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Задача с динамическими граничными условиями для гиперболического уравнения

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Код работы:	K012359
Тема:	Задача с динамическими граничными условиями для гиперболического уравнения
Содержание
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01.05.01 ???????????????



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?????? ? ????????????? ?????????? ????????? ??? ???????????????? ?????????

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????????? ? ??????



???. ????????



?.?.-?.?., ?????????



???????? ?. ?.







(???????)






???????? ??????? ????? 5 ?????? 27501.110 ??????? ??????? ?????????????







(???????)

??????? ???????????? ?.?.-?.?., ????????? ??????? ????????? ?????????????? ?????? ???????? ?. ?.







(???????)

?????? ????????



		?	?	2017 ?.





??????





???????????? ???



?.?.-?.?., ?????????



?????? ?. ?.





		?	?	2017 ?.



(???????)





?????? 2017



??????????



????????

3





	1	?????? ? ????????????? ?????????? ????-????? ??? ???????????????? ????????? 6

1.1

?????????? ?????? . . . . . . . . . . . . . . . . . . . .

6

1.2

?????????????? ??????? . . . . . . . . . . . . . . .

7

1.3

????????????? ???????  . . . . . . . . . . . . . . .

14

2  ?????? ? ???????????? ?????????? ????-



????? ??? ???????????????? ?????????

21

2.1

?????????? ?????? . . . . . . . . . . . . . . . . . . . .

21

2.2

???????? ?????????  . . . . . . . . . . . . . . . . . . .

22

2.3

????????????? ???????  . . . . . . . . . . . . . . .

26



??????????

34



?????? ??????????

35

















































































2



????????



?????????????? ????????????? ?????? ?????????, ???????????? ? ???-????, ???????? ? ????????-??????? ??????? ??? ????????? ? ???????? ????????????. ????????, ????????? ?? ??????? ? ???????????? ? ??-?????? ?????????, ??????????? ???????????????? ???????????. ?????? ???? ??? ?????????? ?????? ?????? ??????? ???????, ??? ?????? ??????? ????? ??????? ?? ??? ???? ???????.





?????????? ????????? ??????? ????? ???:

	u(0; t) =  1(t), u(l; t) =  2(t);

	ux(0; t) =  1(t), ux(l; t) =  2(t);

	ux(0; t) = h[u(0; t)   1(t)], ux(l; t) = h[u(l; t)   2(t)].



???? ? 1 1 ? 2 ????? ????, ?? ??? ??????? ???????? ???????????; ???? ? 2 1 ? 2 ????? ????, ?? ??? ???????, ????? ????? ?????? ????????; ???? ? 3 1 ? 2 ????? ????, ?? ??? ??????? ???????? ???????????. ??? ??? ??????? ?????????? ???????????.





????? ??????? ????????? ??????? ?????????? ??? ???????????? ??? ??? ???? ??????? [1]. ????????, ??? ??????? ???????????, ?? ????????-????? ?????? ????, ????? ????????? ?? ????? ???????? ?????????? ????-???? ???????? u(l; t), ??? ??? ux(l; t) = k1 F [u(l; t)].





????? ????????? ??????? ????? ???????????, ???? ? ?????? ????? ?????????? ???????????? ??????????.





??????? ???????, ?????????? ??????????? ?? t, ?????????? ??????-



???? ????????????? ?????????. ???????? ?????? ??????? ???????? ??-????, ????????????? ? [1]. ????????? ??????? ????? ???????????, ???? ?

?????? ????? ??????????? ???????????? ??????????. ???? ? ????? x = l



??????? ?????????? ???? ????? m, ?? ??? x = l ?????? ???????????



????????? ??????????? ???????: mutt(l; t) = kux(l:t) + mg [1]. ? [9] ???? ???????? ????????? ???????? ???????? ????????? ??????? ? ???????????-?? ?????????.





