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“„Š517.958

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ƒ. ‹. ‹ãª ­ª¨­, €. ‚. ‹ âë襢, ‘. ‚. ë­¤¨­ 

„®ª § ­ â¥®à¥¬ ®à §«®¦¥­¨¨à¥è¥­¨ï®¡é¥©£à ­¨ç­®©§ ¤ ç¨¤«ïŠ‚-ãà ¢­¥­¨ï¯®

ᮡáâ-¢¥­­ë¬¢¥ªâ®à ¬­¥¯à¥à뢭®£®¨¤¨áªà¥â­®£®á¯¥ªâ஢. “á«®¢¨ïà §à¥è¨¬®á⨯®§¢®«ïîâ

®¤-­®§­ ç­®®¯à¥¤¥«¨âì­¥¨§¢¥áâ­ë¥ª®íää¨æ¨¥­âëà §«®¦¥­¨ï¨á¢®¡®¤­ë¥¯ à ¬¥âàëà¥è¥­¨ï

¢¥ªâ®à­®©ªà ¥¢®©§ ¤ ç¨¨¬ ­ |ƒ¨«ì¡¥àâ .

Œ®¤¥«ì­®¥ ãà ¢­¥­¨¥ ®«ì欠­  á ¨­â¥£à «®¬ á⮫ª­®¢¥­¨© ¢ ä®à¬¥ Š‚

(®«ì欠­, Šàãª,‚¥« ­¤¥à)¨á ç áâ®â®©,¯à®¨§¢®«ì­ë¬®¡à §®¬§ ¢¨áï饩®â

¬®«¥-ªã«ïà­®©áª®à®á⨠¡ë«®¯®áâ஥­®¢ [1]. ‚­ áâ®ï饩ࠡ®â¥ ¯à®¤®«¦ îâáï

¨áá«¥¤®-¢ ­¨ï ­ ç â륢[2], [3]¨ [4].

1. ®áâ ­®¢ª  § ¤ ç¨. ‹¨­¥ à¨§®¢ ­­®¥ Š‚-ãà ¢­¥­¨¥, ®¤­®¬¥à­®¥ ¨

áâ æ¨-®­ à­®¥ ¨¬¥¥â¢¨¤

x @h

@x =

1

Z

,1 d

0 1

Z

0

exp(, 02

)k(;; 0

; 0

)h(x; 0

; 0

)d 0

,h(x; 0

; 0

); (1)

£¤¥ k(;; 0

; 0

) = 1+ 3

2

0

0

+ 1

2 (

2

,2)( 0 2

,2) | ï¤à® ¨­â¥£à «ì­®£® ®¯¥à â®à .

”ã­ªæ¨ïh 㤮¢«¥â¢®àï¥â á«¥¤ãî騬£à ­¨ç­ë¬ãá«®¢¨ï¬

h(0;;)=h

0

(;); 0<<1; (2)

h(x;;)=h

as

(x;;)+o(1); x!+1; ,1<<0: (3)

‡¤¥áìh

0

(;)2H(0;1)¯®¯¥à¥¬¥­­®©(H(0;1) |ª« ááä㭪権,㤮¢«¥â¢®àïîé¨å

ãá«®¢¨î ƒñ«ì¤¥à  ­  ®â१ª¥ [0,1]);  á¨¬¯â®â¨ç¥áª ï ç áâì ä㭪樨 h ¯à¨ x ! 1

ï¥âáï «¨­¥©­®©ª®¬¡¨­ æ¨¥©¤¨áªà¥â­ëåà¥è¥­¨© ãà ¢­¥­¨ï(1).

”ã­ªæ¨îhà §« £ ¥¬¯®â६®à⮣®­ «ì­ë¬­ ¯à ¢«¥­¨ï¬(1;; 2

,2).

