Directory UMM :Journals:Journal_of_mathematics:VMJ:

‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
ˆî«ì{ᥭâï¡àì, 2000, ’®¬ 2, ‚ë¯ã᪠3

“„Š 539.377

ˆ‡Œ……ˆ… …‹ˆ…‰ŽƒŽ ’…Œ…€’“ŽƒŽ Ž‹Ÿ,
‘‚Ÿ‡€Ž… ‘ ŠŽ””ˆ–ˆ…’ŽŒ ’…‹ŽŽ‚Ž„Ž‘’ˆ

’. ‡. —®ç¨¥¢

‚ ­ áâ®ï饩 áâ âì¥ ã¤ «®áì ¯®áâநâì ä®à¬ã«ã, ¢ëà ¦ îéãî § ª®­ ¨§¬¥­¥­¨ï ­¥«¨­¥©­®£® ⥬¯¥à âãà­®£® ¯®«ï, ãáâ ­®¢«¥­  ¥¥ ­¥¯®á।á⢥­­ ï á¢ï§ì á § ¤ ­¨¥¬ ­ ç «ì­®£® ¨ ªà ¥¢®£® ãá«®¢¨©,   â ª¦¥ 䨧¨ç¥áª¨¬¨ ᢮©á⢠¬¨ á।ë.

‚ à ¡®â¥ [4], ¢¢¨¤ã á«®¦­®á⨠­¥«¨­¥©­®£® ⥬¯¥à âãà­®£® ¯®«ï, ­¥ ¤ ­ 
®¡é ï á奬  ä㭪樨 ®â­®è¥­¨ï

p(x; t),

¯®í⮬㠭¥ ¢áªàëâ  ¥¥ á¢ï§ì á ä¨-

§¨ç¥áª¨¬¨ ãá«®¢ï¬¨ § ¤ ç¨. ‚ ­ áâ®ï饩 áâ âì¥ ã¤ «®áì ¯®áâநâì ä®à¬ã«ã,
¢ëà ¦ îéãî § ª®­ ¨§¬¥­¥­¨ï ­¥«¨­¥©­®£® ⥬¯¥à âãà­®£® ¯®«ï, ãáâ ­®¢«¥­  ¥¥ ­¥¯®á।á⢥­­ ï á¢ï§ì á § ¤ ­¨¥¬ ­ ç «ì­®£® ¨ ªà ¥¢®£® ãá«®¢¨©,  

â ª¦¥ 䨧¨ç¥áª¨¬¨ ᢮©á⢠¬¨ á।ë.  ¯®¬¨­ ¥¬, çâ® ¥á«¨ ª®íää¨æ¨¥­â
⥯«®¯à®¢®¤­®áâ¨

k

ï¥âáï ä㭪樥© ®â ⥬¯¥à âãàë:

­¥«¨­¥©­®£® ⥬¯¥à âãà­®£® ¯®«ï

@T
c
@t



=

@T
@
k (T )

@x
@x

k = k(T ), â® § ¤ ç 


(1)

¤«ï ®¤­®à®¤­®£® ã¯à㣮£® ¯®«ã¯à®áâà ­á⢠, ®£à ­¨ç¥­­®£® ¯®¢¥àå­®áâìî

x=

0, ¯à¨ ᮡ«î¤¥­¨¨ ãá«®¢¨©

T jt=0 = T0 ;

@T
@x

, k (T , ) = 0

¯à¨

x=0

(2)

¨ ¯à¨ ¯à¨­ï⨨ ®¡®§­ ç¥­¨ï Š¨à壮ä 

F

=

1

k0

ZT

k(T )dT;

T0

¯à¨¢®¤¨âáï ª ãà ¢­¥­¨î ⥯«®¯à®¢®¤­®áâ¨

c @F
k(T ) @t

c 2000 —®ç¨¥¢ ’. ‡.

