O'zbekiston respublikasi oliy va o'rta maxsus talim vazirligi samarqand davlat universiteti haydarov Akram matematik fizika va analizning zamonaviy usullari va nokorrekt masalalari

u(x,y, 0) = g(x,y) = cos 2x, u_z(x,y, 0) = x — 2 siny.

7. 
   Koshi masalasini yeching: u_tt = Au tenglamaning
   

u(x, y, z, 0) = xyz, u_t(x, у, z, 0) = x — 2 sin 2y.

8. 
   Koshi masalasini yeching: u_tt = Au
   

u(x, y, z, 0) = x² + у² + z², u_t(x, y, z, 0) = xy.

9. 
   Koshi masalasini yeching: u_tt = Au
   

u(x,y,z, 0) = 6^X cos у, u_t(x,y,z, 0) = x² -y²

10. 
    Koshi masalasini yeching: u_tt = Au
    

u(x, y, z, 0) = x² + y², u_t(x,y,z, 0) = 1

11. 
    Koshi masalasini yeching: u_tt = Au
    

u(x, y, z, 0) = e^x, u_t(x, y, z, 0) =

12. 
    Koshi masalasini yeching: u_tt = Au
    

1
u(x,y,z, 0) = —, u_t(x,y,z, 0) = 0.
ОС

13. 
    Koshi masalasini yeching: u_tt = Au
    

u(x,y,z, 0) = —, u_t(x,y,z, 0) = 0. У
Koshi masalasini yeching: u_tt = Au
1
u(x,y,z, 0) = —, u_t(x,y,z, 0) = 0. z
Koshi masalasini yeching: u_tt = Au + x + t
u(x, у, z, 0) = xyz, u_t(x, y, z, 0) = xy + z. Koshi masalasini yeching:
u(x, 0) = e~^x, u_t(x, 0) = 1. Koshi masalasini yeching:
— ^xx "l" (uctf 0) — sinx Koshi masalasini yeching: u_tt = u_xx + a sin bt u(x, 0) = cos x, u_t(x, 0) = sin x Koshi masalasini yeching:
u(x, 0) = sin x, u_t(x, 0) = cos x. Koshi masalasini yeching:
u(x, 0) = e~^x, u_t(x, 0) = 3
Garmonik o'lchov yordamida garmonik funksiyani toping _u(_e*) = f ^г> ^0<в<71
^ ^} l-l, 7Г < в < 2n Garmonik o'lchov yordamida garmonik funksiyani toping
0,

u{eⁱ⁽P) =
1,
7Г <6 <71
3n

1. 
   л <6 < — 3n
   
2. 
   —<в<2п A
   

15. 
    
16. 
    
17. 
    
18. 
    
19. 
    
20. 
    21
    

22

7Г
oI
23) Garmonik o'lchov yordamida garmonik funksiyani toping

7Г

-2,

(.eⁱ⁽P) =

и
0 < в <- 4
7Г
-1, - < в < 2л 4

24) Garmonik o'lchov yordamida garmonik funksiyani toping
7Г
<sup>02 7Г
— < в
и
2,

1,

(.ei(P) =
(Д 7Г < в < 2n 25) Garmonik o'lchov yordamida garmonik funksiyani toping</sup>

( ТТ
О, О < в < -
u(eⁱ⁽P) = { , ⁷¹ ^ п ^ ^{v }} 1, — < 6 <л 4
Д 7Г < в < 2л
2. Mustaqil o'zlashtirish uchun savollar

1. 
   u_t = u_xx tenglama uchun minimum qiymat prinsipini isbotlang.
   
2. 
   u_t = u_xx tenglama yechimi uchun maksimum qiymati prinsipini isbotlang.
   
3. 
   u_t = u_xx tenglama uchun aralash masalasi yechimining yagonaligini isbotlang.
   
4. 
   u_t = u_xx tenglama uchun aralash masala yechimining turg'unligini isbotlang.
   
5. 
   u_t = 4u_xx tenglama uchun aralash masala yechimining yagonaligini isbotlang.
   
6. 
   u_t = 4u_xx tenglama uchun aralash masala yechimining turg'unligini isbotlang.
   
7. 
   u_tt = a²u_xx to'lqin tenglama uchun Koshi masalasi mavjudligini isbotlang.
   

8 )u_tt = u_xx to'lqin tenglama uchun Koshi masalasi yechimining mavjudligini isbotlang.

9. 
   u_tt = u_xx to'lqin tenglamasi uchun Koshi masalasi yechimining turg'unligini isbotlang.
   
