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“„Š517.946

ŽŽ„Ž‰…‹ŽŠ€‹œŽ‰Š€…‚Ž‰ ‡€„€—…

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†. ’. Š àá ­®¢ , ”. Œ.  åã襢 

à¨ ®¯à¥¤¥«¥­­ëå ãá«®¢¨ïå £« ¤ª®á⨠¤®ª § ­   ¯à¨®à­ ï ®æ¥­ª  ¤«ï ®¡®¡é¥­­ëå à¥è¥­¨©

­¥«®ª «ì­®©ªà ¥¢®©§ ¤ ç¨¤«ïãà ¢­¥­¨ï€««¥à ¢¯à®áâà ­á⢠呮¡®«¥¢ . áᬠâਢ ¥¬ ï

§ ¤ ç  ।ãæ¨àã¥âáï ª § ¤ ç¥ ƒãàá . ®«ã祭® â ª¦¥ ¨­â¥£à «ì­®¥ ¯à¥¤áâ ¢«¥­¨¥ à¥è¥­¨ï

§ ¤ ç¨ƒãàá .

1. ®áâ ­®¢ª  § ¤ ç¨. ‚ ®¡« áâ¨Q

T

f(x;t) :0<x<l; 0<t<Tg

à áᬮâ-ਬ § ¤ çãá ­¥«®ª «ì­ë¬ãá«®¢¨¥¬

@u

@t =

@

@x

k(x;t) @u

@x

+ @

@x

(x;t) @

2

u

@x@t

,q(x;t)u+f(x;t); (1)

@

@t l

Z

0

u(x;t)dx=B(t); (2)

@u

@x

x=l

=0; (3)

u(x;0) =u

0

(x); (4)

£¤¥ 0 < c

0

6 (x;t) 6 c

1

, jg(x;t)j, jk(x;t)j, j

t

(x;y)j 6 c

2

. ‡ ¤ ­­ë¥ ¢

ãà ¢­¥-­¨¨ (1) ¨ ãá«®¢¨ïå (2){(4) ä㭪樨 㤮¢«¥â¢®àïîâ á«¥¤ãî騬 ãá«®¢¨ï¬ £« ¤ª®áâ¨:

;

x ;

t ;

xt ;k;k

x

;g;f 2C(Q

T

); B(t)2 C[0;T], u

0

(x)2 C 1

[0;l]. “à ¢­¥­¨¥(1)

­ §ë¢ -¥âáï ãà ¢­¥­¨¥¬€««¥à  ¨«¨¬®¤¨ä¨æ¨à®¢ ­­ë¬ ãà ¢­¥­¨¥¬¢« £®¯¥à¥­®á  ¢

¯®ç¢®-£àã­â å.

¥«®ª «ì­®¥ãá«®¢¨¥¢¨¤ (2)¯à¥¤«®¦¥­®¢¯¥à¢ë¥¢à ¡®â å[1,2]. ”¨§¨ç¥áª¨®­®

¢ëà ¦ ¥â à á室, ­ ¯à¨¬¥à,¢« £¨ ¢ ¯®ç¢¥­­®¬ á«®¥ ®â 0 ¤® l. ¥«®ª «ì­ë¥ ãá«®¢¨ï

⨯ (2)¤«ï ãà ¢­¥­¨ï⥯«®¯à®¢®¤­®áâ¨à áᬠâਢ «¨áì â ª¦¥¢ à ¡®â å[3, 4].

2. ’¥®à¥¬  ¥¤¨­á⢥­­®á⨠à¥è¥­¨ï. „®ª ¦¥¬ á¯à ¢¥¤«¨¢®áâì  ¯à¨®à­®©

®æ¥­ª¨à¥è¥­¨ï § ¤ ç¨ (1){(4). „«ïí⮣®ã¬­®¦¨¬ãà ¢­¥­¨¥(1) ­ u(x;t) ¨

¯à®¨­-⥣à¨à㥬 ¯® ¯¥à¥¬¥­­®©x:

l

Z

0 u

t udx=

l

Z

0

(k(x;t)u

x )

x udx+

l

Z

0

((x;t)u

xt )

x udx,

l

Z

0

g(x;t)u 2

dx+ l

Z

0

f(x;t)udx: (5)

c

à¥®¡à §ã¥¬¨­â¥£à «ë,¢å®¤ï騥¢â®¦¤¥á⢮(5)¨­¥«®ª «ì­®¥ãá«®¢¨¥(2)

