МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
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Maksimal printsip va xatolar tahlili
Quvvatov Behruz
Osiyo Xalqaro Universiteti
“Umumtexnik fanlar” kafedrasi o‘qituvchisi
ulughbekovich.bekhruz@mail.ru
Kirish.
Ushbu maqolaning maqsadi chegaraviy masalalarning oddiy va qisman
differensial tenglamalarini yechish uchun chekli ayirma va chekli element usullari
bilan tanishtirib o`tamiz.
Asosiy e'tibor chekli farq va chekli elementlar metodlari va ularni oqilona
matematik nazariya bilan amalga oshirishda muhim tafsilotlarni tushunishga
qaratilgan. Biz ikki yoki undan yuqori o'lchamlarni ko'rib chiqishdan oldin bir
o'lchovli muammolarni har tomonlama muhokama qilishdan boshlaymiz.
Shuningdek, biz tegishli sohalarda ko'proq ma'lumot olishni istaganlar uchun ba'zi
foydali ma'lumotnomalarni sanab o'tamiz. Dastlab, maksimal prinsip tushunchasini
ko`rib o`tamiz.
,
y
x,
for
,
0
b
,
2
2
2
2
2
2
2
ac
y
c
y
x
h
x
a
L
elliptik differentsial operatorini ko'rib chiqamiz va umumiylikni
yo'qotmasdan
0
,
0
c
a
deb faraz qilamiz. Maksimal printsip quyidagi teoremada
berilgan.
1 teorema.
Agar
3
,
C
y
x
u
chegaralangan
sohada
0
,
y
x
Lu
ni
qanoatlantirsa, u holda.
y
x
u
,
soha chegarasida maksimal qiymatga ega bo'ladi.
Isbot:
Agar teorema to'g'ri bo'lmasa, u holda
0
0
,
y
x
lokal nuqtasi
mavjud bo'lib,
y
x
u
y
x
u
,
,
0
0
barcha
y
x
,
uchun, lokal ekstremum
0
0
,
y
x
uchun zarur shart – bu:
0
,
,
0
,
0
0
0
0
y
x
x
u
y
x
x
u
Endi
0
0
,
y
x
soha chegarasida bo'lmagani va
u(x, y)
uzluksiz bo'lganligi
sababli,
Q
sohaida
0
0
,
y
x
ning qo'shnisi mavjud bo'lib, u yerda biz Teylor
kengaytmasiga ega bo’lamiz,
()
ning yuqori belgisi bilan
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
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Том 2, Выпуск 9, 30 Сентябрь
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,
,
2
2
1
,
,
3
3
0
2
0
0
2
0
0
0
0
y
x
O
u
y
u
y
x
u
x
y
x
u
y
y
x
x
u
yy
xx
xx
funksiyalar baholanganligini bildiradi.
0
0
,
y
x
da
,
ya'ni
0
0
2
2
0
,
y
x
x
u
u
xx
ifoda
0
0
,
y
x
da
baholanadi va hokazo.
0
0
0
0
,
,
y
x
u
y
y
x
x
u
barcha yetarlicha kichik
x
va
y
, uchun
0
2
2
1
0
2
0
0
2
yy
xx
xx
u
y
u
y
x
u
x
(1)
Boshqa tomondan, berilgan shartdan
,
0
2
0
0
0
0
0
0
0
yy
xx
xx
u
c
u
b
u
a
Lu
(2)
Bu yerda
0
0
0
,
y
x
a
a
va boshqalar, Teylor kengayishiga qarama –
qarshilikni olishi uchun yuqoridagi tengsizlikni quyidagicha qayta yozamiz.
,
0
2
0
2
0
0
0
0
2
0
0
0
0
0
0
0
0
a
b
c
M
u
u
M
a
b
u
M
a
b
M
a
u
M
a
yy
yy
xy
xx
(3)
Bu yerda
M >
0 doimiy hisoblanadi.
M
ning vaziasi etarlicha kichik bo'lgan
x
va
y
ni tanlashdir.
Endi quyidagi ifodani ko`rsak,
M
a
b
y
M
a
0
0
0
,
x
(4)
(1) ifodadan quyidagini bilamiz.
(5)
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
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Natijada quyidagi kelib chiqadi.
(6)
va yana (1) dan,
(7)
Shunday qilib, (4) va ( 5) dan ( 3) chap tomoni ijobiy bo'lmasligi kerak, bu
(8)
shartga zid keladi va bu bilan isbot to`ldiriladi.
Boshqa tomondan, agar
Lu < 0 bo'lsa, u ning
m
minimal qiymati
chegarasida bo'ladi. Umumiy elliptik tenglamalar uchun maksimal printsip
quyidagicha.
(9)
bo'lsin, bu erda
chegaralangan soha hisoblanadi. U holda 1 teoremadan
u(x,
y)
ifoda
ning
lokal qismida musbat lokal maksimal yoki manfiy
m
lokal minimal
bo‘lishi mumkin emas.
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ТЕОРИЯ И ПРАКТИКА
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ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
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ТЕОРИЯ И ПРАКТИКА
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