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Задача с динамическими граничными условиями для гиперболического уравнения
| Код работы: | K012359 |
| Тема: | Задача с динамическими граничными условиями для гиперболического уравнения |
Содержание
???????????? ??????????? ? ?????
?????????? ?????????
??????????? ??????????????? ?????????? ??????????????? ?????????? ??????? ??????????? ?????????? ???????????? ????????????????? ??????????? ????? ????????? ?.?. ????????? ????????? ??????????
??????? ????????? ??????????????
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01.05.01 ???????????????
?????????? ? ????????
?????? ? ????????????? ?????????? ????????? ??? ???????????????? ?????????
????????? ???????????????? ??????
????????? ? ??????
???. ????????
?.?.-?.?., ?????????
???????? ?. ?.
(???????)
???????? ??????? ????? 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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