Riss teoremasi

Yechilishi: u(x,y)=e^ax+by U(x,y), v(x,y)=e^ax+by V(x,y) deb olaylik. U holda

_aeaxby_{U  e}axby_{U x}

• 
  _(beaxby_V
  

• 
  _eaxby_{Vy )  2e}axby_V
  

 g₁


_be
Bu yerdan
axby_{U  e}
axby_{U y}

• 
  ae
  

axby_V

• 
  _eaxby_Vx
  
• 
  2e
  

axby_{U  g 2}

_eaxby_{U x}

• 
  _eaxby_Vy
  

• 
  ae^axbyU  (b  2)e^axbyV
  

 g₁


_e
axby_{U y}

• 
  _eaxby_Vx
  
• 
  ae
  

axby_V
 (b  2)e
axby_{U  g 2}

Endi, b=-2, a=0 olsak va G₁(x,y)=e^-2y g₁(x,y), G₂(x,y)=e^-2y g₂(x,y) deb belgilasak, u holda quyidagi tenglamalar sistemasiga kelamiz:


_U _x  V_y
U _y  V_x
 G₁
 G₂

Quyidagi almashtirishlarni kiritamiz:


W=U+iV, 2G=G₁+G₂, z=x+iy, t=  i ,


 _ 1 ₍ 



 i ^ ),


 _ 1 ₍ 

 i ^ ).

z 2 x
y z
2 x y

Natijada berilgan tenglamalar sistemasi ushbu ko’rinishga keladi:

^W  G(z)
z

Ushbu tenglamaning xususiy yechimi

W (z)   ¹
0 _
G(t)




t  z
D
d d

ekanidan,

umumiy yechimni
W  W₀ W₁
ko’rinishida izlash mumkin. Bu yerda
W₁ bir jinsli

yoki Koshi-Riman tenglamalar sistemasi
W _{ 0}

z

ning umumiy yechimi. Bu

yechim ixtiyoriy
 (z)
analitik funksiyadan iborat bo’lgani uchun (1.4.1)

tenglamaning umumiy yechimi quyidagi ko’rinishda bo’ladi:

W  (z)  ¹ _ ^G(z)
d d

 t  z
D
W=e^2y (u(x,y)+v(x,y))= e^2yw(z), 2G(x,y)=e^-2y (g₁(x,y)+ g₂(x,y)) ekanidan
w(z)  e^2y [(z)  ¹ _ ^g(t) e^2 d d],
 t  z
D
bu yerda 2g(x,y)= g₁(x,y)+ g₂(x,y), ekanligi kelib chiqadi. Bundan esa
u(x,y)=Re(w(z)), v(x,y)=Im(w(z)).
Javob: u(x,y)=Re(w(z)), v(x,y)=Im(w(z)), bu yerda
w(z)  e^2y [(z)  ¹ _ ^g(t) e^2 d d].
 t  z
D

3. 
   misol. Chegaralangan va chegarasi D bo’lakli silliq bo’lgan ^D sohada
   

quyidagi elliptik tenglamalar sistemasining umumiy yechimini toping [3].

_u_x  v_y


• 
  av  g₁
  

_u _y  v_x  au  g₂
Yechilishi: u(x,y)=e^cx+by U(x,y), v(x,y)=e^cx+by V(x,y) deb olaylik. U holda

_cecxby_{U  e}cxby_{U x}

• 
  _(becxby_V
  

• 
  _ecxby_{V y )  ae}cxby_V
  

 g₁


_be
Bu yerdan,
cxby_{U  e}
cxby_{U y}

• 
  ce
  

cxby_V

• 
  _ecxby_Vx
  

• 
  ae
  

cxby_{U  g 2}

_ecxby_{U x}

• 
  _ecxby_{V y}
  

• 
  ce^cxbyU  (b  a)e^cxbyV
  

 g₁


_e
cxby_{U y}

• 
  _ecxby_Vx
  
• 
  ce
  

cxby_V

• 
  (b  a)e
  

cxby_{U  g 2}

Endi, b=a, c=0 olsak va G₁(x,y)=e^ay g₁(x,y), G₂(x,y)=e^ay g₂(x,y) deb belgilasak, u holda quyidagi tenglamalar sistemasiga kelamiz:


