Directory UMM :Journals:Journal_of_mathematics:VMJ:

‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
€¯à¥«ì{¨î­ì, 2002, ’®¬ 4, ‚ë¯ã᪠2

“„Š 517.956

Š€…‚€Ÿ ‡€„€—€ „‹Ÿ ‘Œ…˜€ŽƒŽ ƒˆ…Ž‹Ž€€Ž‹ˆ—…‘ŠŽƒŽ “€‚…ˆŸ ’…’œ…ƒŽ ŽŸ„Š€
‚. €. …«¥¥¢, ‡. Œ. ¥«å à®¥¢ 

“áâ ­®¢«¥­ë áãé¥á⢮¢ ­¨¥ ¨ ¥¤¨­á⢥­­®áâì à¥è¥­¨ï ®¤­®© ªà ¥¢®© § ¤ ç¨ ¤«ï ᬥ蠭­®£®
£¨¯¥à¡®«®-¯ à ¡®«¨ç¥áª®£® ãà ¢­¥­¨ï âà¥â쥣® ¯®à浪 .
 áᬠâਢ ¥âáï ãà ¢­¥­¨¥

 u , u +  u; y > 0;
0 = @xx y 1
(1)
@x (uxx , uyy + 2 ux + 3 u); y < 0
¢ ®¡« áâ¨
, ®£à ­¨ç¥­­®© ®â१ª ¬¨ AA0 ; BB0 ; A0 B0 ¯àï¬ëå x = 0; x = 1; y = h ᮮ⢥âá⢥­­® ¨ å à ªâ¥à¨á⨪ ¬¨ AC : x + y = 0; BC : x , y = 1 ãà ¢­¥­¨ï (1). ãáâì
1 =
\ (y > 0),

2 =
\ (y < 0), I = (0; 1).
‡ ¤ ç  1. ’ॡã¥âáï ­ ©â¨ äã­ªæ¨î u(x; y ) á® á«¥¤ãî騬¨ ᢮©á⢠¬¨:
1) u(x; y ) | ॣã«ïà­®¥ à¥è¥­¨¥ ãà ¢­¥­¨ï (1) ¢ ®¡« áâ¨
¯à¨ y 6= 0;
2) u ­¥¯à¥à뢭  ¢ § ¬ª­ã⮩ ®¡« áâ¨
;
3) ç áâ­ë¥ ¯à®¨§¢®¤­ë¥ ux ; uy ­¥¯à¥àë¢­ë ¢ ®¡« áâ¨
¨ ¢ â®çª å O (0; 0) ¨ A(1; 0) ¬®£ãâ
®¡à é âìáï ¢ ¡¥áª®­¥ç­®áâì ¯®à浪  ¬¥­ìè¥ ¥¤¨­¨æë;
4)

u 㤮¢«¥â¢®àï¥â ªà ¥¢ë¬ ãá«®¢¨ï¬:

u(0; y) = '1 (y); u(1; y) = '2 (y); u(x; ,x) = 1 (x); u(x; x , 1) = 2 (x);
£¤¥

'1 ; '2 ; 1 ;

2 | ¤®áâ â®ç­® £« ¤ª¨¥ ä㭪樨, ¯à¨ç¥¬

 1
 1
'1 (0) = 1 (0); '2 (0) = 2 (0); 1 , 2 = 2 , 2 :
ãáâì ¢ ­ ç «¥ 2 6= 0; 3 = 0. ‚ ®¡« áâ¨
2 ®¡é¥¥ à¥è¥­¨¥ ãà ¢­¥­¨ï (1) ¬®¦¥â ¡ëâì

¯à¥¤áâ ¢«¥­® ¢ ¢¨¤¥ [1]

£¤¥

 

(2)

u(x; y) =  (x; y) + !(y);

(3)

xx , yy + 2 x = 0;

(4)

 (x; y) | ®¡é¥¥ à¥è¥­¨¥ ãà ¢­¥­¨ï

!(y) | ¯à®¨§¢®«ì­ ï ¤¢ ¦¤ë ­¥¯à¥à뢭® ¤¨ää¥à¥­æ¨à㥬 ï äã­ªæ¨ï. ¥§ ®£à ­¨ç¥­¨ï ®¡é­®á-

⨠¬®¦­® ¯à¥¤¯®«®¦¨âì

!(0) = ! (0) = 0:
Ž¡®§­ ç¨¬  (x) = u(x; 0); v (x) = uy (x; 0).
0

c 2002 …«¥¥¢ ‚. €., ¥«å à®¥¢  ‡. Œ.