? ??????? ? ????????????? ?????????? ????????? ???????? ??? ??-







3



???????????? ????????????? ?????????, ????????? ? ????? ????????, ??? ????????, ????????, ????? ? ??????, ? ????????? ??????, ????????? ? ???-??????. ???????? ?????? ???????????? ??? ????????? ??? ???????????? ?????????? ? ????????, ?????????????? ????????, ????????? ?????????-?? ?????, ?????????? ?????????, ? ????? ????? ?????? ?????? ???????-???. ?????????? ??????????????? ???????????? (???), ??? ???????? ?????-??? ??????? ????????? ??????? ??????? ? ??????? ??????????? ? ?????-????? ????? ??????? ??????? ????????????. ?????????? ??????????? ?? ?????????????? ???????????????? ??????, ?? ????? ??????? ?????????? ? ??????????????? ???????????, ????????, ? ????????????. ??????????? ?????????? ??????????? ???????? ? ???? ????????????? ??????? ?????-???????? ???????????????? ??????????? ? ??????????????? ??????? ?????-???, ????? ?? ???? ???????? ?????????? ??????, ??????????? ????????? ????? ? ?????? [2].





?????? ????? ??? ???????? ???????????????? ????????? ? ???????? ???????????? ???????? ?????? ? ???????????? ?????????. ??????????-?? ????????? ?????????? ???????????, ??????????? ???????? ???????? ??????? ? ??? ??????????? ? ????????? ????????? ? ?????????? ?????? ???????, ? ??????? ?????? ??????? ??????. ???????????? ??????????? ??-??? ???????? ?? ?????? ??.?.??????? [10] , ?.?.???????? [11]. ? ???? ??-????? ???? ??????????? ??????????? ?????? ? ????????????? ????????? ??? ?????????????? ?????????. ???????? ??????????? ????? ??? ?????-?????????? ????????? ???????? ???? ? ????? XX ????, ??????? ???????? ????? ??????? ?????? ?.?.????????? [12], [13].





???????????? ??????? ????? ???? ????????? ?????. ????????, ????-???????? ??????? ?? ???????????????? ??????????, ?? ???????. ??????



	???????????? ????????? ?????????? ??????????? ? ??????? ??????? ??? ???????????? ?????. ??????? ?????????? ????? ?????? ??????? ??? ?? ????????. ? ????????? ??????? ??? ????? ???? ??????, ????????, ??-?? ???????????? ??????? ???????? ???????????????? ????????? ? ??????? ??????????? ?? ???????????.





	?????? ??????????? ?????? ??? ???????????????? ????????? ? ????-????????? ?????????? ?????????. ????? ??? ?????? ???? ??????????? ?





4



[8], ?? ? ?????? ?????? ??????????? ????????? ??????? ?? ??????? ??????. ????? ??????????? ?????? ? ??????????? ????????? ???????? ? ???????? ????????????? ????????????? ??????????? ??????? ???? ??????.











































































































































5



	?????? ? ????????????? ??????-???? ????????? ??? ?????????-??????? ?????????







1.1  ?????????? ??????



?????????? ?????????



utt   (a(x; t)ux)x + c(x; t)u = f(x; t);

(1)



	???????? ??????: ????? ? ??????? QT = (0; l) (0; T ) ??????? (1), ???-???????????? ????????? ??????



u(x; 0) =  (x);  ut(x; o) =  (x)

(2)



? ????????? ????????



ux(0; t) =  (t)ut(0; t);  ux(l; t) =  (t)ut(l; t):

(3)



?????? ??????? ??????????? ??????? ?????? (1) - (3). ?????? ????????? ????????? [7], ??????? ?????????, ??????? ????? ??????? ?????? ??????-

?????. ????? u - ???????????? ???????  (1) ? ??????????? ???????

(2),

(3). ????? ???????

(1) ?? ???????????? ??????? ??????? v ? ???????-

??????? ?? QT















T



l







T



l



Z0

Z0

(utt   (a(x; t)ux)x + c(x; t)u)vdxdt = Z0

Z0

f(x; t)vdxdt:

(4)

??????????? ?????? ????????? ????? ????? ?????????? ?????????







T

l

T

l

l











Z0

Z0

uttvdxdt =  Z0

Z0

utvtdxdt + Z0

utvj0T dx:





??????? ?? ??????? v ??????? v(x; T ) = 0, ????? ? ???? ????????? ????-??? ???????



T

l

T



l

l

Z0

Z0

uttv =  Z0

Z0

utvtdxdt  Z0

(x)v(x; 0)dx:





6





v(x; T ) = 0

???????? ?? ??????? ??????????



T





l







T

l





T



Z0



Z0

(a(x; t)ux)xvdxdt =  Z0

Z0

auxvxdxdt + Z0

auxvj0ldt:

???????? ????????? ???????, ????????? ????????? ?????? ???









