INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
Volume 11, issue 1, April 2025
https://wordlyknowledge.uz/index.php/IJSR
worldly knowledge
Index:
google scholar, research gate, research bib, zenodo, open aire.
https://scholar.google.com/scholar?hl=ru&as_sdt=0%2C5&q=wosjournals.com&btnG
https://www.researchgate.net/profile/Worldly-Knowledge
https://journalseeker.researchbib.com/view/issn/3030-332X
338
Choriyeva Sanam Tojiyevna
f.-m.f.f.d. (PhD), dots. TerDU.
Xamidova Sabina Zikirovna
Termiz davlat universiteti,2-kurs magistranti,
Ergashev Farxod Abdumurodovich
TerDU,Matematika ta’lim yo’nalishi 3-kurs talabasi.
NOKLASSIK MASALALAR
Annotatsiya:
Singulyar koeffitsientli aralash tipdagi tenglama uchun noklassik masalalarning
qo’yilishi, ta’riflangan masalalarning yechimlarining yagonaliklari isbotlanishi keltirilgan,
klassik shartlarning buzilishi haqida so’z yuritilgan.
Kalit so’zlar:
aralash tipdagi tenglama, klassik masala, noklassik masala, singulyar integral
tenglama, Protter sharti.
Abstract:
The formulation of non-classical problems for a mixed-type equation with singular
coefficients, the proof of the uniqueness of the solutions of the described problems, and the
violation of classical conditions are discussed.
Keywords:
mixed-type equation, classical problem, non-classical problem, singular integral
equation, Protter condition.
Buziluvchan giperbolik va aralash tipdagi tenglamalar nazariyasining rivojlanish tarixi G.
Darbu, F. Trikomi YE. Xolmgren va S.Gellerstedtlarning mos ravishda 1894, 1923, 1927 va
1935 yillarda chop etilgan fundamental ishlari bilan bog‘liq.
Aralash tipdagi tenglamalar uchun chegaraviy masalalar bо‘yicha dastlabki fundamental
tadqiqotlar 1920 yili italyan matematigi Franchesko Trikomi tomonidan olib borilgan. Bu ishdan
keyin aralash tipdagi tenglamalar uchun chegaraviy masalalar nazariyasi asosan uchta yо‘nalish
bо‘yicha rivojlana boshladi: birinchi yо‘nalish - Trikomi masalasini umumiyroq aralash tipdagi
tenglamalar uchun о‘rganish bо‘lib, ularga S. Gellerstedt; A.V.Bitsadze; K.I.Babenko; L. I.
Karol; S.P. Pulkin va boshqalarning ishlari bag‘ishlangan; ikkinchi yо‘nalish - Trikomi
masalasining har xil modifikatsiyalariga bag‘ishlangan; uchinchi yо‘nalish esa aralash tipdagi
tenglamalar uchun spektral masalalarni tadqiq etishdan iborat.
Aralash tipdagi tenglamalar uchun chegaraviy masalalarning rivojlanishida shved matematigi
Sven Gellerstedt tomonidan ishlab chiqilgan potensiallar nazariyasi muhim о‘rin egallaydi. S.
Gellerstedt yaratgan usul yordamida buziluvchan elliptik tipdagi tenglama uchun Dirixle va
Xolmgren masalalarining yechimini qulay integral shaklda yozish mumkin va aralash tipdagi
tenglama uchun chegaraviy masalani tadqiq etish juda qulay bо‘ladi. Shuningdek aralash tipdagi
tenglama uchun chegaraviy masalalar nazariyasining rivojlanishiga A.V.Bitsadzening
ekstremum prinsipi katta turtki bergan. Bu prinsip masala yechimining yagonaligini isbotlashda
juda keng qо‘llaniladi. Aralash tipdagi tenglamalar uchun chegaraviy masalalar nazariyasining
rivojlanishida muhim о‘rin tutuvchi yana bir natijalardan biri bu S.G. Mixlin tomonidan ishlab
chiqilgan Karlemanning Trikomi singulyar integral tenglamasini regulyarlashtirish
usuli
hisoblanadi va bu usul F.Trikomi integral tenglamasini yechishda qо‘llanilgan.