=

@ 2F
;
@x2

(3)

ˆ§¬¥­¥­¨¥ ­¥«¨­¥©­®£® ⥬¯¥à âãà­®£® ¯®«ï
£¤¥

c

k0

| ®¡ê¥¬­ ï ⥯«®¥¬ª®áâì,

3{43

| ª®íää¨æ¨¥­â ⥯«®¯à®¢®¤­®áâ¨, ª®-

â®àë© á®®â¢¥âáâ¢ã¥â «¨­¥©­®¬ã ⥬¯¥à âãà­®¬ã ¯®«î,
¯¥à âãà ,
à ­á⢠,



| ­ ç «ì­ ï ⥬-

(t) | ª®íää¨æ¨¥­â ⥯«®®â¤ ç¨ ­  ¯®¢¥àå­®á⨠x = 0 ¯®«ã¯à®áâ| ⥬¯¥à âãà , ãáâ ­®¢¨¢è ïáï ­  ¯®¢¥àå­®á⨠¢ १ã«ìâ â¥ ⥯-

«®®¡¬¥­ . ‚¢®¤ï ¯à®¬¥¦ãâ®ç­ë¥ ä㭪樨

Z dF


¯®«ãç ¥¬

£¤¥

T0

@F
@t

p t

=

T ;

=

c dx @F
p0 R0x pk
;
e
p
@x



V

=

(T ) ¨ T  [4] ä®à¬ã«®©



( )

=

p0 e

R x c 

p0 (t)e

dx
0 pk

R x c dx
0 pk

(4)

;

(5)

0 ( ) | ¯à®¨§¢®«ì­ ï äã­ªæ¨ï,   (4) 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î (3). ”ã­ªæ¨ï ®â­®è¥­¨ï (
) ®¯à¥¤¥«ï¥âáï à ¢¥­á⢮¬

p x; t

p(x; t) =
‚ á¢ï§¨ á ⥬, çâ®

@F
@x

 @F

=

@t

@T 

@x

 @T 
@t

=

@T
@x

 @T
@t

:

F (x; t) ⥬¯¥à âãà­ ï äã­ªæ¨ï, â ª¦¥ ¤®«¦­® ¨¬¥âì ¬¥áâ®

 

@ V

@x p

=

@V
@t

@V
) @V
,
p
@x
@t

=

@p
V;
p @x

1

à¥è¥­¨¥ ª®â®à®£® ¥áâì
1
V = '( )e, 2

£¤¥

R  @ p1 d
0

@x

(

'( ) | ¯à®¨§¢®«ì­ ï äã­ªæ¨ï.

d = pdx + dt; d = pdx , dt);

(6)

‘à ¢­¨¬ ¥¥ ¯à ¢ãî ç áâì á ¯à ¢®© ç áâìî

ä®à¬ã«ë (5),

p0 e

R x c dx
0 pk

,1
= '( )e 2



R  @ p1 d
0

@x

'

j

( ( ) t=0 =

p0 (t)):

Ž¡®§­ ç¥­¨¥ Š¨àå£®ä  ¨ ( ) ¯à¨¢®¤ïâ ª ᮮ⭮襭¨ï¬ (á¬. [4])
c dx
@T  p0 R0x pk
=
=
e
;
@t
p
c dx
@T  p0 p @ R0x pk
=
0 e
;
@x
c @x

@F
@t

R x c
@T 
dx
=
= p0 e 0 pk ;
@x
c dx
@T  p0 0 @ R0x pk
=
e
:
@t
c @x

@F
@x

(7)

3{44

’. ‡. —®ç¨¥¢

‡¤¥áì, â ª ¦¥, ª ª ¨ ¢ëè¥, ¯à ¢ë¥ ç á⨠¤®«¦­ë 㤮¢«¥â¢®àïâì ãá«®¢¨î ¯®â¥­æ¨ «ì­®á⨠¯®«ï

@ (pV0 )
@t

=

¨«¨ ãà ¢­¥­¨î

à¥è¥­¨¥¬ ª®â®à®£® ï¥âáï



@V0
@x
@V0
@x

V0 =

V0 = '1  e
'1 ( ) ¯à®¨§¢®«ì­ ï äã­ªæ¨ï.

R  @ ln p
@t

0

‘à ¢­¨¢

R x c dx

0 p0 @
e
c @x



(8)

0 pk

=

V0

d

(9)

;

á ¯à ¢®© ç áâìî ¢ëà ¦¥­¨ï (8)

1
( ) 2

'1  e

R  @ ln p
0

¨ ¯à¨­ï¢ ¢® ¢­¨¬ ­¨¥ (7), ¯®«ã稬
1
@
'( )e, 2
@x

0 pk

0 p = @p V ;
, @V
@t
@t 0
1
( ) 2

£¤¥

R x c dx 

p0 0 @
e
c @x

R x @ p1 
0 @x

dx

=

'1 ( )

c 12
e
0

@t

d

(10)

R  @ ln p
0

@t

d

:

(11)

â® ¨ ¥áâì à ¢¥­á⢮, ª®â®à®¬ã ¤®«¦­  㤮¢«¥â¢®àïâì äã­ªæ¨ï ®â­®è¥­¨ï

p(x; t).