T



l





T



l









Z0

Z0

(a(x; t)ux)xvdxdt =  Z0

Z0

auxvxdxdt

T















T









Z0



a(0; t) (t)ut(0; t)v(0; t)dt + Z0

a(l; t) (t)ut(l; t)v(l; t)dt:

?????????? ?????????? ?????????? ?

(4), ???????





T



l











l



Z0

Z0

(  utvt   auxvx + cuv)dxdt = Z0

(x)v(x; 0)dx+

T















T









+ Z0

a(0; t) (t)ut(0; t)v(0; t)dt  Z0

a(l; t) (t)ut(l; t)v(l; t)dt+













T

l























+ Z Z

f(x; t)vdxdt:



(5)

	0	0



???????????. ??????? u(x; t) 2 W21(QT ) ?????????? ?????????? ??-?????? ?????? (1) - (3), ???? ??? ????????????? ?????????? ???????



u(x; 0) = (x) ? ????????? (5) ??? ????? ??????? v(x; t) 2 W21(QT ); ??? ??????? ??????????? ???????











W21(QT ) - ???????????? ???????? - ???????????? ??????? ?? L2 ? ????-?????? ??????????? ??????? ???????, ??????? ????? ??????????? L2.









1.2

?????????????? ???????













???????. ????? ??????????? ????????? ??????? c(x; t) 2 C(QT ); at(x; t) 2







2

[0; T ];  (t) > 0 ?  (t) < 0 ??-



C(QT ); a(x; t) 2 C(QT ); f 2 L2

(QT );  ;  2 C





??? ?????????? ?? ????? ?????? ??????????? ??????? ?????? (1) - (3).





7



??????????????. ???????????, ??? ?????????? ??? ?????????? ??-????? ?????? (1) - (3) u1(x; t) ? u2(x; t): ?????????? ?? ???????? u(x; t) = u1(x; t) u2(x; t): ??? ??? ??????? u(x; t) ???????????? ????? ???????? ???? ?????????? ??????? ??????, ?? ??? ? ???? ???????? ?????????? ???????? ???? ??????. ????? ??? ??????? ????? ????????????? ????????



	u(x; 0) = 0;	ut(x; 0) = 0



? ?????????



T



l



T















Z0

Z0

(  utvt   auxvx + cuv)dxdt = Z0

a(0; t) (t)ut(0; t)v(0; t)dt









T























Z0

a(l; t) (t)ut(l; t)v(l; t)dt:



(6)



???????  2 [0; T ] ? ?????????? ???????















v(x; t) =

8

t u(x; )d ;

0   t   ;

(7)









>

R





















<

0;





t



T:











>























:

















??????? v(x; t) 2 W21(QT ), ??? ??? ??? ??????? ????? ?????????? ?????-?????? ??????? ???????, ????????????? L2(QT )

(

		u; 0	t	;

vt(x; t) =

		0;	t	T;



?????? v(x; T ) = 0;

v(0; t) = 0:

















?? ?????? ??????????? ???????

v(x; t) ???????, ??? u(x; t) = vt(x; t)



? ux(x; t) = vxt(x; t). ??????????? ?????????

(6), ????????? ??? ?????



??????????? ??????? v(x; t)

(7) ? ?????????????? ?? ??????. ????? ??????



????????? ?????? ???

























T



l







l









l



l



Z0

Z0

utvtdxdt =  Z0

Z0

utudxdt = Z0

Z0

uutdxdt  Z0

u2j0dx;



??? ??? u(x; 0) = 0 ? u(x; t) = vt(x; t), ????????? ????????? ?????? ???







T

Z0

l









Z0

l













Z0



utudxdt =  2



vt2(x; )dx:



















1



















































8



??????? ?????????????? ??????? ?????????? (6)







T







l















l



Z0

auxvxdxdt =  Z0

Z0

avxtvxdxdt = Z0

Z0

avxvxtdxdt+









l









l













+ Z0

avxj0dx  Z0

Z0

atvx2dxdt;





??????































T

auxvxdxdt = 2

Z0

l











l

atvx2dxdt:



Z0





avx2j0dx   2

Z0

Z0













1







1











































????? ??????????? ?????? ????? ?????????