Quyidagi singulyar koeffitsiyentli buziluvchan giperbolik tipdagi tenglamani
0
Im
,
<
+
=
z
iy
x
z
kompleks yarim tekislikda о‘rganamiz
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
Volume 11, issue 1, April 2025
https://wordlyknowledge.uz/index.php/IJSR
worldly knowledge
Index:
google scholar, research gate, research bib, zenodo, open aire.
https://scholar.google.com/scholar?hl=ru&as_sdt=0%2C5&q=wosjournals.com&btnG
https://www.researchgate.net/profile/Worldly-Knowledge
https://journalseeker.researchbib.com/view/issn/3030-332X
339
( )
( )
0
1
0
1
2
0
=
+
-
+
+
-
-
-
-
y
x
m
yy
xx
m
u
y
u
y
u
u
y
b
a
,
(1)
bu yerda
m
,
0
a
va
0
b
- haqiqiy sonlar hamda ular ushbu
2
/
)
4
(
2
/
0
+
-
m
m
b
,
2
/
)
2
(
0
+
m
a
,
shartlarni qanoatlantiradi
0
D
soha
iy
x
z
+
=
komplekis tekislikning bir bog‘lamli sohasi
bо‘lib, u (1) tenglamaning
1
)
(
2
2
:
2
2
-
=
-
+
-
+
m
y
m
x
AC
,
1
)
(
2
2
:
2
2
=
-
+
+
+
m
y
m
x
BC
xarakteristikalari hamda
0
=
y
о‘qining
AB
kesmasi bilan chegaralangan bir bog‘lamli
sohasi bо‘lsin.
(1.1) tenglama shu narsa bilan e’tiborliki birinchidan bu tenglamaning kichik hadlari
oldidagi koeffitsiyentlari singulyar maxsuslikka ega, ikkinchidan bu yerda
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
y
x
f
u
y
x
c
u
y
x
b
u
y
x
a
u
u
y
x
h
y
x
K
y
x
yy
xx
=
+
+
+
+
(2)
buziluvchan umumiy giperbolik tipdagi tenglama uchun Koshi masalasini normal yechilishining
0
)
(
)
,
(
lim
0
=
-
-
®
y
K
y
x
ya
y
,
(3)
Protter sharti [1] buziladi, bu yerda
0
)
,
(
>
y
x
h
,
0
)
0
(
K
,
0
)
(
<
y
K
,
0
<
y
da. (1.3) shart
bajarilmasligiga qaramasdan, agar
2
/
0
m
a
,
0
0
=
b
bо‘lsa (1) tenglama uchun Koshi
masalasi korrekt qо‘yilgan [1].
Bundan (1) tenglama uchun Koshi masalasini normal yechilishida (3) shart zaruriy shart
emasligi kelib chiqadi. Endi (1.1) tenglamada
0
0
=
b
,
2
0
m
-
=
a
bо‘lsin:
( )
( )
0
)
2
/
(
1
2
=
-
-
-
-
-
+
x
m
yy
xx
m
u
y
m
u
u
y
,
(4)
(4) tenglama uchun Darbu masalasini ta’riflaymiz.
Darbuning ikkinchi masalasi:
0
D
sohada (4) tenglamaning ushbu
)
(
)
0
,
(
x
v
x
u
y
=
,
I
x
:
)
(
x
u
BC
y
=
,
[ ]
1,
0
x
,
(5)
shartlarni qanoatlantiruvchi regulyar
)
(
)
(
)
,
(
0
2
0
D
C
D
C
y
x
u
yechimi topilsin, bu
yerda
( )
( )
I
C
x
v
2
,
( )
( )
( )
I
C
I
C
x
2
1
I
y
,
(
)
1,
1
-
=
I
-
0
=
y
о‘qining intervali.
1-teorema.
Darbuning ikkinchi masalasiga mos bir jinsli masala cheksiz kо‘p chiziqli
bog‘liq bо‘lmagan yechimlarga ega, bir jinsli bо‘lmagan masala esa faqat va faqat,
(
)
(
) (
)
)
(
1
2
2
)
1
2
(
x
x
m
x
v
y
b
b
-
+
=
-
,
( )
1,
0
x
,
shart bо‘lgandagina yechimga ega bо‘ladi, bu yerda
(
)
2
+
=
m
m
b
.