‚ (11) ¢®è«¨ ­®¢ë¥ ¯¥à¥¬¥­­ë¥

¯à¥®¡à §®¢ ­¨ï ª®®à¤¨­ â

@
@x





@
=p
@
(d =

 ¨  , ¯®í⮬㠢®§­¨ª ¥â ­¥®¡å®¤¨¬®áâì

+

@
@
;
@
@t



@
=p
@

,

@
@



pdx + dt; d = pdx , dt):

¥§ã«ìâ â ¯¥à¥å®¤  ¤ ¥â

p'e

1
2

R

’ ª ª ª

@ ln p
( @
0

"
 @ 2 R  ln p d @ 2 R  ln p d
ln p d '0 ( )
1
+ @@
)
0
0
+
+2
'( ) 2
@ 2
@@
R
# c
R
1  ( @ ln p , @ ln p ) d
@ 2 0 ln p d
2
+
:
=
' ( )e 0 @ @
@ 2
0 1

1
0 ¥áâì ­¥ª®â®à®¥ à¥è¥­¨¥ (1:11) ¨§ [4] § ¢¨áï饥 ®â  , â® ¡¥§ ®£à ­¨-

祭¨ï ®¡é­®á⨠¬®¦¥¬ ¤®¯ãáâ¨âì

'1 ( ) = 0 ( )' ( )





'( )
' ( ) =
:
c

ˆ§¬¥­¥­¨¥ ­¥«¨­¥©­®£® ⥬¯¥à âãà­®£® ¯®«ï

3{45

‘«¥¤®¢ â¥«ì­®, ¯®á«¥¤­¥¥ ¯à¨¢®¤¨¬® ª á«®¦­®¬ã ­¥«¨­¥©­®¬ã ãà ¢­¥­¨î

@2

R

R

ln p d
@2
+2
@ 2

ln p d

+

@@

R

@2

ln p d
2
=
2
@
p2

'0 ( )
, 2 ' ( ) ;

(12)

ª®â®à®¥ ¬®¦­® § ¯¨á âì ¢ ¢¨¤¥ á¨á⥬ë

@

R

ln p d

@
¨«¨, áç¨â ï

 (x; t)

R

@ 0

@

+

R

ln p d

@

ln p d

@

R

+

@ 0

ln p d

@
+

@V 
@

V  ,  = 2l
l

+

@V 
@

=






= +2
=
1
p2

1
p2

V

R 
0

ld

(13)

,



'0 ( )
;
' ( )

V  , 
;
l

,

(14)



'0 ( )
;
' ( )

®¯à¥¤¥«¨¬ ¯®§¤­¥¥. ˆ§ ¢â®à®£® ãà ¢­¥­¨ï ­ å®¤¨¬

1
= e2

0

, 2 '' (()) ;

2
p2

§ ¤ ­­®©, ¬®¦­® ¯®áâநâì à ¢­®á¨«ì­ãî á¨á⥬ã

@V 
@

£¤¥

@V 
@

V;

=

C1 ( ) ,

1
2

Z

,1
e 2

R
0

ld




0


d
l



( =



V

+ ;

¢ ¢¨¤¥



=



,  );

(15)

¨ ¯à ¢ãî ç áâì ¯¥à¢®£® ãà ¢­¥­¨ï § ¬¥­ï¥¬ âà¥â쨬

@
@

Z

ln pd =

0

V

)p

2

1 @
= e 2 @

R  V d
0

:

(16)

® ãáâ ­®¢«¥­­ë¬ ä®à¬ã« ¬ (15) ¨ (16) âà¥âì¥ á®®â­®è¥­¨¥ ¨§ (14) ¯¥à¥¯¨á뢠¥âáï ª ª

V
2l

,


2l

,@
= e @

R
0

V  d

0

, '' (()) ;