(6)

T









Z0

(t)a(0; t)ut(0; t)v(0; t)dt =  Z0

(t)a(0; t)u(0; t)vt(0; t)dt









Z0

( (t)a(0; t))tu(0; t)v(0; t)dt =  Z0

(t)a(0; t)u2(0; t)dt



Z

( (t)a(0; t))tvt(0; t)v(0; t)dt:



0



?????????????? ?? ?????? ?????? ????????? ?????????? ?????????



 

	Z	Z

	( (t)a(0; t))tvt(0; t)v(0; t)dt =	( (t)a(0; t))tv(0; t)vt(0; t)dt+



	0	0



Z

	( (t)a(0; t))ttv2(0; t)dt + (( (t)a(0; t))tv2(0; 0):



0



? ???? ????? ?????????????? ?????? ????????? ?????? ????? (6) ?????? ???

T





Z0

(t)a(0; t)ut(0; t)v(0; t)dt = 2 Z0

(t)a(0; t)vt2(0; t)

Z

( (t)a(0; t))ttv2(0; t)dt  v2(0; 0)( (t)a(0; t))tjt=0:



0





9



?????????? ?????????? ?????????????? ??????? ?????????? ?????? ????? (6) ? ?????????



T

	Z	Z

		a(l; t) (t)ut(l; t)v(l; t)dt =	2	(t)a(l; t)vt2dt+



	0	0



Z

	( (t)a(l; t))ttv2(l; t)dt + v2(l; 0)( (t)a(l; t))tjt=0:



0



????? ????? ???? ?????????????? (6) ?????? ???







l

















Z0

(vt2(x; ) + a(x; 0)vx2(x; 0))dx+



















+2 Z0

a(0; t) (t)vt2(0; t)dt  2 Z0

a(l; t) (t)vt2(l; t)dt =





















= Z0

(a(0; t) (t))ttv2(0; t)dt  Z0

(a(l; t) (t))ttv2(l; t)dt









l







l







Z0

Z0

atvx2dxdt + 2

Z0

Z0

cvtvdxdt+



+v2(0; 0)(a(0; t) (t))tjt=0    v2(l; 0)(a(l; t) (t))tjt=0:

(8)









??? ?????????????? ?????????????? ????? ???????? ????????? ????-



????? ??????. ??????? ?????? ?????? ????????? ??? ????????? ????? ??-









x

v xd )2 + 2

x

v d v + v2. ????????

x





???. ???????? v2(0; t) = (R0



R0











l v



(0; t) = 2(R0















???????????



xx



2



v d )

2

+ 2v

2

(x; t), ? ???? ??????? R0

v d



????, ???????





























	d R v2d   l R vx2dx: ?????

		0	0	0



l

l

l





l

Z0

v2(0; t)dx   2l Z0

Z0

vx2dxdx + 2

Z0

v2dx;





10



??? ??? ? ????? ????? ??????????????? ????????? ?? ??????? ?? x, ?? ?????????????? ?? x





l



Z

l



lv2(0; t)   2l2

Z

vx2dx + 2



v2dx:





0



0







? ???????? ????? ????????





l

vx2dxdt + l



Z0

l



Z0

v2(0; t)dt   2l Z0

Z0



Z0



v2dxdt:









2































??????????? ???????? ???????





l

vx2dxdt + l



Z0

l



Z0

v2(l; t)dt   2l Z0

Z0



Z0



v2dxdt:









2































	??????? ?????????? ???? ?????????? ? ??????? ???????, ?? ??????? ???????, ??? j(a(0; t) (t))ttj A0; j(a(l; t) (t))ttj B0, ?????





l

vx2dxdt + l



l

v2(x; t)dxdt);



j Z0

v2(0; t)(a(0; t) (t))ttdtj  A0(2l Z0

Z0



Z0

Z0











2











l











l

v2(x; t)dxdt):

























j Z0

v2(l; t)(a(l; t) (t))ttdtj  B0(2l Z0

Z0

vx2dxdt + l Z0

Z0











2





































??????? ?????? ?????????, ??????? ?? ???????? ?????????. ??? ?????

?????????? ?????????



0

v(0; t) =

0

vxdx+v(x; 0)











Rx







??????????, ???????





. ???-



??? ?? ?????? jv(0; t)j

j R vxdxj + jv(x; 0)j. ???? x ?? 0 ?? l, ??????-







x

l

R

		??? ??????????? ?????? ??? jv(0; t)j	l	jvxjdx +

0






l

1l R jv(x; 0)jdx. ??? ???