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
Volume 11, issue 1, April 2025
https://wordlyknowledge.uz/index.php/IJSR
worldly knowledge
Index:
google scholar, research gate, research bib, zenodo, open aire.
https://scholar.google.com/scholar?hl=ru&as_sdt=0%2C5&q=wosjournals.com&btnG
https://www.researchgate.net/profile/Worldly-Knowledge
https://journalseeker.researchbib.com/view/issn/3030-332X
340
Bir jinsli Darbuning ikkinchi masalasining barcha notrivial yechimlar
( )
)
1
(
2
2
)
,
(
0
2
2
0
t
t
-
-
+
+
=
+
m
y
m
x
y
x
u
,
formula bilan beriladi, bu yerda
)
(
)
(
)
(
2
0
I
C
I
C
x
t
sinfdagi ixtiyoriy funksiya. Endi (4)
tenglama uchun (5) Darbu shartlarini ushbu
)
(
)
0
,
(
x
v
x
u
y
=
,
I
x
;
)
(
x
u
AC
y
=
,
[
]
0
,
1
-
x
(6)
shaklda beramiz.
2-teorema.
(4) tenglama uchun (6) masala yagona yechimga ega.
1-teorema va 2-teoremalardan ushbu xulosa kelib chiqadi: qat’iy giperbolik tenglamalar
uchun qо‘yilgan Koshi masalasining korrektligidan Darbu masalasining korrektligi kelib chiqadi,
buziluvchan giperbolik tenglamalarda esa umuman olganda Koshi masalasi korrektligidan Darbu
masalasining korrektligi kelib chiqmaydi. Buning ustiga (4) buziluvchan giperbolik tenglama
uchun umuman olganda xarakteristikalar, chegaraviy shartlarning ularda qо‘yilishi ma’nosida
teng huquqli emas.
(1) tenglamada
0
0
=
a
bо‘lsin:
( )
(
)
0
0
=
+
+
-
-
y
yy
xx
m
u
y
u
u
y
b
(7)
bu tenglama juda kо‘p matematiklar tomonidan о‘rganilgan [2,3,4]. Umuman olganda, (7)
tenglama uchun oddiy Koshi masalasi korrekt bо‘lmasligi mumkin. A. V. Bitsadze [2] (7)
tenglama uchun boshlang‘ich shartlari bir jinsli bо‘lgan:
0
)
0
,
(
=
x
u
,
I
x
;
0
lim
0
=
¶
¶
-
®
y
u
y
,
I
x
;
Koshi masalasi
2
0
m
-
=
b
bо‘lganda Ushbu
( )
( )
-
+
-
-
-
+
+
=
+
+
2
2
0
2
2
0
)
,
(
0
2
2
2
2
m
m
y
x
y
m
x
y
m
x
u
t
t
,
kо‘rinishdagi notrival yechimlarga ega ekanligini kо‘rsatgan, bu yerda
)
(
0
x
t
ikki marta
uzluksiz hosilaga ega bо‘lgan ixtiyoriy funksiY. Shu holatdan kelib chiqib A. V. Bitsadze [5]
boshlang‘ich shartlari
)
(
)
0
,
(
x
x
u
t
=
,
I
x
;
)
(
)
(
lim
0
0
x
y
u
y
y
n
b
=
¶
¶
-
-
®
,
I
x
,
(8)
kо‘rinishda bо‘lgan shakli о‘zgargan Koshi masalasini о‘rgangan va uni korrekt ekanligini
kо‘rsatgan, bu yerda
(
)
1
2
0
<
-
b
m
.
Agar
1
0
b
bо‘lsa, (7) tenglamaning yechimlari buzilish chizig‘i atrofida
chegaralangan bо‘lmaydi. Haqiqatdan ham ushbu
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
Volume 11, issue 1, April 2025
https://wordlyknowledge.uz/index.php/IJSR
worldly knowledge
Index:
google scholar, research gate, research bib, zenodo, open aire.
https://scholar.google.com/scholar?hl=ru&as_sdt=0%2C5&q=wosjournals.com&btnG
https://www.researchgate.net/profile/Worldly-Knowledge
https://journalseeker.researchbib.com/view/issn/3030-332X
341
-
-
=
-
,
,
)
ln(
)
(
)
,
(
0
1
0
y
y
y
x
u
b
агар
агар
булса
булса
1
1
0
0
,
=
b
b
xususiy yechimlar yuqoridagi fikrimizni tasdiqlaydi.