¨«¨, § ¬¥­¨¢ «¥¢ãî ç áâì ¤ ­­®£® à ¢¥­á⢠ «¥¢®© ç áâìî ¢â®à®£® ãà ¢­¥­¨ï
¨§ (14) | ¢ ä®à¬¥

@V 
@

,@
= e @

R  V  d
0

0

, '' (()) :

3{46

’. ‡. —®ç¨¥¢

®á«¥ 㬭®¦¥­¨ï ­ 
@
e @

@

e @
R
0

R

V  d @
@ ¨¬¥¥¬

0

V  d @V



@
'0 ( ) @
e
' ( )

+

@

R

V  d @

0

=

@

@
;
@

®âªã¤  ¯®«ãç ¥¬ ãà ¢­¥­¨¥ ®â­®á¨â¥«ì­® íªá¯®­¥­âë
@
@ @
e
@

R 
V d
0

+

@
'0 ( ) @
e

' ( )

R 
V d
0



=1

@
@


=1

;

¥è ï ¯®á«¥¤­¥¥ ãà ¢­¥­¨¥ ¯®«ãç ¥¬

R

@
e @

0

‘«¥¤®¢ â¥«ì­® (á¬. (16)),
1

=

p
  â ª¦¥ ãç¨â뢠ï, çâ® V 

R

V  d



=

s

' ( )
C ( ) +

=

Z '2 ( )

h

, '0 ( )

0

C ( ) +

h

' ( ) C ( ) +

0

R

' ( )d

(17)

;

(18)

¤®¯ã᪠¥â ­¥¯à¥àë¢­ë¥ ¯à®¨§¢®¤­ë¥ (á¬. (15)), ¨§

(17) ¢ë¢®¤¨¬

V

C ( ) + 0 ' ( )d
:
' ( )

R
0

R
0

' ( )d

' ( )d

i

i

d + V0 ( ):

(19)

¥§ã«ìâ â ¯à¨à ¢­¨¢ ­¨ï ¯à ¢ëå ç á⥩ (15) ¨ (19) ¯®§¢®«ï¥â ®¯à¥¤¥«¨âì

 (;  )

¢ ¢¨¤¥


2l
@
, @

Z '2

=

Z '2
1
2

0

, '0
h

h

h

h

R

R
0

0

R

C ( ) +

' C ( ) +

C ( ) +

' C ( ) +

0

, '0

' d

' d

i

R
i

0

0

' d

' d

d ,

i

i

d



@
,
V0 e
@

R
0

ld



:

(20)

p; V  ;  ¨§ (18){(20) ¢­¥á¥¬ ¢ âà¥âì¥ à ¢¥­á⢮ ¢ëà ¦¥­¨ï (14)
çâ® V0  0 (íâ® ¤®¯ã饭¨¥ ã¯à®é ¥â ­ å®¦¤¥­¨¥ l)

„ «¥¥, §­ ç¥­¨ï
¯à¨ ãá«®¢¨¨,

l,
@
@
=2

1

l



R
' ( )
0

d
R

C ( )+

0



,

@
@



'0 ( )
' ( ) 

' d
R
' (R )d
0 C ( )+  ' d
0



,

'

, ''0 (()) 



R
C ( )+ ' d
0


R
' C ( )+ ' d

2

,'0

0

(21)

:

ˆ§¬¥­¥­¨¥ ­¥«¨­¥©­®£® ⥬¯¥à âãà­®£® ¯®«ï

3{47

Žâá ¨ ®¯à¥¤¥«ï¥¬ l. “áâ ­®¢«¥­­ë¥ ¢ëè¥ ä®à¬ã«ë (18){(20) 㤮¢«¥â¢®àïî⠢ᥬ à ¢¥­á⢠¬ (14). ’ ª ª ª ¢ ä®à¬ã«¥ (19) ¯à¨  = 0; V  = 0, â®
¢ ¯à ¢®© ç á⨠(15) C1 () = 0. ‚ᥠ¢ëè¥ ¯¥à¥ç¨á«¥­­ë¥ ä㭪樨 § ¢¨áïâ ®â
C ( ) ¨ ' ( ) ¨, á«¥¤®¢ â¥«ì­®, ᮣ« á­® (10) ¡ã¤¥â
R @ ln p
c dx
1
p0 @ R pk