ƒà ­¨ç-­ ï § ¤ ç  (1){(3) à §¡¨¢ ¥âáï ­  âà¨áª «ïà­ë¥, ª®â®à륬®¦­® ®¡ê¥¤¨­¨âì¢ ®¤­ã

¢¥ªâ®à­ãî. à¨¬¥­¥­¨¥ § ª®­®¢ á®åà ­¥­¨ï (ç¨á«  ç áâ¨æ, x ª®¬¯®­¥­âë ¨¬¯ã«ìá ,

í­¥à£¨¨)¯®§¢®«ï¥â ã¯à®áâ¨âì ï¤à®¢¥ªâ®à­®© £à ­¨ç­®© § ¤ ç¨. ®«ãç ¥¬

@h(x;)

@x

+h(x;)= 1

2 1

Z

,1

K(; 0

)h(x; 0

)d 0

; K(; 0

)= 0

@

1 4 0

0 3c

0

0

0 0 1

1

A

; (4)

£¤¥x2(0;+1); 2(,1;1); 2R ; c=1,9 2

. ƒà ­¨ç­ë¥ãá«®¢¨ï(2){(3)¯à¨­¨¬ îâ

¢¨¤

h(x;)=h

0

()=(h

01 ();h

02 ();h

03 ())

T

h(x;)=h

as

(x;)+o(1); x!1; ,1<<0; (6)

£¤¥T|®§­ ç ¥ââ࠭ᯮ­¨à®¢ ­¨¥,h

0i

()2H(0;1)¯à¨i=1;2;3, äã­ªæ¨ïh

as (x;)

®¯à¥¤¥«ï¥âáï­¨¦¥,ª ª«¨­¥©­ ïª®¬¡¨­ æ¨ïᮡá⢥­­ëåà¥è¥­¨©¤¨áªà¥â­®£®

ᯥª-âà  ãà ¢­¥­¨ï(4). ¥è¥­¨¥ £à ­¨ç­®© § ¤ ç¨ (4){(6) ¡ã¤¥¬ ¨áª âì ¢ ª« áá¥

¢¥ªâ®à-ä㭪権 h(x;) = (h

1

(x;);h

2

(x;);h

3 (x;))

T

â ª¨å, çâ® h

i

(x;) (i = 1;2;3)

­¥¯à¥-àë¢­ë ¯® x ­  ¯®«ã®á¨ x 2 [0;+1] ¯à¨ ¢á¥å 2 (,1;1), ¨¬¥îâ ª®­¥ç­ë¥ ¯à¥¤¥«ë

¯à¨ ! 1 ¨ ! 0 ¤«ï «î¡®£® x 2 (0;+1), ­¥¯à¥à뢭® ¤¨ää¥à¥­æ¨àã¥¬ë ¯® x

­  ¯®«ã®á¨ x2(0;+1) ¯à¨ ¢á¥å 2(,1;1) ¨ ¨­â¥£à¨àã¥¬ë ¯® ­  (,1;1) ¯à¨¢á¥å

x2(0;+1).

2. Šà ¥¢ ï § ¤ ç  ¨¬ ­  | ƒ¨«ì¡¥àâ . ‚ ãà ¢­¥­¨¨ (4) ¯¥à¥¬¥­­ë¥ x ¨

|­¥§ ¢¨á¨¬ë¥. à¨¬¥­ïﬥ⮤ࠧ¤¥«¥­¨ï¯¥à¥¬¥­­ëå”ãàì¥,¯®«ã稬¢¥ªâ®à­®¥

å à ªâ¥à¨áâ¨ç¥áª®¥ ãà ¢­¥­¨¥

(,)(;)= 1

2

1

Z

,1

K(; 0

)(;)d 0

; (7)

¢ ª®â®à®¬ ¯à®¨§¢®¤­ë¥ ¯® ¯à®áâà ­á⢥­­®© ¯¥à¥¬¥­­®© ®âáãâáâ¢ãîâ. ” ªâ¨ç¥áª¨

§ ¤ ç  (7) | § ¤ ç  ­  ᮡá⢥­­ë¥ §­ ç¥­¨ï¤«ï ¨­â¥£à «ì­®£® ®¯¥à â®à . ‘¯¥ªâà

¤ ­­®£® ®¯¥à â®à  ¢ª«î砥⠢ á¥¡ï ­¥¯à¥à뢭ãî ç áâì, á®áâ®ïéãî ¨§ ª®­â¨­ã㬠

§­ ç¥­¨© ᯥªâà «ì­®£® ¯ à ¬¥âà  2 [,1;1], ¨ ¤¨áªà¥â­ãî, á®áâ®ïéãî ¨§ ­ã«¥©

¤¨á¯¥àᨮ­­®©ä㭪樨

c

(z)=det

c

(z)= 2

0 (z)!

