=
(
)
[0, ],
n
PC
T R
chiziqli fazoni quyidagicha belgilaymiz:
(
)
(
(
)
1
0,
,
:[0, ]
; ( )
,
,
,
1,...,
,
n
n
n
i
i
PC
T R
x
T
R
x t
C t t
R
i
p
+
=
→
=
bu yerda
( )
i
x t
+
va
( )
i
x t
−
(
0,1,..., )
i
p
=
mavjud va chegaralangan;
( ) ( )
i
i
x t
x t
−
=
.
Malumki,
(
)
[0, ],
n
PC
T R
chiziqli fazo Banax fazosi bo‘lib, bu fazoda norma
quyidagicha kiritiladi:
1
(( ,
])
max
,
1, 2,...,
.
i i
PC
C t t
x
x
i
p
+
=
=
Izlanayotgan
(
)
( )
[0, ],
n
x t
PC
T R
funksiya, barcha
[0, ],
,
1,2,...,
i
t
T
t
t i
p
=
lar
uchun
(1)
integro-differensial
tenglamani
qanoatlantiradi,
1
2
1, 2,..., , 0
...
i
p
t
t i
p
t
t
t
T
=
=
lar uchun (2) nolokal integral shartni va (3)
chegaraviy shartni qanoatlantiradi.
Aytaylik,
(
)
( )
[0, ],
n
x t
PC
T R
funksiya (1)-(3) masalaning yechimi bo‘lsin.
(
)
1
2
1
0
( )
( )
( ) ( )
, ( ), max
( ) |
;
.
T m
k
k
k
x t
a
t b
s x s ds
f t x t
x
h h
=
=
+
yuqoridagi tenglamada quyidagicha belgilash kiritamiz:
0
( ) ( )
T
k
k
c
b
s x s ds
=
so‘ng integro-differensial tenglama quyidagi ko‘rinishga keladi:
(
)
1
2
1
( )
( )
, ( ),max
( ) |
;
.
m
k
k
k
x t
a
t c
f t x t
x
h h
=
=
+
U holda oxirgi tenglamani
(
1
0,
,
i
t
t
+
intervalda integrallash orqali quyidagi
tenglikka ega bo‘lamiz:
1
0
( )
( , ( ), )
t
m
k
k
k
a
s c
f s x s
ds
=
+
=
( )
( )
( )
( )
( )
( )
1
2
1
0
( )
0
...
t
i
x s ds
x t
x
x t
x t
x t
x t
+
+
+
=
=
−
+
−
+ +
−
=
International scientific journal
“Interpretation and researches”
Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2
50
( )
( )
( )
( )
( )
( )
( )
( )
1
1
2
2
0
...
.
i
i
x
x t
x t
x t
x t
x t
x t
x t
+
+
+
= −
−
−
−
−
− −
−
+
(2) integral shartni oxirgi tenglikka qo‘llasak, quyidagi tenglikka ega bo‘lamiz:
( )
(
)
1
0
0
( )
(0)
( )
( , ( ), )
.
i
t
m
k
k
i
i
k
t t
x t
x
a
s c
f s x s
ds
I x t
=
=
+
+
+
(4)
(4) dagi
(
)
( )
[0, ],
n
x t
PC
T R
funksiya (2) chegaraviy shartni qanoatlantiradi:
0
( )
(0)
T
A
K t dt x
+
=
( )
(
)
1
0
0
0
0
( )
( )
( , ( ), )
( )
.
i
T
t
T
m
k
k
i
i
k
t t
B
K t
a
s c
f s x s
dsdt
K t
I x t
dt
=
= −
+
−
(5)
det
0
Q
dan va (5) tenglikdan biz quyidagi tenglamaga ega bo‘lamiz:
( )
(
)
1
1
0
0
0
0
(0)
( )
( )
( , ( ), )
( )
.
i
T
t
T
m
k
k
i
i
k
t
t
x
Q
B
K t
a
s c
f s x s
dsdt
K t
I x t
dt
−
=
=
−
+
−
(6)
(6) tenglikni (4) tenglikka qo‘yib, quyidagi
1
1
0
0
0
0
( )
( )
( )
( , ( ), )
( )
i
T
t
T
m
k
k
k
t
t
x t
Q
B
K t
a
s c
f s x s
dsdt
K t
dt
−
=
=
−
+
−
+
( )
(
)
0
0
( , ( ), )
.
i
t
i
i
t t
f s x s
ds
I x t
+
+
(7)
tenglamaga ega bo‘lamiz. Quyidagi
1
1
0
0
0
( )
( )
( , ( ), )
( )
( )
( , ( ), )
,
T
t
T T
m
m
k
k
k
k
k
k
t
K t
a
s c
f s x s
dsdt
K s ds
a
t c
f t x t
dt
=
=
+
=
+
( )
(
)
( )
(
)
0
0
0
( )
( )
,
i
i
i
T
T
i
i
i
i
t t
t T t
K t
I x t
dt
K t dt I x t
=
tengliklarni (7) tenglikka qo‘yib
1
1
1
0
( )
( )
( )
( , ( ), )
T T
m
k
k
k
t
x t
Q B
Q
K s ds
a
t c
f t x t
dt
−
−
=
=
−
+
−
( )
(
)
( )
(
)
1
0
1
0
0
( )
( )
( , ( ), )
.
i
i
i
T
t
m
i
i
k
k
i
i
t t
k
t t
t
Q
K t dt I x t
a
s c
f s x s
ds
I x t
−
=
−
+
+
+
(8)
tenglamaga ega bo‘lamiz.