0





v(x; t) = u2(x; t), ?? vx(x; t) = j(u2(x; t))xj = j2uuxj. ????????? ?????? v u:

u2(x; t)   l

l













1

l

u2(x; t)dx. ????????????? ???????????? ????







j2uuxjdx +











0



l

0





?

:

R

2



1



2

.



R











1





1























?????



?

















"

2juuxj



+ "1

u







" = l"1



c(") = l

+ "1 , ?????







"1ux















































l







l

















v2(0; 0)   " Z0

vx2dx + c(") Z0

v2(x; 0)dx:







11



???????? ??????????? ????, ???????





l

l

v2(l; 0)   " Z0

vx2dx + c(") Z0

v2(x; 0)dx:



	???? ?????????? ???? ? ,????????? ??????? ???????, ?? ??????? ????-?????? ????? ????????????? ????????? A ? B, ??? j(a(0; t) (t))t)jt=0j A; j(a(l; t) (t))t)jt=0j B, ????????



l

Z

		v2(0; 0)	A	("vx2(x; 0) + c(")v2(x; 0))dx;



0



l

Z

		v2(l; 0)	B	("vx2(x; 0) + c(")v2(x; 0))dx:



0



???????, ??? ? ???? ??????? ??????? ?????????? ????? ????????????? ????? c0, ??? jc(x; t)j c0. ???????? ??????????? ????, ???????



l







l

2j Z0

Z0

cvtvdxdtj  c0

Z0

Z0

(v2 + vt2)dxdt:



????????? (6), ???????? ??? ?????????? ??????, ??????? ??? ???? ??-??????







l



















l















Z0

(u2(x; ) + a(x; 0)vx2(x; 0))dx   c1

Z0

Z0

(u2 + vx2)dxdt+

















+c2" Z0

l











Z0

l































vx2(x; 0)dx + c3





v2(x; 0)dx;













??? c1

= maxf2l(A0 + B0);

2(A0+B0)

; c0g; c2

= (A + B); c3

= c2c("). ?????







l







































a0







a0



???????, ??? a(x; t)   a0 > 0; ? ??????? " =

2c2

: ????? a0    c2" =



2

> 0.



???????? ?????? ?????? ????????? ?????? ????? ? ?????, ???????









l

(u2(x; ) + 2 vx2(x; 0))dx   c1





Z0

l



+ vx2)dxdt + c3

Z0

l









Z0



Z0



(u2





v2(x; 0)dx:









a0



















































12



	? ???? ????????????? (7) v2(x; 0)	R u2(x; t)dt; ?? ???????? ? ???????-

????







0































l

(u2(x; ) + 2 vx2

(x; 0))dx   c4



Z0

l

(9)



Z0





Z0



(u2 + vx2)dxdt:









a0





















??????????? ?????????, ?????????? vx2(x; t), ? ????? ? ?????? ?????? (9). ??? ????? ?????? ??????? w(x; t), ??????? ????? ????????????



t









w(x; t) =

R0

uxd . ??????? vx(x; t) ????? w(x; t). ????????t

t

(7), ????-

??? v(x; t) ????? ??????????? ? ???? v(x; t) =   R

ud  + R0

ud : ?????

		vx(x; t) =	w(x; ) + w(x; t), ?????? v(x; 0) =	w(x; ): ????????, ???



	vx2(x; t) = w2(x; )	2w(x; )w(x; t)+ +w2(x; t): ?? ??????????? ???? ?



	???? ??????????? ????????? vx2(x; t) 2w2(x; ) + 2w2(x; t): ????????? ?????????? ?????????? ? (9)

l

(u2(x; ) + 20 w2

(x; t))dx   c4



Z0

l



Z0





Z0



(u2 + 2w2(x; t) + 2w2(x; ))dxdt:







a



































l

???????, ??? ? ?????? ????? ?????????? ??????????? ???????? R R w2(x; )dxdt







0

0





l





?? ??????? ?? t. ??????????????? ??? ?? t, ???????

R0

R0

w2(x; )dxdt =

l

R w2(x; )dx. ?????



0

















Z0

l

(x; t))dx   c4



Z0

l

l





(u2(x; )+ 20 w2



Z0



(u2 +2w2(x; t)dxdt)+2c4  Z0

w2(x; )dx:







a







































	????????? ????????, ?????????? w2(x; ), ? ????? ????? ? ???????	???,

?????

a0

2c4   > 0: ????? ??? ??????????????

a0

2c4

a0







2



2



4

. ????? ???