c (z);

£¤¥

t(z)= 1

Z

,1 d

,z ;

0

(z)=1+ 1

2

zt(z); !

c

(z)=1+3cz 2

0 (z);

 

c

(z) | ¤¨á¯¥àᨮ­­ ï¬ âà¨æ ,¨¬¥îé ï¢¨¤

c (z)=

0

@

0

(z) 2zt(z) 0

0 !

c

(z) 0

0 0

0 (z)

1

A

:

‘®¡á⢥­­ë¥ ¢¥ªâ®àë, ®â¢¥ç î騥 ­¥¯à¥à뢭®¬ã ᯥªâàã, ïîâáï

ᨭ£ã«ïà-­ë¬¨¨ ¯à¨­ ¤«¥¦ â¯à®áâà ­áâ¢ã®¡®¡é¥­­ëå ä㭪権[5]

(;)= 1

2

K(;)P 1

,

n()+

c

()n()( ,); ;2(,1;1); (8)

n()= 1

Z

(; 0

)d 0

‘®¡á⢥­­ë¥ §­ ç¥­¨ï¤¨áªà¥â­®£®á¯¥ªâà  ï¢«ïîâáïà¥è¥­¨ï¬¨ ᮢ®ªã¯­®áâ¨

"

2

0

(z) =0;

!(z)=0:

à¨ «î¡®¬c 2(,1;1] ¨¬¥¥¬ç¥âëॠᮡá⢥­­ëå à¥è¥­¨ï ãà ¢­¥­¨ï(7),

®â¢¥-ç îé¨å¤¨áªà¥â­®¬ãᯥªâàã:

h (1)

(x;)=(1;0;0) T

;

h (2)

(x;)=(0;0;1) T

;

h (i)

(x;)=(x,)h (i,2)

(x;); i=3;4:

‚á«ãç ¥ 0 <c 61 ª ­¨¬ ¤®¡ ¢«ï¥âáï ¥é¥¤¢  à¥è¥­¨ï, ®â¢¥ç îé¨å §­ ç¥­¨ï¬

¤¨áªà¥â­®£®á¯¥ªâà 

0

h

0

(x;)=exp

,x

0

(

0

;); 0<c<1;

£¤¥

(

0 ;)=

1

2 (

0 )

K(;

0 )

0 ,

n(

0

); n(

0 )=

0

@ 2

0 t(

0 )

,

c (

0 )

0 1

A

;

¨ ¯à¨c=1 (

0 =1):

h (5)

(x;)=(0;;0) T

;

h (6)

(x;)=(x,)h (5)

(x;):

ˆá¯®«ì§ãï ¨¤¥î Š. Š¥©§ , ª®â®àë© ¯à¥¤«®¦¨« ¨áª âì ®¡é¥¥ à¥è¥­¨¥

᪠«ïà-­ëåãà ¢­¥­¨©¯¥à¥­®á ¢¢¨¤¥á¨­£ã«ïà­®£®¨­â¥£à «ì­®£®®¯¥à â®à â¨¯ Š®è¨[6],

­ ©¤¥¬à¥è¥­¨¥§ ¤ ç¨ (4){(6)¢¢¨¤¥¨­â¥£à «ì­®£®à §«®¦¥­¨ï¯®á¨á⥬¥

ᮡá⢥­-­ëå ä㭪権. „®ª § â¥«ìáâ¢ã áãé¥á⢮¢ ­¨ï ¨ ¥¤¨­á⢥­­®á⨠⠪®£® à §«®¦¥­¨ï

¯à¥¤¯®è«¥¬à¥è¥­¨¥á®®â¢¥âáâ¢ãî饩 ãà ¢­¥­¨î(7) ¢á¯®¬®£ â¥«ì­®© ªà ¥¢®©

§ ¤ -ç¨ ¨¬ ­ |ƒ¨«ì¡¥àâ 

X +

()=G()X ,

(); 2(0;1); (9)

£¤¥X(z)| ­¥¨§¢¥á⭠ﬠâà¨æ -äã­ªæ¨ï, ­ «¨â¨ç¥áª ï¢ª®¬¯«¥ªá­®©¯«®áª®áâ¨á

ࠧ१®¬ [0;1], X

() | £à ­¨ç­ë¥ §­ ç¥­¨ï ᮮ⢥âá⢥­­® ᢥàåã ¨ á­¨§ã ­ 

à §-१¥ (0;1), G() = [ +

c ()]