(5)

2{24

‚. €. …«¥¥¢, ‡. Œ. ¥«å à®¥¢ 

‹¥¬¬  1.

ɇǬ

1 6 0 ¨
lim u(x; 0)ux (x; 0) = xlim
!1, u(x; 0)ux (x; 0) = 0;

(6)

x!0+

â® ¤«ï «î¡®£® ॣã«ïà­®£® ¢ ®¡« áâ¨
¢¥­á⢮

Z1

0

1

à¥è¥­¨ï ãà ¢­¥­¨ï

á¯à ¢¥¤«¨¢® ­¥à -

Z1

 (x)v(x) dx = , [ 02 (x) , 1  2 (x)] dx 6 0:

(7)

0

C „®ª § â¥«ìá⢮ «¥¬¬ë 1 ¯à¨¢¥¤¥­® ¢ [2]. B
‹¥¬¬  2. …᫨ u = 0 ­  OC [ AC , â® ¤«ï «î¡®£®

à¥è¥­¨ï ãà ¢­¥­¨ï

(1)

(1) á¯à ¢¥¤«¨¢® à ¢¥­á⢮
Z1

0

ॣã«ïà­®£® ¢ ®¡« áâ¨

 (x)v(x) dx = 0:


2
(8)

C „®ª § â¥«ìá⢮ «¥¬¬ë 2 ¯à¨¢¥¤¥­® ¢ [3]. B
‘à ¢­¨¢ ï (7) ¨ (8) ¨¬¥¥¬, çâ®  (x)  0.
’¥®à¥¬  1. ãáâì u(x; y ) | ॣã«ïà­®¥ à¥è¥­¨¥ ®¤­®à®¤­®© § ¤ ç¨ 1. ’®£¤ 
u(x; y)  0 ¢
1 .
C ‚ ®¡« áâ¨

1 à áᬮâਬ ⮦¤¥á⢮
u(uxx , uyy + 1 u) = (uux )x , uuy , u2x + 1 u2  0:
à®¨­â¥£à¨à®¢ ¢ ¥£® ¯à¨ ®¤­®à®¤­ëå £à ­¨ç­ëå ãá«®¢¨ïå (2) ¨  (x)  0, ¯®«ã稬
Z1

0

u2 (x; 1) dx +

Z


1

(u2x , 1 u2 ) dx dy = 0:

Žâá á«¥¤ã¥â, çâ® u(x; y)  0 ¢ ®¡« áâ¨
1 . ‚ ᨫã à ¢¥­á⢠ (x; 0) = u(x; 0),
y (x; 0) = uy (x; 0), ¨¬¥¥¬, çâ®  (x; y)  0 ¢ ®¡« áâ¨
2 . ˆ§ ãá«®¢¨ï  (x; ,x) = ,!(y)

§ ª«îç ¥¬, çâ® !(y) = 0 ¯à¨ , 21 6 y 6 0. ‘«¥¤®¢ â¥«ì­®, u(x; y)  0 ¢
2 , â. ¥.
u(x; y)  0 ¢
¨ à¥è¥­¨¥ § ¤ ç¨ 1 ¥¤¨­á⢥­­®. B
’¥®à¥¬  2. ‚ ®¡« áâ¨
áãé¥áâ¢ã¥â à¥è¥­¨¥ § ¤ ç¨ (1).
C ˆ§ ¯ à ¡®«¨ç¥áª®© ç á⨠®¡« áâ¨
¯à¨ y ! 0+, ¯®«ã稬 § ¤ çã

