ULANISH CHIZIQLARI PARALLEL BO‘LGAN PARABOLO-GIPERBOLIK TENGLAMALAR UCHUN CHEGARAVIY MASALA

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-

GIPERBOLIK TENGLAMALAR UCHUN CHEGARAVIY MASALA

1

kafedrasi o

1

tel: 99 515-08-98, ermatovjamoldin578@gmail.com

Imomnazarova Nurjaxon Toktaboyevna

2

2

tel: 99 471-49-19, imomnazarovanurjaxon@gmail.com

3

3

tel: 93 751-04-06, tzahro455@gmail.com

-

Volterraning ikkinchi tur integral tenglamasiga keltirib topilgan.

lar:

Parabolo-giperbolik tenglama, chegaraviy masala, Dalamber

formulasi, Grin funksiyasi, Dirixle formulasi, Abel tenglamasi, Volterraning ikkinchi tur
integral tenglamasi.

-

1

,

University of business and science

1

,

-08-98, ermatovjamoldin578@gmail.com

2

University of business and science

2

,

-49-19, imomnazarovanurjaxon@gmail.com

3

University of Business and Science,

3

3 751-04-06, tzahro455@gmail.com

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-

BOUNDARY VALUE PROBLEM FOR PARABOLIC-HYPERBOLIC

EQUATIONS WITH PARALLEL CONNECTING LINES

1

Lecturer, Department of Innovative Technologies, University of Business and

Science

1

phone: 99 515-08-98, ermatovjamoldin578@gmail.com

Imomnazarova Nurjaxon Toktaboyevna

2

Lecturer, Department of Innovative Technologies, University of Business and

Science

2

phone: 99 471-49-19, imomnazarovanurjaxon@gmail.com

3

Student of the Mathematics, University of Business and Science

3 751-04-06, tzahro455@gmail.com

Abstract:

In this article, we study the solution of boundary value problems for a

parabolic-hyperbolic equation with parallel lines of type change. The solution of the
problem is found by reducing to the Volterra integral equation of the second kind.

Keywords:

Parabolic-hyperbolic equation, boundary value problem, d'Alembert

formula, Green's function, Dirichlet formula, Abel equations, Volterra integral equations
of the second kind.

I. KIRISH

1-

Dastlab elliptik-

-yillarda italiyalik matematik

F. Trikomi tomonidan olib borilgan va S. Gellerstedt, A. V. Bitsadze, K. I. Babenko, I. L.
Karol, F.I. Frankl, M.M.Smirnov, M.S. Salaxiddinov, T.D. Jurayev va boshqalar
tomonidan rivojlantirilgan.

Keyin aralash tipdagi tenglama tushunchasi ikki yoki uchta klassik tipdagi

tenglamalarning har
kengaytirildi. Aralash elliptik-parabolik va parabolik-giperbolik tipdagi tenglamalarni

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jihatdan kam

Bizga quyidagicha

1

,

,

0

,

,

2,3,

xx

y

xx

yy

i

u

u

D

u

u

D

i

(1.1)

1

D

-soha,

0

0

0

,

,

AB BB

B A

va

0

A A

segmentlar mos ravishda

0,

1,

1

y

x

y

va

0

x

kvadrat

0

1,0

1

x

y

;

2

D

-soha,

0

A A

A

va

0

A

nuqtalardan chiquvchi

0

:

0,

:

1

AC

x

y

A C

y

x

1 1

,

2 2

C

3

D

-soha,

0

BB

segment

hamda

B

va

0

B

nuqtalardan chiquvchi

0

:

1,

:

2

BE

x

y

B E

x

y

chiziqlarning

3 1

,

2 2

E

1

2

,

,

D D

va

3

D

0

AA

va

0

BB

ochiq segmentlarni

D

bilan

belgilaymiz.

Masala.

Quyidagi berilgan shartlarni qanoatlantiruvchi

,

u x y

funksiyani toping:

1)

0

AA

va

0

BB

segmentdagi nuqtalardan tashqari

D

sohaning hamma joyida

,

u x y

2)

1

1

1

1

0

0

2

0

3

0

,

,

j

u x y

C D

C D

AA

BB

C D

AA

C D

BB

1, 2,3;

j

3) Shartlarni qanoatlantiradi

1

1

, 0

,

2

AC

u

y

y

(1.2)

0

2

1

,

1,

2

B E

u

y

y

(1.3)

, 0

1;

AB

u

x

x

(1.4)

(1. 4) va bular uchun ulashish shartlarini qanoatlantiradi

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1

1

1

1

2

2

2

2

0,

0,

,

0,

0,

,

1 0,

1 0,

,

1 0,

1 0,

,

x

x

x

x

u

y

y u

y

y

u

y

y u

y

y

u

y

y u

y

y

u

y

y u

y

y

(1.5)

bu yerda

, , , , ,

1,2,

i

i

i

i

i

y

i

x

oldindan berilgan funksiyalar,

,

,

,

,

,

i

i

i

i

i

uzluksiz.

II.ADABIYOTLAR TAHLILI

Parabola-

-

-

Ushbu masalaning boshqalardan farqi ([1-5]) parabolik sohada ikkinchi chegaraviy

masala yechimidan foydalanib,

i

y

,

0,1

i

ga nisbatan

0

x

va

1

x

da

Volterraning ikkinchi tur integral tenglamalari sistemasi olinadi va uning yechimining
yagonaligi integral tenglamalar nazariyasidan kelib chiqadi.

tenglamalari sis

III.XULOSA

Differensial tenglamalar va matematik fizika inson faoliyatining deyarli barcha

sohalariga kirib boradi, bu ilmiy-

n, aralash tipdagi differensial tenglamalar hozirgi kunda xususiy

biriga aylandi.

-giperbolik tenglama

uchun c

yoritishga harakat qilindi.

-giperbolik tenglama

tur integral tenglamasiga keltirib topilgan.

Umuman olganda, olingan natijalar ushbu maqolaning tadqiqot maqsadlariga

erishish haqida gapirishimizga imkon beradi.

-giperbolik tenglama uchun

ini tushunib yetdim, yana shuningdek, bu

mavzuga jiddiy yondashish kerakligini tushundim. Matematikaning ozgina hisob-
kitobdagi xatosi hayotdagi jarayonlarni anchagina tang ahvolga olib kelishi mumkinligini

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anglash mumkin. Men maqola mavzusiga oid bilim va
va ishonamanki, bu maqoladan matematikaning maxsus kurslarida foydalanish mumkin.

chiqishga va ularni hisoblashga imkon beradi.

IV.FOYDALANILGAN ADABIYOTLAR

- 1973. -

-

-148.

-

-

- 1985. -

-

-36.

-

- 1987. -

-87.

-

-

1987. -

-82.

.-

.

, -

-

. 6-

20.