O`zbekiston respublikasi oliy va o`rta maxsus ta`lim vazirligi
Integral tenglama yadrosi
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Intagral tenglamalar nazariyasi
- Bu sahifa navigatsiya:
- 4. Abel’ tenglamasi
- O’zingizni sinab ko’ring.
- Boshlang’ich shartlari bilan birga berilgan quyidagi differensial tenglamalarga mos bo’lgan integral tenglamalar tuzilsin
- Aynigan yadroli integral tenglamani yeching.
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3. Integral tenglama yadrosi t x chizig’ida birinchi tartibli maxsuslikka ega bo’lgan hol. iy x z
kompleks tekislikda yotuvchi yopiq kontur bo’lsin
, ) ( d t (14) (14) integral oddiy ma’noda uzoqlashuvchi bo’ladi, lekin bu integral “integralning bosh qiymati ma’nosida” yaqinlashuvchi bo’ladi. t nuqtani
chiziqdan markazi t nuqtada radiusi bo’lgan aylana bilan ajratib olamiz va qolgan qismini
orqali belgilaymiz. Singular integral ushbu
d t d t ) ( lim ) ( 0 tenglik bilan aniqlanadi.
z
chiziqda yotmaydigan ixtiyoriy nuqta bo’lsin.
d z i z ) ( 2 1 ) ( funksiyani kiritamiz
t z
ga integrallansin ) (t t
va ) (t e
orqali ) (z
ning t z
ga mos ravishda
kontur ichidan yoki tuchqarisida integragandagi limit qiymatin belgilaymiz.
U holda Soxotskiy-Plemel formulasiga k’ra d t i t t i ) ( 2 1 ) ( 2 1 ) (
(15) d t i t t e ) ( 2 1 ) ( 2 1 ) ( (16)
tengliklar o’rinlidir.
) (z
funksiya kontur ichida regulyar funksiya bo’lsin va ga
qadar uzluksiz bo’lsin. Agar z kontur tashqarisda bo’lmasa.
0
( z va
0 ) ( t e
yoki Soxotskiy-Plemel formulasiga ko’ra 0 ) ( 1 ) (
d t i t
(17) Bu tenglik 1
sonu 0
( ) (
d t i t
(18) singulyar integral tenglamaning echimi ekanligini bildiradi va bu xarakteristik songa
kontur ichida
chiziqqacha regulyar funksiyaning
chiziqdagi qiymati xos funksiyaga mos keladi.
Endi ) (z
chiziqdan tashqari sohada regulyar bo’lib chiziqgacha uzluksiz bo’lsin va 0 )
.
chiziq ichda 0 )
z
va
0 ) ( t i
bu tenglikdan Soxotskiy- Plemel formulasiga ko’ra 0 ) ( 1 ) ( d t i t (19) tenglikka kelamiz.
Bundan (19) tenglama yani bir 1 xarakteristik songa ekanligi kelib chiqadi va bu songa cheksiz ko’p xos funksiyalar mos keladi. Shunday qilib Fredgol’mning har qanday xarakteristik songa chekli sondagi xos funksiyalar mos kelishi haqidagi teorema buzuladi.
Ushbu integral tenglamani o‘rganamiz.
x x x f y x dy y 0 0 , 1 0 ), ( ) ( ) ( (20) Bu tenglamani ushbu ko‘rinishda yozib t t f y x dy y 0 ) ( ) ( ) (
uni 1 ) ( 1 t x ga ko‘payiramiz va ) ,
( x oralig‘ida integrallaymiz
x t x t x dt t f y t dy y t x dx 0 0 0 1 1 ) ( ) ( ) ( ) ( ) ( (21) (21) tenglikda integrallash tartibini o‘zgartiramiz
у x t y t y t y x t 0 0 0 t х
x x y x t x dt t f y t t x dx dy y 0 0 1 1 ) ( ) ( ) ( ) ( ) ( (22) Ichki integralda ushbu almashtirishni bajaramiz
) ( y x y t
U holda sin
) 1 ( ) ( ) 1 ( ) , 1 ( ) 1 ( ) 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) ( 1 0 1 1 1 1 0 1 0 1 1 1 1 B d d y x y x d y x y t t x dx x y (23)
Shunday qilib (22) tenglama (23) asosan ushbu ko‘rinishni oladi
x t x dt t f dy y 0 0 1 ) ( ) ( sin ) (
yoki
x x x x t x dt t f x f dt t f t x t x t f dx d t x d t f dx d t x dt t f dx d y 0 1 1 0 0 0 0 1 ) ( ) ( ' ) 0 ( sin ) ( ' ) ( ) )( ( sin 1 ) ( ) ( sin 1 ) ( ) ( sin ) (
O’zingizni sinab ko’ring. 1.
Fur’e almashtirishni yozing. 2.
Abel’ tenglamsini yozing.
Misol va masalalar to`plami Boshlang’ich shartlari bilan birga berilgan quyidagi differensial tenglamalarga mos bo’lgan integral tenglamalar tuzilsin: 1.
, 1 ) 0 ( , 0
y y 2. 1 )
( , 0 ) 0 ( , 0 6 5 y y y y y
3. 1 ) 0 ( , 0 ) 0 ( , 0 y y y y 4. 1 )
( , 0 ) 0 ( , 0
y xy y
5. 0 ) 0 ( , 1 ) 0 ( , cos
y y x x y y 6. 1 )
( ) 0 ( , 1 sin y y y x y y
7. n p b y a y px h y n y , ) 0 ( , ) 0 ( , sin 2 8.
b y a y y n y h y ) 0 ( , ) 0 ( , 0 2 2
Volterra integral tenglamalarini ketma-ket yaqinlashish usulidan foydalanib yeching. 1.
0 ) ( , ) ( ) ( ) ( 0 0
dt t t x x x x 2.
0 ) ( , ) ( ) ( 1 ) ( 0 0 x dt t t x x x
3. 1 ) ( , ) ( ) ( 1 ) ( 0 0 x dt t t x x x 4. 1 )
, ) ( 1 ) ( 0 0 x dt t x x x
5. 1 ) ( , ) ( 1 ) ( 0 0
x dt t x x x 6. x x dt t x x x x ) ( , ) ( 2 ) ( 0 0 2
7. 1 ) ( , ) ( 2 ) ( 0 0 2
dt t x x x x 8. 1 )
, ) ( ) ( 1 ) ( 0 0 x dt t t x x x x
1. Aynigan yadroli integral tenglamani yeching. 2 0 2 2 ) ( sin
4 ) ( x dt t x x
2. Aynigan yadroli integral tenglamani yeching. 1 1 arcsin ) ( ) (
dt t e x x
3. Aynigan yadroli integral tenglamani yeching. 4 4 ) ( ) ( ctgx dt t tgt x
4. Aynigan yadroli integral tenglamani yeching. 1 0 2 1 1 ) ( arccos ) (
dt t t x
5. Aynigan yadroli integral tenglamani yeching. ) 4 1 ( 5 6 ) ( ) ln ln ( ) ( 1 0 x dt t x t t x x
6. Aynigan yadroli integral tenglamani yeching. 2 0 sin ) ( cos sin ) ( x dt t t x x
7. Aynigan yadroli integral tenglamani yeching. 2 0 ) ( sin ) ( ) ( x dt t x t x
8. Aynigan yadroli integral tenglamani yeching. 0 cos ) ( ) sin( ) ( x dt t t x x
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