 00 (x) , v(x) + 1  (x) = 0;

(9)

 (0)'1 (0);  (0) = '2 (0):

(10)

 (x; ,x) = 1 (x) , !(,x);  (x; x , 1) = 2 (x) , !(x , 1);

(20 )

¥è¥­¨¥ ãà ¢­¥­¨ï (4), 㤮¢«¥â¢®àïî饥 ªà ¥¢ë¬ ãá«®¢¨ï¬

2{25

Šà ¥¢ ï § ¤ ç  ¤«ï ᬥ蠭­®£® £¨¯¥à¡®«®-¯ à ¡®«¨ç¥áª®£® ãà ¢­¥­¨ï

®¯à¥¤¥«ï¥âáï ä®à¬ã«®©

 (x; y) =


1

x,y




+

2

2

x+y+1



2

,!



y,x



2

,!



x+y,1



2

:

(11)

‡ ¤ ç  (9), (10) § ¬¥­®©

 (x) = z (x) + x2 + ('2 (0) , '1 (0) , 1)x + '1 (0) = z (x) + h(x)
¯à¨¢®¤¨âáï ª ¢¨¤ã

z (x) + 1 z (x) = f (x);

(12)

z (0) = 0; z (1) = 0;

(13)

00

£¤¥

f (x) = v(x) , h(x) , 2.

¥è¥­¨¥ § ¤ ç¨ (12), (13) ¨¬¥¥â ¢¨¤

z (x) =

Z1

G(x; t; 1 )f (t) dt;

(14)

0
£¤¥

G(x; t; 1 ) =

p
p
8
sin ,1 t sin ,1 (1 , x)
>
>
p, sin p,
;

>
: , sin

1

| äã­ªæ¨ï ƒà¨­ . ‚®§¢à é ïáì ª ä㭪樨

 (x)

0

p, x sin p, (1 , t)
p,1  sin p,1
;
1
1
1

Z1

6 t 6 x;

x6t61

 (x), à ¢¥­á⢮

(14) ¯¥à¥¯¨è¥¬ ¢ ¢¨¤¥

G(x; t; 1 )v(t) dt + f~(x);

(15)

0
£¤¥

f~ = h(x) ,

Z1

G(x; t; 1 )f2 + 1 h(t)g dt:

0
ˆ§ à ¢¥­á⢠ (11) ¨¬¥¥¬

 (x; 0) =  (x) =
y (x; 0) = v(x) = ,

1
2

0

1

x

1

2

x
2

+


+

1
2

2


0

2

x+1



2

x+1
2



 x

, ! ,2 , !
 x



x,1
2

, 2! ,2 , 2!
1

0



1


0

;

x,1
2

(16)



:

(17)

®¤áâ ¢«ïï (16), (17) ¢ à ¢¥­á⢮ (15), ¯®«ã稬 ¨­â¥£à®-ä㭪樮­ «ì­®¥ ãà ¢­¥­¨¥

!

 x

,2

+!



x
2

,2
1


=

1
2

Z1

 
t
t
+!
G(x; t; 1 ) ! ,
0

0

0

2

2

,2

1



dt + g(x);

(18)

2{26
£¤¥

‚. €. …«¥¥¢, ‡. Œ. ¥«å à®¥¢ 

Z1


t 1 t


 
1
1
0
0
g(x) = G(x; t; 1 ) 2 1 2 , 2 2 2 + 2 dt + 1 x2 + 2 x +2 1 + f~(x):


0

„¨ää¥à¥­æ¨àãï à ¢¥­á⢮ (18), ¯®«ã稬

!0



1



 
Z
t



x
1
1
x
t
0
0
0
, 2 , ! 2 , 2 = , Gx(x; t; 1 ) ! , 2 + ! 2 , 2 dt = ,2g0 (x): (19)