??????  2

[0;

a0

]

























8c4



































l







l















a1

Z0

(u2 + w2)dx   c5

Z0

Z0

(u2 + w2)dxdt;











13



??? ?5 = max[c4; 2c4]; a1 = max[1; a20 2c4 ]: ? ?????????? ??????????? ????? ????????? ????? ????????, ????? ?? ?????????? ???????





l

Z

	(u2(x; ) + w2(x; ))dx	0:



0



??? ???????? ?????? ??? ????????????? ????????? ???? ???????

	u2(x; ) = 0 ? w2(x; ) = 0. ?????????????, u(x; ) = 0 ???	2 [0; a0 ]:



8?4



?????????? ?????? ???? ?????? ? ?????????? ??????? t = 8a?04 , ???????,



	??? u(x; ) = 0 ???	2 [0; a0 ]: ????? ???????, ?? ???????? ????? ?????



4c4



???????, ??? u = 0 ?? ???? ???????? QT . ? ??? ??? u = u1 u2 ? u = 0, ?? u1 = u2. ?????????????, ?????? ? ????????????, ?????? ?????????? ???????????? ??????? ???????????? ??????.





1.3  ????????????? ???????



??? ?????????????? ????????????? ??????????? ??????? ???????? ????? ????????????.





???????? ?????????????????? ???????????? ??????? ???????????? ??-????. ??? ????? ????? ???????? ????? ?????????.





?????????? ?????????????????? fwn(x)g, ??? wn  2

2







C

(  ) \ C(  ) ?



wn(x) = 0 ??? x 2 : wn ???????? ??????? ??????????? ? ???????? ?????? ??????? ? W21( ). ?????? ????? ????? ????? ???????, ??? ??? ????????-???? W21 ???????? ?????????????.



????? ?????? ???????????? ??????? ?????? ? ????



m



Xk



um(x; t) =    dk(t)wk(x):

(10)

=1





?? ???????????

Z

(umttwj + aumxwj0 + cumwj)dx + a(0; t) (t)umt(0; t)wj



a(l; t) (t)utm(l; t)wj = Z

f wjdx

(11)











14



???????





Z

m







m

















(k=1 dk00

(t)wk(x)wj(x) + a k=1

dk(t)wk0

(x)wj0(x)+









X







X













m









m















Xk









X















+c   dk(t)wk(x)wj(x))dx +  (t)

a(0; t)dk0(t)wk(0)wj(0)dt







=1









k=1

















m



(t)wk(l)wj(l)dt = Z











(t) k=1 a(l; t)dk0



f wj(x)dx:









X



















????? Akj =  wk(x)wj(x)dx, Bkj =

wk0(x)wj0(x)dx,

, ????? (11) ??????







R,





?

R

R









???



























= wk(0)wj(0) Dkj











f wj(x)dx







Ckj



= wk(l)wj(l)  fj(t) =







































m



m







m











X

X







Xk













dk00(t)Akj + a

dk(t)Bkj + c

dk(t)Akj+







k=1

k=1







=1













m







m

















Xk







X















+ (t)

a(0; t)dk0(t)Ckj     (t)

a(l; t)dk0

(t)Dkj = fj(t):

(12)







=1







k=1















???????? ??????? ???????????????? ????????? ??????? ??????? ??-?????????? dk. ??????? ????????? ???????, ??????? ???????? ?? (2)

dk(0) =  k;  dk0(0) =  k;

(13)



m

???  k ? k ???????????? ????????? ????  m(x) =

kwk(x),  m(x) =



=1

m

kP

P

		kwk(x). ??? m ! 1 ??? ????? ??????????????	(x) ?	(x) ? ???-

k=1

??? W21 ? L2 ?????????????? m(x) ! (x) ? m(x) ! (x). ???????? ?????? ???? (12), (13). ????? ?????? ????????? ????? ? ?????? ?????, ????? ??????? Akj ???????? ?????????. ? ???? ??????? ?? wj(x) ?????-?? Akj - ?????????????. ?? wk ???????? ?????????????? ?????????? ?? ??????????????? ? L2. ? ???? ?????? ???????????? ????????????????