1
0
>
b
bо‘lganda Koshi masalasi korrekt bо‘lishi uchun boshlang‘ich shartlar
)
(
)
,
(
)
(
lim
1
0
0
x
y
x
u
y
y
t
b
=
-
-
-
®
;
( )
( )
-
¶
¶
-
-
-
-
®
y
x
u
y
y
y
y
,
)
(
lim
1
0
0
2
0
b
b
kо‘rinishda bо‘lishi kerak;
1
0
=
b
bо‘lganda esa Koshi masalasi korrekt bо‘lishi uchun
boshlang‘ich shartlar
)
(
)
ln(
)
,
(
lim
2
/
)
2
(
0
x
y
y
x
u
m
y
t
=
-
+
-
®
,
)
(
)
ln(
)
,
(
)
,
(
)
(
ln
)
(
lim
2
/
)
2
(
2
/
)
2
(
2
0
x
y
y
x
A
y
x
u
y
y
y
m
m
y
n
=
-
-
¶
¶
-
-
+
+
-
®
,
kо‘rinishda bо‘lishi kerak, bu yerda
-
)
,
(
y
x
A
aniq kо‘rinishga ega bо‘lgan maxsus kiritilgan
funksiY.
Shunday qilib, (1) tenglama yechiminning tuzilishi va differensial xossalari uning kichik
hadlari oldidagi koeffitsiyentlar
0
a
va
0
b
ga bog‘liqdir. (1) tenglama uchun masalalar
0
a
va
0
b
parametrik tekislikda
)
,
(
0
0
b
a
P
nuqtaning о‘zgarishiga qarab qо‘yiladi.
0
>
y
yarim tekislikda
0
)
/
(
0
=
+
+
y
yy
xx
m
u
y
u
u
y
b
(9)
tenglamani о‘rganamiz.
(9) tenglama shu bilan xarakterliki, uning uchun oddiy N masalasi korrekt emas.
Haqiqatdan ham
0
W
- yuqori
0
>
y
yarim tekislikda yotuvchi va uchlar
)
0
,1
(
-
A
,
)
0
,1
(
B
nuqtada bо‘lgan (9) tenglamaning normal chizig‘i
1
)
2
(
4
2
2
2
0
:
=
+
+
+
-
m
y
m
x
s
chizig‘i
hamda
0
=
y
о‘qining
AB
kesmasi bilan chegaralangan bir bog‘lamli bо‘lsin. Ushbu
masalani ta’riflaymiz.
N
masalasi
.
0
W
sohada (9) tenglamaning ushbu
)
,
(
0
0
y
x
u
j
s
=
,
0
)
,
(
s
y
x
,
)
(
0
x
y
u
y
n
=
¶
¶
=
,
)
1
,1
(
-
=
I
x
,
shartlarni qanoatlantiruvchi regulyar yechimi
( )
( )
)
(
,
0
2
0
W
W
C
C
y
x
u
topilsin.
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
Volume 11, issue 1, April 2025
https://wordlyknowledge.uz/index.php/IJSR
worldly knowledge
Index:
google scholar, research gate, research bib, zenodo, open aire.
https://scholar.google.com/scholar?hl=ru&as_sdt=0%2C5&q=wosjournals.com&btnG
https://www.researchgate.net/profile/Worldly-Knowledge
https://journalseeker.researchbib.com/view/issn/3030-332X
342
Bevosita tekshirish yordamida kо‘rsatish mumkinki ushbu
2
2
2
2
2
2
2
2
2
2
2
1
2
2
1
)
2
(
4
1
)
,
(
+
+
+
+
+
+
-
+
-
-
=
+
+
+
m
m
m
y
m
x
y
m
x
y
m
x
y
x
u
funksiya bir jinsli
N
masalaning notrivial yechimi bо‘ladi, ya’ni (9) tenglama uchun
N
masalasi korrekt emas. Shu munosabat bilan A.V.Bitsadze (9) tenglama uchun ushbu shakli
о‘zgargan
N
masalasini о‘rgangan:
0
W
sohada (9) tenglamaning ushbu
)
,
(
0
0
y
x
u
j
s
=
,
0
)
,
(
s
y
x
,
),
(
lim
0
0
x
y
u
y
y
n
b
=
¶
¶
+
®
)
1
,1
(
-
=
I
x
shartlarni qanoatlantiruvchi regulyar yechimi topilsin.