á«¥-¤ãî騬®¡à §®¬:

k(0;t)u

x

(0;t)+(0;t)u

xt

(0;t)=, l

Z

0

g(x;t)dx+(t); (t)= l

Z

0

f(x;t)dt,B(t); (6)

l

Z

0 u

t udt=

1 2 @ @ kuk 2 L2 ; kuk 2 L2 = l Z 0 u 2

(x;t)dx; (7

1 ) l Z 0 (k(x;t)u x ) x

udx=,k(0;t)u

x

(0;t)u(0;t), l Z 0 k(x;t)u 2 x

(x;t)dx; (7

2 ) l Z 0 ((x;t)u xt ) x

udx=,(0;t)u

xt

(0;t)u(0;t), 1 2 @ @t l Z 0 (x;t)u 2 x dx+ 1 2 l Z 0 t u 2 x dx: (7 3 )

„«ï᪠«ïà­®£®¯à®¨§¢¥¤¥­¨ïä㭪権f(x;t) ¨u(x;t)á¯à ¢¥¤«¨¢  ®æ¥­ª 

(f;u)= l

Z

0

f(x;t)u(x;t)dx6 1 2 kfk 2 L 2 + 1 2 kuk 2 L 2 : (7) ®¤áâ ¢«ïï (7 1 );(7 2 );(7 3

) ¢ ¨­â¥£à «ì­®¥â®¦¤¥á⢮ (5), á ãç¥â®¬ ­¥«®ª «ì­®£®

ãá«®-¢¨ï (6)¨­¥à ¢¥­á⢠ (7), ¯®«ãç ¥¬

1 2 @ @t kuk 2 L2 + , l Z 0

gudx+(t)

u(0;t)+ 1 2 @ @t l Z 0 u 2 x dx 6 1 2 kfk 2 L2 + 1 2 kuk 2 L2 + 3c 2 2 ku x k 2 L2 +c 2 kuk 2 L2 : Žâá ­ ®á­®¢ ­¨¨ ­¥à ¢¥­á⢠ Š®è¨¨¬¥¥¬ 1 2 @ @t kuk 2 L 2 + @ @t l Z 0 u 2 x dx 6 1 2 kfk 2 L 2 + 1 2 kuk 2 L 2 + 3c 2 2 ku x k 2 L 2 +c 2 kuk 2 L 2 +u 2

(0;t)+ 1 2 c 2 2 lkuk 2 L 2 + 2 (t) 2 : (8)

‘¯à ¢¥¤«¨¢  ®æ¥­ª (á¬. [5])

u 2

(0;t)6"ku

x k 2 L2 +c(")kuk 2 L2

; ">0; c(")>0:

’®£¤  ­  ®á­®¢ ­¨¨ í⮩®æ¥­ª¨¨§ (8)­ å®¤¨¬

@ @t kuk 2 L2 + @ @t l Z u 2 x

£¤¥

1

= max(3

c

2

+ 2

";

1 +

c 2

2 l

+ 2(

c

2

+

c

(

"

))). ‡ ¬¥­¨¬ ¢ (9)

t

­ 

¨ ¯à®¨­â¥£à¨à㥬

¯®

¢ ¯à¥¤¥« å ®â 0 ¤®

t

:

ku

(

x;t

)

k 2

L2

+

l

Z

0

(

x;t

)

u

(

x;t

)

dx6 t

Z

0

kf

(

x;

)

k 2 L2 d

+

t Z 0 2

(

)

d

+

1 t Z 0

ku

(

x;

)

k 2

L

2

+

ku

x

(

x;

)

k

2 L 2 d

+

l Z 0

(

x;

0)

u

(

x;

0)

dx

+

ku 0

(

x

)

k 2

L

2

¨«¨, á ãç¥â®¬ ãá«®¢¨ï

(

x;t

)

>c 0 >

0,

kuk 2 L2

+

c 0 ku x k 2 L2 6 t Z 0

kf

(

x;

)

k 2 L 2 d

+

t Z 0 2

(

)

d

+

ku 0

(

x

)

k 2 L 2

+

c 1 ku x

(

x;

0)

k

2 L2

+

1 t Z 0

ku

(

x;