????????? ?????? (12) ? ?????????? ????????? (13) ?????????? ?????-????, ? d00k 2 L1(0; T ). ?????????????, ?????????????????? ???????????? ??????? fum} ?????????.











15



?????? ??????? ?????????????? ???? ?????????????????? ? W21. ??? ????? ????? ???????? ????????? ??????. ??????? (11) ?? d0j , ??????-

?????? ?? j ?? 1 ?? m ? ?????????????? ?? t ?? 0 ??

Z

m



m

(t)wj0(x) + cum

m





(uttm j=1 dj0

(t)wj(x) + auxm j=1 dj0



j=1 dj(t)wj(x))dx





X



X



X







m



m









X



Xj







+ autm(0; t)

dj0(t)wj(0)    autm(l; t)

dj0(t)wj(l) =







j=1



=1





	m

X

		=	f	d0j(t)wj(x)dx:

j=1



???????? (10), ?????????????? ????????? ????????? ?? t ?? 0 ??





l



l





l









Z0

Z0

uttmutmdxdt + Z0

Z0

auxmuxtmdxdt + Z0

Z0

cumutmdxdt+





















l



+ Z0

a(utm(0; t))2dt  Z0

a(utm(l; t))2dt =

Z0

Z0

f utmdt:

(14)

????????? ? ?????? ????????? ??????. ??????????? ?????? ????????? ????? ????? ?????????? ?????????



l

uttmutmdxdt = 2

l

(utm(x; ))2dx   2

Z0

l



Z0

Z0



Z0





(utm(x; 0))2dx:







1



1



































???????? ?? ??????? ??????????





l

auxmuxtmdxdt =  2





l

Z0

l



Z0

Z0



Z0

Z0

at(uxm)2dxdt + 2



a(uxm(x; ))2dx











1







1

























l





















2



a(uxm(x; 0))2dx:

















Z0























1





























????????? (14) ? ???? ????????? ??????????????





2 Z0

l



















((utm(x; ))2 + a(x; )(uxm(x; ))2)dx + Z0

a(utm(0; t))2dt



1



































































16











a(utm(l; t))2dt = 2

l

(utm(x; 0))2dx+





Z0



Z0



















1



















l













l









+2





at(uxm)2dxdt + 2



a(uxm(x; 0))2dx







Z0

Z0



Z0







1







1





















































l





l











Z0

Z0

cumutmdxdt +

Z0

Z0

f utmdxdt:

(15)



? ??????? ??????????? ?? a(x; t); at(x; t) ? c(x; t), ?????????? ??? ??-???????????? ?????????????? ??????????? ???????, ?????? ?????? ?????? ????? (15). ??????? ?????? ??????? ??????????





















l













l





























Z0

Z0

at(uxm)2dxdt   a0

Z0

Z0



(uxm)2dxdt:





?????? ?????? ????????????? ? ????????? ?????????

















l

















l





































Z0

Z0

cumutmdxdt   c0

Z0

Z0

((um)2 + (utm)2)dxdt;















l

futmdxdt   2



l

f2dxdt + 2



Z0

l















Z0

Z0



Z0

Z0



Z0



(utm)2dxdt:























1













1

















??????? ???????????



























































































Z0

l

























































((utm(x; ))2 + a(x; )(uxm(x; ))2)dx+









































































+2 Z0

a(utm(0; t))2dt  2 Z0

a(utm(l; t))2dt











l



















l

















l













Z0

(utm(x; 0))2dx + a0

Z0

Z0

(uxm)2dxdt + Z0



a(uxm(x; 0))2dx













l





















l



















l





c0

Z0

Z0

((um)2 + (utm)2)dxdt + Z0

Z0

f2dxdt +

Z0

Z0

(utm)2dxdt:

(16)







17



















(

)











(









t







+.??????? ?????? ??????







































) = R0















?????????? um x;



? ???? um



x;





umdt



um(x; 0), ?????



(u



(x; ))



= = (R0

t

)

+2

R0 ut



+(

(



0))



















m







2





umdt

2





mdtum



um



x;







2













???????????? ??????????? ?? x ??R00 ??t l



























?????????? ????????? (um)2

2



(umdt + 2(um(x; 0))2: ??????????????















l















l





















l



















Z0

(um(x; ))2dx   2  Z0

Z0



(utm)2dxdt + 2



Z0

(um(x; 0))2dx:

(17)





??????