Shakli о‘zgargan
N
masalasi korrekt qо‘yilgan. Ushbu qо‘llanmada asosan singulyar
koeffitsiyentli
(
)
0
/
0
1
2
/
0
=
+
+
+
-
y
x
m
yy
xx
m
u
y
u
y
u
u
y
signy
b
a
(10)
tenglama ham о‘rganilgan. (10) tenglama
iy
x
z
+
=
, kompleks tekisligining
0
Im
>
z
yuqori
yarim tekisligida uchlari
)
0
,1
(
-
A
va
)
0
,1
(
B
nuqtalarda va yuqori yarim tekislikda joylashgan
Г
:
)
(
x
f
y
=
chizig‘i bilan,
0
Im
<
z
pastki yarim tekislikda esa (10) tenglamaning
AC
va
BC
xarakteristikalari bilan chegaralangan bir bog‘lamli
D
sohada о‘rganildi.
Asosiy e’tibor (6) tenglama uchun
{
}
0
<
=
-
y
D
D
sohada shakli о‘zgargan Koshi
masalasini о‘rganishga,
{
}
0
>
=
+
y
D
D
sohada Dirixle va shakli о‘zgargan
N
masalasini,
aralash
D
sohada esa Trikomi masalasini hamda Frankl turidagi nolokal masalalarni
о‘rganishga qaratilgan.
Foydalanilgan adabiyotlar:
1. Пулькин С.П.Задачи Трикоми для обобщенного уравнения Лаврентьева-
Бицадзе//Докл. АН CCСР.1958. Т.118.№1.С.38-41.
2. Бицадзе А.В., Салахитдинов М.С. К теории уравнений смешанно-составного
типа//Сибирский математический журнал. 1961, Т.2. №1.С. 7-19.
3. Ильин В.А., Моисеев Е.И. Нелокальная краевая задача для оператора Штурма-
Лиувилля в дифференциальной и разностной трактовках//Докл. АН CССР. 1986. Т.
291. № 3. С. 534-539.
4. Shaymardanova,
A.
R.
(2021).
O
‘ZBEK
TILINING
INTRALINGVAL
LAKUNALARI.“. FILOLOGIK TADQIQOTLAR: MUAMMO VA YECHIM” mavzusida
xalqaro ilmiy-nazariy anjuman materiallari, 234.
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impact factor: 8,293
Volume 11, issue 1, April 2025
https://wordlyknowledge.uz/index.php/IJSR
worldly knowledge
Index:
google scholar, research gate, research bib, zenodo, open aire.
https://scholar.google.com/scholar?hl=ru&as_sdt=0%2C5&q=wosjournals.com&btnG
https://www.researchgate.net/profile/Worldly-Knowledge
https://journalseeker.researchbib.com/view/issn/3030-332X
343
5. Shaymardanova, А. (2020). THE ROLE AND POSITION OF LACUNAS OF CULTURAL
LOCATION IN THE PROCESS OF COMMUNICATION (CAN THE WORD “YANGA”
BE THOUGHT TO BE A LACUNA IN UZBEK LANGUAGE?). Theoretical & Applied
Science, (12), 322-325.
6. Shaymardanova, A. (2021). Компьютерный Перевод Культурных Характеристик
Проблема Изготовления. Computer Linguistics: Problems, Solutions, Prospects, 1(1).
7. Makhamadievna, A. M., & Tulkunovna, R. N. (2021). Teaching foreign language by using
effective methods. Asian Journal of Research in Social Sciences and Humanities, 11(12),
47-50.
8. Makhamadievna, A. M. (2022). Effective Teaching English Language to Non-Linguistic
Students by Using Project Based-Learning. The Peerian Journal, 6, 48-51.
9. Кумыкова С.К., Нахушева Ф.Б. Дифференц.уравнения. 1978. Т. 14, №1. С. 50-65.
10. Салахитдинов М.С., Мирсабуров М. Нелокальние задачи для уравнений смешанного
типа с сингулярными коэффициентами. Ташкент 2005. “Universitet”. “Yangiyo’l
poliyrafservis” -224 c.