0,

0

1


Ž¡®§­ ç ï ! , 2 = (z ), ¯®«ã稬 ¨­â¥£à®-ä㭪樮­ «ì­®¥ ãà ¢­¥­¨¥ ¢¨¤ 


1







Z


1
1
(z) ,  ,z , 2 = , Gx(x; y; 1 ) (t) +  ,t , 2 dt , 2g(,2z):

0

(20)





”ã­ªæ¨î (z ) ¡ã¤¥¬ ¨áª âì ¢ ¢¨¤¥ (z ) = r(z ) , r ,z , 21 .
“ç¨âë¢ ï ¯®á«¥¤­¥¥ á« £ ¥¬®¥ ¢ à ¢¥­á⢥ (20), ¯®«ã稬 ä㭪樮­ «ì­®¥ ãà ¢­¥­¨¥


(21)
r(z) , r ,z , 12 = ,2g(,2z);
ª®â®à®¥ ï¥âáï ç áâ­ë¬ á«ãç ¥¬ «¨­¥©­®£® ä㭪樮­ «ì­®£® ãà ¢­¥­¨ï [4]
Af [ (t)] + Bf ['(t)] = F (t);
(22)
£¤¥ t | ¯¥à¥¬¥­­ ï, A; B; (t); '(t); F (t) | § ¤ ­­ë¥ ä㭪樨 t, f | ­¥¨§¢¥áâ­ ï
®¯¥à æ¨ï, ®¡à é îé ï ãà ¢­¥­¨¥ (22) ¢ ⮦¤¥á⢮.
ˆá¯®«ì§ãï ⥮à¨î ¨â¥à æ¨®­­®£® ¨áç¨á«¥­¨ï, ¤®ª §ë¢ ¥âáï, çâ® ä㭪樮­ «ì­®¥
ãà ¢­¥­¨¥ (21) ¨¬¥¥â à¥è¥­¨¥ [4]. ˆ§ ¥¤¨­á⢥­­®á⨠à¥è¥­¨ï § ¤ ç¨ 1 á«¥¤ã¥â, çâ®
äã­ªæ¨ï !(y) ®¯à¥¤¥«ï¥âáï ¥¤¨­á⢥­­ë¬ ®¡à §®¬. ’ ª¨¬ ®¡à §®¬, äã­ªæ¨ï u(x; y)
¯®«­®áâìî ®¯à¥¤¥«¥­  ¢ ®¡« áâ¨
.
ãáâì ⥯¥àì 2 = 0; 3 > 0. „®ª § â¥«ìá⢮ ¥¤¨­á⢥­­®á⨠à¥è¥­¨ï § ¤ ç¨ 1 ¢
í⮬ á«ãç ¥ ¯à®¢®¤¨âáï  ­ «®£¨ç­® ¯à¥¤ë¤ã饬ã.
‚ ®¡« áâ¨
2 ®¡é¥¥ à¥è¥­¨¥ ãà ¢­¥­¨ï (1) ¨¬¥¥â ¢¨¤ (3), £¤¥  (x; y) | ®¡é¥¥
à¥è¥­¨¥ ãà ¢­¥­¨ï
xx , yy + 3  = 0:
(23)
¥è¥­¨¥ § ¤ ç¨ Š®è¨ á ­ ç «ì­ë¬¨ ¤ ­­ë¬¨
 (x; 0) = u(x; 0) =  (x); y (x; 0) = uy (x; 0) = v(x);
¨¬¥¥â ¢¨¤ [5]

 (x; y) =  (x , y) +2  (x , y) + 21
+ 21

xZ+y
x,y

@J
 () @y
0

,p

xZ+y

x,y

p

J0

,p

p




3 (x , )2 , y2 v() d



3 (x , )2 , y2 v() d;

(24)

2{27

Šà ¥¢ ï § ¤ ç  ¤«ï ᬥ蠭­®£® £¨¯¥à¡®«®-¯ à ¡®«¨ç¥áª®£® ãà ¢­¥­¨ï

J0 (z ) | äã­ªæ¨ï ¥áá¥«ï ¯¥à¢®£® த  ­ã«¥¢®£® ¯®à浪 . “¤®¢«¥â¢®àïï (24) ªà ¥¢ë¬ ãá«®¢¨ï¬ (20 ), ¯®«ã稬