_U _x  V_y
U _y  V_x
Quyidagi almashtirishlarni kiritamiz:
 G₁
 G₂

W  U  iV ,
2G  G₁  iG₂,
z  x  iyt=  i ,

 _ 1 ₍ 

 i ^ ),
 _ 1 ₍ 



 i ^ ).

z 2 x
y z
2 x y

Natijada berilgan tenglamalar sistemasi ushbu ko’rinishga keladi:

^W  G(z)
z

Ushbu tenglamaning xususiy yechimi

W (z)   ¹
0 _
G(t)




t  z
D
d d

ekanidan,

umumiy yechimni
W  W₀ W₁
ko’rinishida izlash mumkin. Bu yerda W₁
bir jinsli

yoki Koshi-Riman tenglamalar sistemasi
W _{ 0}

z

ning umumiy yechimi. Bu

yechim ixtiyoriy
 (z)
analitik funksiyadan iborat bo’lgani uchun (1.4.1)

tenglamaning umumiy yechimi quyidagi ko’rinishda bo’ladi:

W  (z)  ¹ _ ^G(z)
d d

 t  z
D

W  e^y ux, y  vx, y  e^y wz
ekanidan
2Gx, y  e^y g₁ x, y  ig₂ x, y

w(z)  e^ay[(z)  ¹ _ ^g(t) e^a d d],
 t  z
D

bu yerda
2gx, y  g₁x, y ig₂ x, y, ekanligi kelib chiqadi. Bundan esa

ux, y  Rewz, vx, y  Imwz
Javob: ux, y  Rewz, vx, y  Imwz, bu yerda


w(z)  e^ay[(z)  ¹ _ ^g(t) e^a d d],
 t  z
D

3. 
   misol. Chegarasi bo’lakli silliq bo’lgan D chegaralangan tekis sohada quyidagi elliptik sistemaning umumiy yechimini toping [3].
   

_au_x  bv_y


• 
  acu  g₁
  

_au _y  bv_x  bcv  g₂

_{u  e}xy_{U, v  e}xy_V
bo’lsin.

U holda
_aexy_{U  e}xy_{U x  be}xy _aexy_{U  e}xy_{U x  bae}xy

• 
  e^xyV_y  ace^xyU  g₁
  
• 
  e^xyV_x  bce^xyV  g ₂
  

aU_x

• 
  bV_y
  

 aa  cU  bV  e ^{xy} g₁

2
  0,
  0
deb olamiz.
aU _y

• 
  bV_x
  
• 
  ba  cV
  

 bV  e ^{ y} g

G₁  e^cx g₁, G₂  e^cx g₂
aU_x  bV_y  G₁

P  aU,
Q  bV
olsak,
aU _y

• 
  bV_x
  

 G₂

_P_x  Q_y  G₁
^P  Q  G
_^ y x 2

W  P  iQ,
2G  G₁  iG₂ ,
z  x  iy,
t    i
almashtirishlarni olsak,

^W  Gz ^{tenglama hosil bo’ladi. Buning xususiy yechimi:}
z

W   ¹
0 _
^{G t } dd. Berilgan sistemaning umumiy yechimini quyidagi


t  z
D

W  W₁


 W₀

ko’rinishda izlaymiz. Bunda

W ^W  0
¹ z

tenglamaning umumiy

yechimi. Bu istalgan Demak
W₁   z  analitik funksiyadan iborat.

W   z   ¹ _ ^{G t } dd
 t  z
D

W  P  iQ  U  ibV
 e^cx u  ibV   e^cx

2G  G₁  iG₂
 e^cx g₁  ig₂   e^cx 2g

D 


z  au  ibv  e^{cx } z   ¹ _ ^{G t } e^cx dd ^

_^ _
t  z ^

U holda

ux, y  ¹ Rezvx, y  ¹ Imz.

a b

2.