(16) ? (17)

























































l































































Z0

((um)2 + (utm)2 + a(uxm)2)dx + 2 Z0



a(utm(0; t))2dt

































l





















l





















2

Z0

a(utm(l; t))2dt   Z0

(utm(x; 0))2dx + Z0

a(uxm(x; 0))2dx+











l

















l











l















l





+2

Z0

(um(x; 0))2dx + a0

Z0

Z0

(uxm)2dxdt + Z0



Z0



(utm)2dxd  c0

Z0

Z0

((um)2+































l























l























+(utm))dxdt + 2  Z0

Z0

(utm)2dxdt + Z0

Z0

f2dxdt:











????? M = max[a0; 1  c0 + 2 ;  c0], ?????



































l































































Z0

((um)2 + (utm)2 + a(uxm)2)dx + 2 Z0



a(utm(0; t))2dt



















































l































2 Z0

a(utm(l; t))2dt   M Z0

Z0

((um)2+





































l























l



















+(utm)2 + (uxm)2)dxdt + 2

Z0

(um(x; 0))2dx + Z0

(utm(x; 0))2dx+





















l



















l







































+ Z0

a(uxm(x; 0))2dx + Z0



Z0



f2dxdt:













18



?? ????????? ????????? ???? ? ????????????? L2 ? W21 ???????









l



































Z0

((um)2 + (utm)2 + a(uxm)2)dx + 2 Z0

a(utm(0; t))2dt





























l

















2 Z0

a(utm(l; t))2dt   M Z0

Z0

((um)2 + (utm)2+













+(uxm)2)dxdt + jjfjjL2

2



+ jj

jjL2

2

+ mjj jjW2

21 :







??? ??? f ???????? ???????????? ???????? ? L2,  ? W21

, ?? L2,??





2



2



2

K





K

















jjfjjL2

+ +jj jjW21

+ jj jjL2



, ???



????????????? ?????????. ?????



























????????? ??????????? ?????? ???

























l

















l















Z0

((um)2 + (utm)2 + (uxm)2)t= dx   M Z0

Z0

((um)2 + (utm)2 + (uxm)2)dxdt + K:





? ????? ??????????? ????????? ????? ????????, ????? ?? ?????????? ???????

l

Z

	((um)2 + (umt)2 + (umx)2)t= dx	eM K:



0



?????????????? ???????????? ??????????? ?? t ?? 0 ?? T



l

Z Z

	((um)2 + (umt)2 + (umx)2)t= dxdt	M1 K(eM T    1);





	0	0



?????

1



M T

1) = N

, ?????

jju

m  2

N

. ?????? ?????????-

























M K(e









jjW21(QT )







????????? ??????? ?????????? ? W21

, ?????????????, ????? ???????? ???-





??????????????????, ????? ?????????? ? ???????? ?? ????? ?? ????????-????, ?? ???? ? u 2 W21. u ??????? ?????????? ??????? ?????? (1) - (3). ????? ???????? ???, ????? ????????, ??? ??? ????? ??????? v(x; t) 2 W21 ??????????? ???????????? ????????? (5) ? (x) = 0





T

l

T



Z0

Z0

(  utvt   auxvx + cuv)dxdt = Z0

a(0; t) (t)ut(0; t)v(0; t)dt







19



T

T



l



Z0

a(l; t) (t)ut(l; t)v(l; t)dt + Z0

Z0

f(x; t)vdxdt:

(18)

??? ????? ????? ????????, ??? ??? ?????????? ????? ???????? ?

W21



????????? ??????? ??????????? ????????? (18). ??????? ??????? m(x; t) 2 W21(0; T ) ? (x; T ) = 0. ??????? (11) ?? m. ????? ????????? ???????-??????? ?? t ?? 0 ?? T ? ???????????? ?? j ?? 1 ?? m



	l

Z Z

(umt  tm + aumx  xm + cum  m)dxdt+



	0	0



	T	T

	Z	Z

		+	aumt(0; t) mdt	aumt(l; t) mdt =



	0	0



	l

Z Z

	f mdxdt:



	0	0

	????????? ????????? ??????????? ??? ????? ???????	(x; t) =

m

P

	dj(t)wj(x). ???????? ? ??????? ??? m ! 1 ??? ?????????????

j=1

(x; t)

T



l



l





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