£¤¥

x

 (x) = 2
1

+

+

0

1

,p

p




3  ( , x) v( ) d

0



2

 ( )

J0

(25)

 x


@ ,p p
 ( ) J0 3  ( , x) d , 2! , ;
@x
2

x+1

2

Zx

Zx

2



 (x) = 2

2

, 2'(0) +

Zx

, 2'2 (0) ,

Zx

J0

,p

3

p

(




, 1)( , x) v(x) d

1

@
J
@x 0

,p

3

p

(




, 1)( , x)

d , 2!



, x ,2 1



(26)

:

®á«¥ ®¡à é¥­¨ï ¨­â¥£à «ì­ëå ãà ¢­¥­¨© (25), (26) ®â­®á¨â¥«ì­®

 (x),

ᮮ⢥â-

á⢥­­® ¯®«ã稬 [3]

 (x) = 2

,2

x

1

Zx

2

J0



x

, 2! , 2

,p

Zx

J0

+

,p

0

p




3 x(x ,  )


1



 (x) =2

2

,2

x+1



2

Zx
1

@
J
@x 0

, 2!

,p

3



x,1



2

p

(1

,

Zx

J0

1




, x)( , x)





, ! ,2

2

0






3 (x ,  ) v( ) d


,p

(27)

d;



3 (,x +  )v( ) d


2

 + 1
2

, ! ,2



1

(28)

d:

 ¢¥­á⢠ (27), (28) ᮮ⢥âá⢥­­® ¯¥à¥¯¨è¥¬ ¢ ¢¨¤¥

 (x) =

Zx

v( )J0

0

,p

Zx
+

 
1

0

 (x) = ,

Zx




3 (x ,  ) d

v( )J0



2

J0

,p

p

3 x(x ,  )




Zx
+

!


0

0

,p




3 ( , x) d +

Zx


0

2
1

1

Zx
+

1

!



2


0

+1

 ,1
2

, 2



J0
J0



J0

,p

,p

,p

3

3

p

p

p




3 x(x ,  ) d;

(1

(1




, x)( , x)
, x)( , x)




(29)

d
(30)

d:

2{28

‚. €. …«¥¥¢, ‡. Œ. ¥«å à®¥¢ 

 (x) ¯®¤áâ ¢¨¬ (29) ¨ ¯®«ã祭­®¥ à ¢¥­á⢮

‚ «¥¢ãî ç áâì à ¢¥­á⢠ (15) ¢¬¥áâ®
¯à®¤¨ää¥à¥­æ¨à㥬 ¯®
­¥­¨¥

v(x) +

Zx
0

x.

‚ १ã«ìâ â¥ ¡ã¤¥¬ ¨¬¥âì ᬥ蠭­®¥ ¨­â¥£à «ì­®¥ ãà ¢-



@ ,p
v( ) J0 3 (x ,  ) d =
@x

x



, ! ,2 ,
0

Zx

!


0

0

£¤¥

g(x) = f~(x) ,



,2



J0

2

0

0

,p

,3

p




x(x ,  ) d:

R(x;  ) १®«ì¢¥­âã ãà ¢­¥­¨ï (31)

…᫨ ®¡®§­ ç¨âì ç¥à¥§

(31)

0



1

Gx (x; ; 1 )v( ) d



@ ,p p
J0 3 x(x ,  ) + q (x);
@x

 

Zx

Z1

¨ ¯à¥¤¢ à¨â¥«ì­® áç¨-

â âì ¥£® ¯à ¢ãî ç áâì ¨§¢¥áâ­®©, â® à¥è¥­¨¥ í⮣® ãà ¢­¥­¨ï ¬®¦­® ¯à¥¤áâ ¢¨âì ¯®
ä®à¬ã«¥

v(x) ,

Z1

Q0 (x;  )v( ) d

=

,!