2
@t d ;
e
=
'e
c @x
’¥¬ á ¬ë¬ ãáâ ­®¢«¥­® ⮦¤¥á⢥­­®¥ à ¢¥­á⢮ (á¬. (11))

R 1 
R
1  @ p d
1  @ ln p d
@
,
2
@x
2
0
'( )e
= '( )e 0 @t ;
@x
­  ®á­®¢ ­¨¨ ª®â®à®£® ç áâ­ë¥ ¯à®¨§¢®¤­ë¥ ä㭪樨 F (x; t) ¢ëà ¦ îâáï
(á¬. (5) ¨ (7)) ä®à¬ã« ¬¨
1
R 1
1  @ p d
@F
'( ) , 12 R0 @@xp d
@F
,
2
@x
0
;
:
= '( )e
=
e
@x
@t p(;  )
®«¥¥ ⮣®, ¯®áª®«ìªã ¯à ¢ë¥ ç á⨠㤮¢«¥â¢®àïîâ ãá«®¢¨ï¬ â¥®à¥¬ë ˜¢ àæ 
(á¬. (5)), â® ¤¨ää¥à¥­æ¨ «ì­®¥ ᮮ⭮襭¨¥
1
R 1
1  @ p d
1
'( ) , 21 R0 @@xp d
,
2
dt + '( )e 0 @x d
(22)
dF = K (T )dT =
e
k0
p(;  )
¯®§¢®«ï¥â ¨áª«îç¨âì ⥬¯¥à âãà­ãî äã­ªæ¨î T (x; t). ãáâì k(T1 ) (T0 < T1 <
T ) | ¥áâì §­ ç¥­¨¥ k(T ) ¢ á®áâ®ï­¨¨ T = T1 (¯® ⥮६¥ ® á।­¥¬). ’®£¤ 
¢¬¥áâ® ¢â®à®£® ãá«®¢¨ï (2) ¡ã¤¥¬ ¨¬¥âì1

 F
@F
T0 , 
F (x; t)jt=0 = 0;
(23)
, (t) k(T ) + k = 0 ¯à¨ x = 0:
@x
1
0
’ ª ª ª
@T
@F
=
k(T ) ;
@x
@x
£¤¥ F ®¡®§­ ç¥­¨¥ Š¨à壮ä , â® ¢ í⮬ á¬ëá«¥ § ¯¨á ­­®¥ ªà ¥¢®¥ ãá«®¢¨¥ (23)
®¡à â­® ¤ ¥â (2). ‘®®â­®è¥­¨¥ (22) ¬®¦­® ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥ ®¯à¥¤¥«¥­­®£®
¨­â¥£à « 
Z '( ) 1 R  @ p1
F=
e, 2 0 @x d d;
p(;  )
0

1

k(T ) | 㤮¢«¥â¢®àï¥â ¢á¥¬ ãá«®¢¨ï¬ â¥®à¥¬ë ® á।­¥¬. à¨ x = 0
limt!0 k(T1 ) = k(T0 ).

3{48
£¤¥
çâ®

’. ‡. —®ç¨¥¢

 jt=0 = 0 .

’ ª¨¬ ®¡à §®¬, ¢ë¯®«­¥­® ­ ç «ì­®¥ ãá«®¢¨¥. „ «¥¥, § ¬¥ç ï,

@F
@x

,1
= '( )e 2

R  @p
1

@x d

0

,



'( ) , 12 R0 @@xp d
e
p
1



t=0

x = 0 ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥

ªà ¥¢®¥ ãá«®¢¨¥ (23) ¬®¦­® ¯à¨

;

3
2

Z
R
R
@p
 @p

'( ) ,
@x d ,  4 1
@x d d + T0 ,  5 = '(0):
'( )e,
e
k (T )
p
k
1

1
2

1

1
2

0

1

0

0

0

‚¢¥¤ï ®¡®§­ ç¥­¨¥

Z '( )
p

0

1
e, 2

R  @p
1

0

@x d d =

¨§ ¤¨ää¥à¥­æ¨ «ì­®£® ãà ¢­¥­¨ï

Q

(¯à¨



@Q

 T0 ,  '(0)
,
Q=
+
@ pk(T1 )
p k0

­ å®¤¨¬

Q:

k0

0

£¤¥ ¢ ᨫ㠮¡®§­ ç¥­¨© ¯®áâ®ï­­ ï
¢ãî ç áâì (23) ¢¬¥áâ®

1
'( )e, 2

R  @ p d

¯à¨

@ , k(T1 1 )
e
 @
1

(23)1

x=0

3
p d d 5 ;

R 
0

Q0 ¯®«ãç ¥âáï à ¢­®© 0.