,1

,

c

() | ¬ âà¨ç­ë© ª®íää¨æ¨¥­â § ¤ ç¨. ƒà ­¨ç­ë¥

§­ ç¥­¨ï ¤¨á¯¥àᨮ­­®© ¬ âà¨æë ­  ࠧ१¥ (,1;1) ­ å®¤ïâáï ᮣ« á­® ä®à¬ã« ¬

‘®å®æª®£® [7].

¥è¥­¨¥ § ¤ ç¨ (9)| ä ªâ®à-¬ âà¨æ 

X(z)= 0

@

U(z) 4(U(z),V(z)) 0

0 V(z) 0

0 0 U(z)

1

A

;

í«¥¬¥­â몮â®à®©®¯à¥¤¥«¥­ëä®à¬ã« ¬¨U(z)=zexp(,u(z))¨V(z)=zexp(,v(z)),

£¤¥

u(z)= 1

1

Z

0

(),

,z

d; v(z) = 1

1

Z

0

"(),

,z d;

() = arg +

0

() | £« ¢­®¥ §­ ç¥­¨¥  à£ã¬¥­â  ä㭪樨 +

0

(), "() = arg! +

x () |

£« ¢­®¥§­ ç¥­¨¥  à£ã¬¥­â ä㭪樨! +

3. ¥è¥­¨¥ £à ­¨ç­®© § ¤ ç¨.

„«ï §­ ç¥­¨© ¯ à ¬¥âà 

c 2

(

,1;

0) ¨

c 2

(0

;

1) ®¯à¥¤¥«¨¬

h as

(

x;

) ª ª ª®¬¡¨­ æ¨î ¯¥à¢ëå ç¥âëà¥å ᮡá⢥­­ëå à¥è¥­¨©,

®â¢¥-ç îé¨å ¤¨áªà¥â­®¬ã ᯥªâàã

h

as

(

x;

) = (

q 1

+

q

2

(

x,

)

;

0

;q 3

+

q

4

(

x,

))

T

;

£¤¥

q 2

¨

q

4

| § ¤ ­­ë¥ ¯®áâ®ï­­ë¥,

q

1

¨

q

3

| ­¥¨§¢¥áâ­ë¥ ª®­áâ ­âë.

„«ï

c

= 1 à¥è¥­¨¥

h as

(

x;

) ®¯à¥¤¥«ï¥âáï ª ª ª®¬¡¨­ æ¨ï ¢á¥å è¥á⨠¤¨áªà¥â­ëå

à¥è¥­¨©, ᮮ⢥âáâ¢ãîé¨å í⮬ã á«ãç î

h

as

(

x;

) =

,

j

1

+

j

2

(

x,

)

;j 3

+

j 4

(

x,

)

;j 5

+

j

6

(

x,

)

T

;

£¤¥

j 2

;j

4

¨

j

6

| § ¤ ­­ë¥ ¯®áâ®ï­­ë¥,

j

1 ;j

3

¨

j

5

| ­¥¨§¢¥áâ­ë¥ ª®­áâ ­âë.

’¥®à¥¬  1. ƒà ­¨ç­ ï § ¤ ç 

(4)

{

(6)

¯à¨

0

< c <

1

¨¬¥¥â ¥¤¨­á⢥­­®¥ à¥è¥-­¨¥, ¯à¥¤áâ ¢¨¬®¥ ¢ ¢¨¤¥à §«®¦¥­¨ï ¯® ᮡá⢥­­ë¬ ¢¥ªâ®à ¬ ¤¨áªà¥â­®£® ᯥªâà 

¨ ᮡá⢥­­ë¬ ¬ âà¨æ ¬­¥¯à¥à뢭®£®á¯¥ªâà :

h

(

x;

) =

h as

(

x;

) +

A 0

h

0

(

x;

) +

1

Z

0

exp(

,x=

)(

;

)

A

(

)

d;

(10)

£¤¥ q

2 ¨q

4

| ¯®áâ®ï­­ë¥, á®¡á⢥­­ë¥ ¢¥ªâ®àë

(

;

)

­ å®¤ïâáﯮ ä®à¬ã«¥

(8)

. ‚ à §«®¦¥­¨¨

(10)

­¥¨§¢¥áâ­ë¬¨ ïîâáï ª®íää¨æ¨¥­âë A

0 , q

1 ;q

3

|

®â¢¥ç -î騥¤¨áªà¥â­®¬ã ᯥªâà㨠¢¥ªâ®à-äã­ªæ¨ï A

(

)

, ïîé ïá类íää¨æ¨¥­â®¬ ­¥-¯à¥à뢭®£® ᯥªâà .