0

Q0 (x;  ) = Gx (x; ; 1 ) +

Z1

0

x
2

,

Zx

Q1 (x;  )!



,2

0



d + q~(x) = (x);

(32)

R(x; 1 )G(1 ; ; 1 ) d1 ;

0

Q1 (x;  ) =


0



@ ,p p
J0 3 x(x ,  ) + R(x;  ) +
@x

q~(x) =

Zx

Zx

R(x; 1 )





@ ,p p
J0 3 1 (1 ,  ) d;
@x

R(x;  )q ( ) d:
0

0
’ ª¨¬ ®¡à §®¬, ¯®«ã稬 ®â­®á¨â¥«ì­®

T (x;  )

‘ ¯®¬®éìî १®«ì¢¥­âë

v(x)

ãà ¢­¥­¨¥ ”।£®«ì¬  ¢â®à®£® த .

ãà ¢­¥­¨ï (32), §­ ç¥­¨¥

騬 ®¡à §®¬

v(x) = (x) +

Z1

v(x)

§ ¯¨á뢠¥âáï á«¥¤ãî-

T (x;  )( ) d:

(33)

0
®¤áâ ¢¨¢ §­ ç¥­¨¥

(x) ¢ à ¢¥­á⢮

v(x) =

Z1
0

M (x;  )!


0

(33), ¯®«ã稬



,2



d , !

0



, x2



+ P (x);

(34)

Šà ¥¢ ï § ¤ ç  ¤«ï ᬥ蠭­®£® £¨¯¥à¡®«®-¯ à ¡®«¨ç¥áª®£® ãà ¢­¥­¨ï

£¤¥
M (x;  )

8
>
>
Q (x;  )
>
< 1

=>
>
>
:

, T (x; ) +

,T (x; ) +

R1



R1



0 6  6 x;

T (x; 1 )Q(1 ;  ) d1 ;

T (x;  )Q1 (1 ;  ) d1 ;

P (x)

=

Z1

2{29

x

6  6 1;

T (x;  )~
q ( ) d:

0

‘ ¤à㣮© áâ®à®­ë, § ¤ çã (1), (2) ¢ ¯ à ¡®«¨ç¥áª®© ç áâ¨
1 ®¡« áâ¨
, § ¬¥­®©
¬®¦­® ᢥá⨠ª § ¤ ç¥

u = e1 y ,

yy

= xx ;

 (0; y )

= e1 y '1 (y);

 (1; y )

= e1 y '2 (y):

(35)

¥è¥­¨¥ § ¤ ç¨ (35) ¨¬¥¥â ¢¨¤
 (x; y )

=

Z1

 ( )G(; 0; x; y ) d

0

,

Zy

+

Zy

e1 h '1 (h)G (0; h; x; y ) dh

0

e1 h '2 (h)G (1; h; x; y ) dh;

0

£¤¥ G(; h; x; y) | äã­ªæ¨ï ƒà¨­  ¯¥à¢®© ªà ¥¢®© ®¤­®à®¤­®© § ¤ ç¨ (35).  ©¤¥¬
¯à®¨§¢®¤­ãî uy , § â¥¬ ¢ ¯®«ã祭­®¬ à ¢¥­á⢥ ¯®«®¦¨¬ y = 0. ã¤¥¬ ¨¬¥âì
v (x)

£¤¥

=

K (x;  )

 (x) =


@u 
@y y=0

=

Z1

K (x;  ) ( ) d

+ (x);

(36)

0

,

= 1 G(; 0; x; 0) + Gx (; 0; x; 0)

8
2 y
Z
<
@

y
1
4
e
e1 h '1 (h)G (0; h; x; y ) dh
@y :

,

0

Zy
0




;

39
=

h
1
5
:
e '2 (h)G (1; h; x; y ) dh
;

®¤áâ ¢«ïï ⥯¥àì (15) ¢ à ¢¥­á⢮ (36), ¯®«ã稬
v (x)