)

k 2

L2

+

ku

x

(

x;

)

k

2 L2 d:

(10)

ãáâì

2

= min(1

;c

0

)

;

3

= max(1

;c

1 ;

1

). ’®£¤  ¨§ (10) á«¥¤ã¥â

2 kuk 2 W 1 2 (0;l) 6 3 t Z 0

ku

(

x;

)

k 2 W 1 2 (0;l) d

+

t Z 0

kf

(

x;

)

k L2 d

+

t Z 0 2

(

)

d

+

ku 0

(

x

)

k 2 W 1 2 (0;l) ;

£¤¥

kuk 2 W 1 2 (0;l)

=

kuk 2 L 2

+

ku x k 2 L

2

.   ®á­®¢ ­¨¨ «¥¬¬ë 1.1 ¨§ [5] ®ª®­ç â¥«ì­® ¯®«ã稬

kuk 2 W 1 2 (0;l)

6M

(

t

)

kfk 2;Q t

+

t Z 0 2

(

)

d

+

ku 0 k 2 W 1 2 (0;l) ;

£¤¥

kfk 2 2;Q t t Z 0 kfk 2 L 2 d

=

t Z 0 l Z 0

jf

(

x;

)

jdx

d:

(11)

ˆ§  ¯à¨®à­®© ®æ¥­ª¨ (11) á«¥¤ã¥â ¥¤¨­á⢥­­®áâì à¥è¥­¨ï § ¤ ç¨ (1){(4) ¨

­¥¯à¥-à뢭 ï § ¢¨á¨¬®áâì à¥è¥­¨ï ®â ¢å®¤­ëå ¤ ­­ëå ­  ª ¦¤®¬ ¢à¥¬¥­­®¬ á«®¥ ¢ ­®à¬¥

¯à®áâà ­á⢠

W

1

2

(0

;l

).

3. ‡ ¤ ç¨ ƒãàá , ¯®áâ஥­¨¥ ä㭪樨 ¨¬ ­ .

‚ ⮩ ¦¥ ®¡« áâ¨

Q T

à á-ᬮâਬ § ¤ çã

L

(

u

)

(

u

xt

)

x

+ (

ku

x

)

x

+

du

t

,qu

=

,f

(

x;t

)

; d<

0

; d t

2C

(

Q T

)

;

(12)

u

(0

;t

) =

'

(

t

)

; u

x

(0

;t

) =

(

t

)

; u

(

x;

0) =

u 0

(

ˆ¬¥¥â ¬¥áâ®á®®â­®è¥­¨¥

vL(u),uM(v)= @Q

@x ,

@P

@t

; (14)

£¤¥

Q=vu

xt

+u(v

x )

t +kvu

x ,kv

x u;

P =v

x u

x

,dvu;

M(v)=,(v

x )

xt +(kv

x )

x

,(dv)

t ,qv:

®âॡ㥬 ­¥¯à¥à뢭®áâì P;Q ¢ Q

T

, ­¥¯à¥à뢭®áâì ¨ ®£à ­¨ç¥­­®áâì P

x ;Q

t ¢

Q

T .

Ž¯à¥¤¥«¨¬  ­ «®£ ä㭪樨 ¨¬ ­  v = v(x;t;;) á«¥¤ãî騬¨ âॡ®¢ ­¨ï¬¨

(á¬. [6]):

M(v)=0;

v(;t;;)=0; v

x

(;t;;) = 1

exp

t

Z

k(;t

1 )

(;t

1 )

dt

1

;

v(x;;;)=w(x;);

(15)

£¤¥ w(x;) |à¥è¥­¨¥ § ¤ ç¨ Š®è¨

(v

x )

x

+dv =0; >c

0

>0; d <0;

v(x;;;)

x=

=0; v

x

(x;;;)

x= =

1

2 :

(16)

à®¨­â¥£à¨à㥬 ᮮ⭮襭¨¥ (14) ¯® ®¡« á⨠f(x;t) : 0 <x<;0 <t <g,

£¤¥ (;) | ¯à®¨§¢®«ì­ ï â®çª  ®¡« áâ¨Q

T .