(24)

®¤áâ ¢¨¬ ¢ ¯à -

Q §­ ç¥­¨¥ ¨§ (24) ¨ ¯à®¤¨ää¥à¥­æ¨à㥬 ®¡¥ ç á⨠¯®

1

0

@x

=

¨«¨



R 
1
T0 , 
k
(
T
)
Q=,
k(T1 ) + e 1 0 p d
k0

2
Z
T
,

 4Q0 + 0 k(T1 ) , '(0)k(T1)

:

x = 0);

1

e k(T1 )

"

2
Z
R   d T , 

p 4 0
, ' (0)
0

1
'( )e, 2

k0

R @p
1

0

@x d

0

, '(0)

#

@ , k(T11 )
e
 @
1

1
e, k(T1 )

R 
0

p d

R   d
0

p

3
d 5 + '(0)

ˆ§¬¥­¥­¨¥ ­¥«¨­¥©­®£® ⥬¯¥à âãà­®£® ¯®«ï


T0 , 
, '(0)
=
k0

Z

3{49



@ , k(T1 0 ) R0 p d
e
d :
 @
1

0

à¥¤¯®«®¦¨¬, çâ®
1
'( )e, 2

R  @p
1

@x d

0

  â ª¦¥

Z

(0) k T
, '(0) = p , p(0)
e

1
( 0)

@ , k(T11 ) R0 p d
e
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3
2

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p
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'(0): '(0) =

1
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1
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C ( ) = p (0;  )
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d
@x
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1
,
2 0
e

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(2 )
,
e
;
p(0; 2 ) p(0; 0)
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R , 2
1
,
2 0
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1

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u
u
=t

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¢®£® ãá«®¢¨ï (23).

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p(;  ), '( ), C ( ), ¨ ®­¨ ¯®«­®áâìî ®¯à¥¤¥«¥­ë ¢ ¯®«ã¯à®áâà ­á⢥, ¢ª«îç ï
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¯®¯ëâ âìáï ® ®¯à¥¤¥«¨âì ⥬¯¥à âãà­ãî äã­ªæ¨î T . „ «¥¥, ¯®áª®«ìªã
' ( ) ­ ©¤¥­  (á¬. (7)), ­ å®¤¨¬ ¨ p0 (t). ‚¥à­¥¬áï ª (), F (x; t) 㦥 ¨§¢¥áâ­ ,
¬¥¦¤ã F (x; t) ¨ T  (x; t) (á¬. (4)) áãé¥áâ¢ã¥â ä㭪樮­ «ì­ ï § ¢¨á¨¬®áâì

T  (x; t) =

F :

( )

‘ ¤à㣮© áâ®à®­ë (á¬. (8))

dF
dT 

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0 ( )

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dT  =

0

dF

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0 ( )
dF:
k (T )

‹¨â¥à âãà 

1.
2.
3.
4.

Š à«á«®ã ƒ., …£¥à „. ’¥¯«®¯à®¢®¤­®áâì ⢥à¤ëå ⥫.| Œ.:  ãª , 1964.
Š®¢ «¥­ª® A. „. Žá­®¢ë â¥à¬®ã¯à㣮áâ¨.|Š¨¥¢:  ãª  ¤ã¬ª , 1970.
‹ëª®¢ €. ‚. ’¥®à¨ï ⥯«®¯à®¢®¤­®áâ¨.|Œ.: ‚ëáè ï 誮« , 1967.
—®ç¨¥¢ ’. ‡. Ž äã­¤ ¬¥­â «ì­®© ä㭪樨 ­¥«¨­¥©­®£® ⥬¯¥à âãà­®£®
¯®«ï // ‚« ¤¨ª ¢ª §áª¨© ¬ â. ¦ãà­.|2000.|’. 2, ‚ë¯. 1.|‘. 32{44.

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‘â âìï ¯®áâ㯨«  13  ¯à¥«ï 2000 £.