International scientific journal
“Interpretation and researches”
Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2
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Berilgan (8) ifodani biroz soddalashtiraylik. U holda quyidagi tenglik hosil
bo‘ladi:
1
1
1
0
0
( )
( , ( ), )
( )
( )
( , ( ), )
t
T T
m
m
k
k
k
k
k
k
t
a
s c
f s x s
ds
Q
K s ds
a
t c
f t x t
dt
−
=
=
+
−
+
=
1
1
0
0
( )
( )
( , ( ), )
t
m
k
k
k
Q
A
K s ds
a
c
f
x
d
−
=
=
+
+
−
1
1
0
( )
( )
( , ( ), )
;
T T
m
k
k
k
t
Q
K s ds
a
c
f
x
d
−
=
−
+
(9)
( )
(
)
( )
(
)
1
0
0
( )
i
i
i
T
i
i
i
i
t t
t T t
I x t
Q
K t dt I x t
−
−
=
( )
(
)
( )
(
)
1
1
1
0
0
( )
( )
.
i
i
i
i
t
T
i
i
i
i
t t
t t
T
t
Q
A
K t dt I x t
Q
K t dt I x t
+
−
−
=
+
−
(10)
(9) va (10) tengliklarni hisobga olib, (8) tenglikni quyidagi integral tenglama
ko‘rinishida yozamiz:
(
1
,
,
0,1,..., ,
i
i
t
t t
i
p
+
=
lar uchun
( )
( )
(
)
1
0
( )
i
i
i
i
t t
x t
Q B
G t I x t
−
=
+
+
( )
1 0
0
( )
( )
,
( , ( ), )
T
T
m
k
k
k
G s a
s c ds
G t s f s x s
ds
=
+
+
(11)
bu yerda
1
0
1
( )
,
0
,
( )
( )
,
.
t
T
t
Q
A
K s ds
s
t
G t
Q
K s ds
t
s
T
−
−
+
=
−
(11) tenglamada
0
( ) ( )
,
T
k
k
c
b
s x s ds
=
belgilashni hisobga olib, quyidagi chiziqli algebraik tenglamalar sistemasini
yozamiz (ChATS)
(
)
1
2
1
,
,
m
k
k j
k j
k
k
i
j
c
c
f I
=
+
= +
1,
,
k
m
=
(12)
bu yerda
International scientific journal
“Interpretation and researches”
Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2
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(
)
0
0
,
( )
( )
( )
,
T
T
k j
i
k
j
f I
b
s
G
a
d ds
=
1
1
0
( )
,
T
k
k
Q B b
s ds
−
=
(
)
(
)
2
1
2
0
0
,
( )
( )
, ( ), max
( ) |
;
T
T
k
i
k
f I
b
s
G
f
x
x
h h
d
=
+
( )
( )
(
)
0
,
1, ,
( , ( )),
1, 2.
i
i
i
i
l
l
t t
G t I x t
ds k
m h
h t x
l
+
=
=
=
(13)
Agar quyidagi Fredgolm sharti bajarilsa, (12) ning (ChATS) o'ng tomoni yagona
chekli yechimga ega bo‘ladi:
11
12
1
21
2 2
2
1
2
1
. . .
1
. . .
( )
0.
. . .
. . .
. . .
. . .
. . .
1
k
k
k m
k
k
k m
k
k m
k m
k m m
+
+
=
+
(14)
Foydalanilgan adabiyotlar.
1.
Benchohra M., Henderson J., Ntouyas S.K. Impulsive differential
equations and inclusions. Contemporary mathematics and its application. New York:
Hindawi Publishing Corporation, 2006.
2.
Yuldashev T.K. Degeneratsiyalangan yadroli Boussinesq tipidagi
integrodifferensial tenglama uchun nolokal aralash qiymatli muammo, Ukraina
Matematik jurnali, 2016, jild. 68, No 8, S. 1278-1296
3.
Xolboyev N. IKKI KARRALI FURE QATORLARINING DOIRAVIY
QISMIY
YIG‘INDISI
UCHUN
UMUMLASHGAN
LOKALIZATSIYA
MASALASI //Журнал математики и информатики. – 2022. – Т. 2. – №. 2.
4.
Xolboyev Nurjon, & Suyarova Gulsora. (2024). MAKSIMAL
NOCHIZIQLI IMPULSLI DIFFERENSIAL TENGLAMALAR. Uz-Conferences,
1(1),
960–964.
Retrieved
from
conference.com/index.php/p/article/view/200
5.
Qahhorov M., Nurjon X. ЧЕКЛИ СТЕРЖЕНДА ИССИҚЛИК
ОҚИМИНИ БОШҚАРИШ //Журнал математики и информатики. –2022. –Т. 2. –
No. 1.
6.
Xolboyev , N. A. o‘g‘li. (2023). MAKSIMAL NOCHIZIQLI
INTEGRO-DIFFERENSIAL
TENGLAMALAR
SISTEMASI
UCHUN
CHEGARAVIY MASALA. Educational Research in Universal Sciences, 2(2), 269–
273. Retrieved from
http://erus.uz/index.php/er/article/view/1725
7.
Xolboyev , N. A. o‘g‘li, & Po‘latov , S. S. o‘g‘li. (2023).
DIFFERENSIAL OPЕRATORIGA MOS SPЕKTRAL YOYILMALAR. Educational
International scientific journal
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Volume 1 issue 3 (49) | ISSN: 2181-4163 | Impact Factor: 8.2
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Research
in
Universal
Sciences,
2(4),
523–525.
Retrieved
from
http://erus.uz/index.php/er/article/view/2185
https://www.researchgate.net/publication/378008469_Differensial_operatorlarga_