=

Z1

e (x;  )v ( ) d
K

+ (x);

0

£¤¥
e (x;  )
K

=

Z1
0

K (x;  )G(; 1 ; 1 ) d:

(37)

2{30

‚. €. …«¥¥¢, ‡. Œ. ¥«å à®¥¢ 

Ž¡®§­ ç¨¬ ç¥à¥§ N (x;  ) १®«ì¢¥­âã ãà ¢­¥­¨ï (37), à¥è¥­¨¥ í⮣® ãà ¢­¥­¨ï
¬ë ¬®¦¥¬ ¯à¥¤áâ ¢¨âì ¯® ä®à¬ã«¥
v (x)

= (x) +

Z1

N (x;  ) ( ) d:

(38)

0



®¤áâ ¢«ïï ¢ëà ¦¥­¨¥ (38) ¤«ï v(x) ¢ ä®à¬ã«ã (34) ¯®«ã稬 ®â­®á¨â¥«ì­® !0 , x2
¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥ ”।£®«ì¬  ¢â®à®£® த 
!0



x

,2 =

Z1

M (x;  )! 0

0



, 2



d +  (x);



(39)

£¤¥ (x) = P (x) , v(x).
’ ª ª ª íª¢¨¢ «¥­â­®áâì ¢áî¤ã á®åà ­ï¥âáï, â® ¨§ ¥¤¨­á⢥­­®á⨠à¥è¥­¨ï § ¤ ç¨ 1 á«¥¤ã¥â ®¤­®§­ ç­ ï à §à¥è¨¬®áâì ¨­â¥£à «ì­®£® ãà ¢­¥­¨ï (39).
€­ «®£¨ç­ë¬
®¡à §®¬ ¬®¦­® ¯®«ãç¨âì ¨­â¥£à «ì­®¥ ãà ¢­¥­¨¥ (39) ®â­®á¨â¥«ì 
x
,
1
0
­® ! 2 . ’ ª¨¬ ®¡à §®¬, ®¯à¥¤¥«¥­ë ä㭪樨  (x); v(x); !(y). ‘«¥¤®¢ â¥«ì­®,
®¯à¥¤¥«¥­  äã­ªæ¨ï u(x; y) ¢ ®¡« áâ¨
. B

‹¨â¥à âãà 
1. ‘ « å¨â¤¨­®¢ Œ. ‘. “à ¢­¥­¨ï ᬥ蠭­®-á®áâ ¢­®£® ⨯ .|’ èª¥­â: ”€, 1974.|156 á.
2. ‘ ¡¨â®¢ Š. . Š ⥮ਨ ãà ¢­¥­¨© ᬥ蠭­®£® ¯ à ¡®«®-£¨¯¥à¡®«¨ç¥áª®£® ⨯  ᮠᯥªâà «ì­ë¬ ¯ à ¬¥â஬ // „¨ää¥à¥­æ. ãà ¢­¥­¨ï.|1989.|’. 25, ü 1.|‘. 117{126.
3.  § à®¢ „. ‡ ¤ ç  „¨à¨å«¥ ¤«ï ®¤­®£® ãà ¢­¥­¨ï ᬥ蠭­®£® ⨯  // ˆ§¢.

€ ’‘‘, á¥à.

䨧.-¬ â. ­ ãª.|1984.|ü 6.|‘. 81{84.
4. ƒ¥àᥢ ­®¢ . Œ. ˆâ¥à æ¨®­­®¥ ¨áç¨á«¥­¨¥ ¨ ¥£® ¯à¨«®¦¥­¨ï.|Œ.: Œ èáâன¨§¤ â, 1950.|69 á.
5. ’¨å®­®¢ €. ., ‘ ¬ à᪨© €. €. “à ¢­¥­¨ï ¬ â¥¬ â¨ç¥áª®© 䨧¨ª¨.| Œ.:  ãª , 1977.|716 á.

 «ì稪

‘â âìï ¯®áâ㯨«  18  ¯à¥«ï 2002 £.