‚ à §«®¦¥­¨¨ à¥è¥­¨ï £à ­¨ç­®© § ¤ ç¨ ¨á¯®«ì§ã¥âáï ­¥¢®§à áâ î饥

ᮡá⢥­-­®¥ à¥è¥­¨¥, ᮮ⢥âáâ¢ãî饥 §­ ç¥­¨î

0

, ¯®í⮬ã à §«®¦¥­¨¥ (10)  ¢â®¬ â¨ç¥áª¨

㤮¢«¥â¢®àï¥â £à ­¨ç­®¬ã ãá«®¢¨î (6).

C

ˆá¯®«ì§ãï £à ­¨ç­®¥ ãá«®¢¨¥ (5), ¯¥à¥©¤¥¬ ®â à §«®¦¥­¨ï (10) ª ᨭ£ã«ïà­®¬ã

¨­â¥£à «ì­®¬ã ãà ¢­¥­¨î á ï¤à®¬ Š®è¨:

h

as

(0

;

) +

A 0

h

0

(0

;

) +

c

(

)

A

(

) + 12

1

Z

0

K

(

;

)

A

(

)

d

,

=

h 0

(

)

:

(11)

‚¢¥¤¥¬ ¢á¯®¬®£ â¥«ì­ãî ¢¥ªâ®à-äã­ªæ¨î

N

(

z

) = 12

1

Z

0

K

(

z;

)

A

(

)

d

,z

;

(12)

 ­ «¨â¨ç¥áªãî ¢áî¤ã ¢ ª®¬¯«¥ªá­®© ¯«®áª®á⨠á ࠧ१®¬ [0

;

1]. …¥ £à ­¨ç­ë¥

§­ ç¥-­¨ï ᢥàåã ¨ á­¨§ã ­  ࠧ१¥ (0

;

1)

N

(

) = lim

y!0

á¢ï§ ­ë ¬¥¦¤ãᮡ®© ä®à¬ã« ¬¨‘®å®æª®£® [7].

“¬­®¦¨¬®¡¥ ç áâ¨ãà ¢­¥­¨ï(11) ­ iK( 2 ). ®«ã稬 [ + (), , ()](h as

(0;)+A

0 h

0

(0;))+ 1 2 [ + (), , ()][N +

()+N , ()] +K( 2 )()K ,1 ( 2 )iK( 2

)A()=iK( 2 )h 0 (): ‡ ¬¥­¨¬ à §­®áâì + () , ,

() ­  à ¢­®¥ ¢ëà ¦¥­¨¥ K( 2 ) + ()K ,1 ( 2 ) , K( 2 ) , ()K ,1 ( 2 ) ¨ 㬭®¦¨¬ ®¡¥ ç á⨠¯®«ã祭­®£® à ¢¥­á⢠ á«¥¢  ­  ¬ âà¨æã-äã­ªæ¨îK ,1 ( 2 ). ‚१ã«ìâ â¥¯®«ã稬¢¥ªâ®à­ãîªà ¥¢ã¤ ç㐨¬ ­ | ƒ¨«ì-¡¥àâ  P + ()(N +

()+h

as

(0;)+A

0 h

0

(0;)),P ,

()(N ,

()+h

as (0;) +A 0 h 0

(0;))=iK( 2

)h

0

(); 2(0;1); (13)

£¤¥ P(z)=(z)K ,1

(z 2

).