’®£¤ , á ãç¥â®¬ £à ­¨ç­ëå ãá«®¢¨© (13) ¨ ®¯à¥¤¥«¥­¨ï ä㭪樨 v =v(x;t;;),

¯®«ã稬¯à¥¤áâ ¢«¥­¨¥

u(;)=(0;)v

x

(0;;;)'(),

Z

0 h

(0;t)v(0;t;;) 0

(t)

+ ,

(v

x

(0;t;;))

t ,kv

x

(0;t;;)

'(t)+kv(0;t;;) (t) i

dt

+

Z

0 h

(x;0)v

x

(x;0;;)u 0

0

(x),d(x;0)v(x;0;;)u

0 (x)

i

dx

+

Z

0

Z

0

v(x;t;;)f(x;t)dxdt:

(17)

’ ª¨¬ ®¡à §®¬, à¥è¥­¨¥ § ¤ ç¨ ƒãàá  (12){(13) ¯à¥¤áâ ¢¨¬® ¢ ®¬ ¢¨¤¥ á

¥¤¨­á⢥­­®áâì  ­ «®£  ä㭪樨 ¨¬ ­ , ®¯à¥¤¥«ï¥¬®£® ãá«®¢¨ï¬¨ (15){(16),

¤®ª -§ ­ë¢à ¡®â¥[6]. ”®à¬ã« (17)¯®§¢®«ï¥â ¨áá«¥¤®¢ âìà §«¨ç­ë¥ªà ¥¢ë¥«®ª «ì­ë¥

¨ ­¥«®ª «ì­ë¥ § ¤ ç¨¤«ï ¯á¥¢¤®¯ à ¡®«¨ç¥áª¨åãà ¢­¥­¨© ¢¨¤ (12).

à¥¤áâ ¢«¥­¨¥(17) § ¯¨è¥¬­¥áª®«ìª® ¨­ ç¥:

u(;)=(0;)v

x

(0;;;)u(0;),(0;)v(0;;;)u

x (0;)

+

Z

0

h

,

(0;t)v(0;t;;)

t

,k(0;t)v(0;t;;) i

u

x (0;t)

+ h

kv

x

(0;t;;), ,

v

x

(0;t;;)

t i

u(0;t)

dt

+

Z

0 h

(x;0)v

x

(x;0;;)u 0

0

(x),d(x;0)v(x;0;;)u

0 (x)

i

dx

+

Z

0

Z

0

v(x;t;;)f(x;t)dxdt,(0;0)v(0;0;;)u 0

0 (0):

(18)

®«ì§ãïáì ¯à¥¤áâ ¢«¥­¨¥¬ (18), áãç¥â®¬ ¤®¯®«­¨â¥«ì­ëåãá«®¢¨©(2){(4), ­ å®¤¨¬

v

x

(0;;l;)u(0;),v(0;;l;)u

x (0;)

+

Z

0 ,

H

1

(;t)u(0;t)+H

2 (;t)u

x (0;t)

dt=

1

(); (19)

l

Z

0 v

x

(0;;;)du(0;), l

Z

0

v(0;;;)du

x (0;)

+

Z

0 ,

K

1

(;t)u(0;t)+K

2 (;t)u

x (0;t)

dt=

2 ();

£¤¥

H

1

(;t)= 1

(0;) h

k(0;t)v

x

(0;t;l;), ,

(0;t)v

x

(0;t;l;)

t i

;

H

2

(;t)= 1

(0;) h

,

(0;t)v

(0;t;l;)

t

,k(0;t)v

(0;t;l;) i

;

1

()=, 1

(0;)

(l;0)v

x

(l;0;l;)u 0

0 (l)

+ l

Z

0 h

(x;0)v

x

(x;0;l;)u 0

0

(x),d(x;0)v

(x;0;l;) i

dx

+ l

Z

Z

v

(x;t;;)f(x;t)ddt,(0;0)v

(0;0;l;)u 0

K

1

(;t)= 1

(0;) Z

0 ,

k(0;t)v

x

(0;t;;),(v

x )

t

d;

K

2

(;t)= 1

(0;) l

Z

0 h

,

(0;t)v(0;t;;)

t

,k(0;t)v i

d;

2

()=, 1

(0;) l

Z

0

Z

0

Z

0

v(x;t;;)f(x;t)dxdt

+

Z

0 ,

(x;0)v

x

(x;0;;)u 0

0

(x)+d(x;0)v(x;0;;)u

0 (x)

dx

,(0;0)v(0;0;;)u 0

0 (0)

d+ l

Z

0 u

0

(x)dx+

Z

0

B(t)dt:

‘¨á⥬㠨­â¥£à «ì­ëåãà ¢­¥­¨©(19) ¯¥à¥¯¨è¥¬¢ ®¯¥à â®à­®¬¢¨¤¥

A~u+B~u=~;

£¤¥

A= 0

B

B

@ v

x

(0;;l;) ,v(0;;l;)

l

Z

0 v

x

(0;;;)d , l

Z

0

v(0;;;)d 1

C

C

A :

  ®á­®¢ ­¨¨ «¥¬¬ë,¤®ª § ­­®© ¢ à ¡®â¥ [6]¨¬¥¥¬ detA(t)6=0, 06t6T. ®í⮬ã

á¨á⥬ ãà ¢­¥­¨©(19)ï¥âáïá¨á⥬®©¨­â¥£à «ì­ëåãà ¢­¥­¨©‚®«ìâ¥àà ¢â®à®£®

த ,ª®â®à ï¡¥§ãá«®¢­® à §à¥è¨¬ .

’ ª¨¬®¡à §®¬,§ ¤ ç (1){(4)।ãæ¨à®¢ ­ ª§ ¤ ç¥ƒãàá (12){(13)®¤­®§­ ç­ ï

à §à¥è¨¬®áâ쪮â®à®©ã¦¥ ãáâ ­®¢«¥­ .

®«ã祭­ë¥ १ã«ìâ âë ¨¬¥îâ ¬¥áâ® ¨ ¢ á«ãç ¥, ª®£¤  ­  ¯à ¢®¬ ª®­æ¥

®â-१ª  [0;l] § ¤ ­® ãá«®¢¨¥ âà¥â쥣® த  ,(l;t) = (t)u(l;t) +(t), £¤¥ (l;t) =

ku

x

(l;t) + (l;t)u

xt

(l;t), ¯à¨ í⮬ §­ ª¨ ª®íää¨æ¨¥­â®¢ k(x;t);q(x;t);(t) ­¥

¨£à -îâ஫¨ ¤«ïª®à४⭮©¯®áâ ­®¢ª¨¨á室­®©§ ¤ ç¨,áãé¥á⢥­­ë¬ ï¥âáï⮫쪮

ãá«®¢¨¥(x;t)>c

0 >0.

‹¨â¥à âãà 

1. Š ¬ë¬¨­‹.€. Ž¡®¤­®©ªà ¥¢®©§ ¤ ç¥â¥®à¨¨â¥¯«®¯à®¢®¤­®áâ¨á­¥ª« áá¨ç¥áª¨¬¨

£à ­¨ç-­ë¬¨ãá«®¢¨ï¬¨//†‚Œ¨Œ”.|1964.|’.4,ü6.|‘.1006{1024.

2. —㤭®¢áª¨©€.”.¥ª®â®à륪®à४⨢뢯®áâ ­®¢ª¥¨à¥è¥­¨¨§ ¤ ç⥯«®-¨¢« £®¯¥à¥­®á 

¢¯®ç¢¥//‘¡. âà㤮¢€”ˆ.|1969,¢ë¯.23.|‘.41{54.

3. ˆ®­ª¨­.ˆ.¥è¥­¨¥®¤­®©ªà ¥¢®©§ ¤ ç¨¢â¥®à¨¨â¥¯«®¯à®¢®¤­®áâ¨á­¥«®ª «ì­ë¬¨

4.  åã襢€.Œ. £à㦥­­ë¥ãà ¢­¥­¨ï//„“.|1983.|’.19,ü1.|C.86{94.

5. ‹ ¤ë¦¥­áª ïŽ.€. Šà ¥¢ë¥§ ¤ ç¨¬ â¥¬ â¨ç¥áª®©ä¨§¨ª¨.|Œ.:  ãª ,1973.|407á.

6. ˜å ­ãª®¢Œ.•. Ž­¥ª®â®àëåªà ¥¢ë姠¤ ç å¤«ïãà ¢­¥­¨ïâà¥â쥣®¯®à浪 ,¢®§­¨ª îé¨å

¯à¨¬®¤¥«¨à®¢ ­¨¨ä¨«ìâà æ¨¨¦¨¤ª®á⨢¯®à¨áâëåá। å//„“.|1982.|’.18,ü4.|‘.689{

699.