10. 
    u_tt = u_xx tenglama uchun aralash masala yechimi mavjudligini isbotlang.
    
11. 
    u_tt = u_xx tenglama uchun aralash masala yechimining turg'unligini isbotlang.
    
12. 
    u_tt = 4u_xx tenglama uchun Koshi masalasi yechimining turg'unligini isbotlang.
    
13. 
    u_tt = u_xx tenglamada to'lqin tarqalishi tenglamasini nimaga teng.
    
14. 
    u_tt = 4u_xx,u(x,0) = (p{x),u_t{x, 0) = ip(x) masala yechimini toping.
    
15. 
    u_tt = u_XX)u(x, 0) = (p{x),u_t{x, 0) = ip(x) Koshi masalasining yechimi qanday cp va ip funksiyalar uchun regulyar bo'ladi?
    
16. 
    Agar u_tt = u_xx tenglama yechimi u(x, t) bo'lsa, xu_x + tu_t funksiya ham shu tenglama yechimi bo'lishini isbotlang.
    

17) Agar u(x,t) funksiya u_tt = u_xx tenglama yechimi bo'lsa, u² + u² funksiya ham shu tenglama yechimi bo'lishini isbotlang.

18

19. 
    
20. 
    21 22 23
    

Koshi - Riman tenglamalar sistemasining turini aniqlash. au_x + v_y = 0, u_y + v_x = 0 sistemaning umumiy yechimini toping. u_x — v_y = 0, u_y — v_x = 0 sistemaning umumiy yechimini toping.
Tenglama umumiy yechimini toping u_yy - и_л, = 4e^:
Tenglama umumiy yechimini toping u_xx Tenglama umumiy yechimini toping u_xy
+ yu_y — и = 0.
3. Mustaqil o'zlashtirish uchun savollar
1 )u(x,y) = xy funksiyaning x²+y²< 1 doirada ekstremumlarini Zaremba - jiro prinsipi orqali aniqlang.

2. 
   u(x,y) = xy funksiyaning D = {(x,y); x² + y² < 1} doirada minimumini toping.
   
3. 
   u(x,y) = xy funksiyaning D = {(x,y); x² + y² < 1} doirada maksimumini toping.
   

с _x2 y2 \

2. 
   u(x,y) = x² - y² funksiyaning D = Ux,y); — + — < lj ellipsda ekstremumlarini Zaremba - jiro prinsipi orqali aniqlang.
   
3. 
   u(x,y) = x² - y² funksiyaning D = j(x,y); + < lj ellipsda minimumini toping.
   
4. 
   u(x,y) = x² + y² funksiyaning D = j(x,y); + < lj ellipsda
   

maksimumini toping.

2. 
   Agar w(x) funksiya D sohada ikkinchi tartibli hosilalari bilan uzluksiz bo'lsa, u holda Aw < 0 bo'lganda w(x) nisbiy manfiy minimumga erishmasligini isbotlang.
   
3. 
   Agar w(x) fnuksiya D sohada ikkinchi tartibligacha uzluksiz hosilaga ega bo'lib, Aw >0 bo'lsa, u holda D sohada musbat nisbiy maksimumga ega boTmasligini isbotlang.
   
4. 
   u(x,y) = xy funksiyaning D = {(x,y); x² + y² < 1} soha chegarasida
   

^ — normal hosilasini toping.

2. 
   и = x² — у² funksiya uchun D = j(x,y); + < lj soha
   

дъi
chegarasida — - normal hosilasini toping.

2. 
   Agar и funksiya D sohada garmonik bo'lib, D ning chegarasi S gacha uzluksiz hosilaga ega bo'lsa, u holda x₀ £ S nuqtada funksiya maksimumga ega
   

du
bo'lsa, bu nuqtada — > 0 bo'lishini isbotlang.

2. 
   Agar и funksiya D sohada garmonik bo'lib, soha chegarasi S gacha uzluksiz hosilaga ega bo'lsa, u holda x₀ £ S minimum nuqta bo'lganda, bu
   

du
nuqtada — < 0 bo'lishini isbotlang.

2. 
   Agar /(z) analitik funksiyaning haqiqiy qismi u(x,y) bo'lsa, u holda quyidagi Gursa formulasi o'rinli ekanligini isbotlang.
   

f(z) ^{= 2u}{^'Yi)~ ^u(°>°) + ^{ci e D}> z = x + iy

2. 
   u(x,y) = xy³ — yx³ bo'lsa, /(z) ni Gursa formulasi orqali toping.
   
3.