®«ì§ãïáì १ã«ìâ â ¬¨ ¯à¥¤ë¤ã饣® ¯ã­ªâ  ¯à¥®¡à §ã¥¬ § ¤ çã (13) ¢

¢¥ªâ®à-­ã¤ çã ¯® ᪠çªã [X + ()] ,1 (N +

()+h

as

(0;)+A

0 h 0 (0;)) ,[X , ()] ,1 (N ,

()+h

as

(0;)+A

0 h 0 (0;)) =i[P + ()X + ()] ,1 h 0

(); 2(0;1): (14)

‚¢¥¤¥¬®¡®§­ ç¥­¨¥B()=[P + ()X + ()] ,1 . “ç¨â뢠ﯮ¢¥¤¥­¨¥¢å®¤ïé¨å¢ªà ¥¢®¥

ãá«®¢¨¥(14) ä㭪権¨ ¢®á¯®«ì§®¢ ¢è¨áì ®¡®¡é¥­­®©â¥®à¥¬®©‹¨ã¢¨««ï,¯®«ã稬

N(z)=,h

as

(0;)+ 1 2 A 0 0 K(z; 0 ) z, 0 n( 0

)+X(z) (z)+ C z, 0 +D : (15) ‡¤¥áì (z)= 0 B @ 1 (z) 2 (z) 3 (z) 1 C A = 1 2 1 Z 0 B()h 0 () d ,z

| ¨­â¥£à «â¨¯ Š®è¨;C ;D| ¢¥ªâ®àë,ª®¬¯®­¥­â몮â®àëå c

i ;d

i

|¯à®¨§¢®«ì­ë¥

ª®­áâ ­âë (i=1;2;3).

„«ïª®à४⭮á⨯®«ã祭­®£® à¥è¥­¨ï(15)­¥®¡å®¤¨¬®, ç⮡ëà ¢¥­á⢠ (12)¨

(15)®¯à¥¤¥«ï«¨®¤­ã¨â㦥äã­ªæ¨î. â®¬®¦­®á¤¥« â짠 áç¥â¢ë¡®à á¢®¡®¤­ëå

¯ à ¬¥â஢|ª®¬¯®­¥­â®¢¢¥ªâ®à®¢C ¨D, â ª¦¥§ áç¥â®¯à¥¤¥«¥­¨ï­¥¨§¢¥áâ­ëå

ª®íää¨æ¨¥­â®¢ A 0 ;q 1 ¨ q 3 á«¥¤ãî騬 ®¡à §®¬. ‚¥ªâ®à­®¥ ãá«®¢¨¥, ãáâà ­ïî饥 ã

à¥è¥­¨ï (15) ¯®«îá ¢ â®çª¥

0

, ®¤­®§­ ç­® ®¯à¥¤¥«ï¥âí«¥¬¥­âë ¢¥ªâ®à  C, 

ãáâà -­ïî饥 ¯®«îá ¢ ¡¥áª®­¥ç­® 㤠«¥­­®© â®çª¥ | í«¥¬¥­âë ¢¥ªâ®à  D. ¥¨§¢¥áâ­ë¥

ª®íää¨æ¨¥­âëA 0 ;q 1 ¨ q 3 ®¤­®§­ ç­® ®¯à¥¤¥«ïîâáï ¨§ à ¢¥­á⢠ ¯à¥¤¥«®¢ ä㭪権

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ƒŠ-ãà ¢­¥­¨ï ¢ § ¤ ç¥ ® ⥬¯¥à âãà­®¬ ᪠窥 // à¨ª«. ¬ â¥¬ â¨ª  ¨ ¬¥å ­¨ª .|1990|’. 54,

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3.

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€­ «¨â¨ç¥áª®¥ à¥è¥­¨¥ ¬®¤¥«ì­®£® ƒŠ-ãà ¢­¥­¨ï ®«ì欠­ 

¢ § ¤ ç¥ ® ⥬¯¥à âãà­®¬ ᪠窥 á ãç¥â®¬  ªª®¬®¤ æ¨¨ í­¥à£¨¨ // Œ â¥¬. ¬®¤¥«¨à®¢ ­¨¥.|

1992.|’. 4, ¢ë¯. 10, ‘. 41{46.

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Boundary value problems for a model Boltzmann equation with

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¨ ­¥ª®â®àë¥ ¨å ¯à¨«®¦¥­¨ï ª ¬ â¥¬ â¨ç¥áª®© 䨧¨ª¥.| Œ.:  ãª , 1968.|512 á.