Msm60 Grapti Ergasia 1 2011-2012 Apantiseis
description
Transcript of Msm60 Grapti Ergasia 1 2011-2012 Apantiseis
-
1
: : 60
. 2011-2012
1
1. :
1 21 2 1 2 1 2
, , 2 , 0x xu x x u x x u x x
1 2 1 1u x ,x x 1 22 1 0x x .
1 2
2 0x xu u u (1)
1 2 1dx dx
ds ds 1 2 1 2x x c x x c (2)
c .
1 2, 1,1x x .
1 22 1 0x x 1 2,x x 1 2, 1,2x x . ,
(1) 3 0 , (1)
1 2 1, 1u x x x (3)
1 22 1 0x x Cauchy.
1 2x x , 1x , ,
1 0 1 2, 0x x . (1)
2 0u u 1 2, ,u u x x 2,u f e (4)
f 1C . (4) 1 2Ox x
121 2 1 2,x
u x x f x x e (5)
1 2,x x 1 22 1 0x x
1 12 21 1 1 11 3 1 3 1 1x x
x f x e f x e x .
1 1
13 1
3
yy x x
2
13
2
3
y yf y e
.
(5) Cauchy (1),(3)
1 2
1
21
1 2 231 2
2,
3
x xxx x
u x x e e
1 22 2 11 2 3
1 2
2,
3
x xx xu x x e
-
2
2. Cauchy
1 2
2 2
1 22 1 1 22 2
2x x
x xx x u u x x u u
, 1x 2 0x
1 1,0u x x , 1x .
1 2
2 2
1 22 1 1 22 2
2x x
x xx x u u x x u u
1x 2 0x (1)
.
3
12 12
dxx x u
ds (2)
21 22
dxx x u
ds (3)
2 2
1 2
2
x xdu
ds
(4)
(2)-(4) . (2) (4)
2 21 22 1 2 1
dx dxx x x x
ds ds (5)
(5) (4)
1 22 1 2 0dx dx du
x xds ds ds
1 2 2 0d
x x uds
1 2 12x x u c (6)
1c . , (2) (3)
2 2
2 2 2 21 2 1 21 2 2 1 1 22 2
2
dx dx x xx x x x u d x x u
ds ds
(7)
(7) (4) 2 2
2 2 2 21 22 1 22 0 4
2
x xd u x x u c
(8)
2c . 1 1 2 1 2, , 2g x x u x x u
2 2 22 1 2 1 2, , 4g x x u x x u , ,
1 2 2 1
1 2
2
82 2
g g x x
ux x
i j k
2 21 2 2 1 2 1 8 4 8 4 2 2ux x ux x x x i j k 0 ,
21 2, ,x x u , 1 28 4 0ux x 2
2 18 4 0ux x 2 2
2 12 2 0x x .
(1)
21 2 2 18 4 8 4 0ux x ux x , (1). (1), :
2 2 21 2 1 24 2x x u G x x u (9)
G 1C .
1 1 1,0 ,u x x x (10)
-
3
(9)
2 21 1 14 2x x G x 2
1 1
32 2
4x G x
23
4
uG u
(9)
22 2 2
1 2 1 2
34 2
4x x u x x u
2 2 2 2 2 2
1 2 1 2 1 24 4 16 3 12 12x x u x x u x x u
2 2 2 2 21 2 1 2 1 24 3 4 12x x x x u x x u
2 2 2 2 2
1 2 1 2 1 2
33 0
4u x x u x x x x
. ,
(10)
2 2 2 21 21 2 1 2 1
1 2
2 2 2 21 21 2 1 2 1
33 , 0
2,
33 , 0
2
x xx x x x x
u x xx x
x x x x x
3.
u ux
t x
, 0x , 0t (1)
0u ,t t , 0t (2)
0u x, sin x , 0x (3) ) (1) (1) . )
2
2
2
2
xx t , x t
u x,tt
xt sin x t , x t
.
) (1)
1dx dt
x t cds ds
c . x t x
, 1 0J , 0, 0x t .
, ,u x t u , (1) u u u u
2
,2
u f
2
,2
xu x t f x t (4)
f 1C .
-
4
) (1)-(3)
f 0x 0t
(2) (3).
f (4)
x t , (2) (3)
,
0x 0t .
, ,x t
x t x t c t 00, t . , 0c t
, x t 0x t t , 0 0t . (4)
2
0,2
xu x t f t (5)
x t (2) 0 0t f t , 0 0x t t .
(5) 2 2
0, ,2 2
x xu x t t x t x t (6)
,
,x t x t . x 0 ,0x
0x t c x (4)
2
0, ,2
xu x t f x x t (7)
(3), 0t 0x x ,
2
00 0 0,0 sin
2
xu x f x x
2
00 0sin
2
xf x x
(7)
22
, sin2 2
x txu x t x t
2
, sin ,2
tu x t x t xt x t (8)
, x t , (4)
2
, 02
xu x t f
0f 0 0
lim 0, lim ,0 0t x
u t u x
.
, 0 0f 2
, ,2
xu x t x t (9)
(1)-(3) ,x t x t
2
2
2
2
xx t , x t
u x,tt
xt sin x t , x t
-
5
4. ) 2 2C
1
2
1 2 23xu x x x
2
3
1 1xu x x
0 0 0u , .
) 2 2C
1
2
1 2 23 001xu , x x x
2
3
1 1xu x x
0 0 0u , . ) ()
2 2C ;
1
2 3
1 2 2 1 2 1 2 1 2 23 ,xu x x x u x x x x x x f x (1)
f 1C .
(1) 2x
2
3
1 1 2'xu x x f x
, 2
3
1 1xu x x . ,
2 2' 0f x f x c c . ,
31 2 1 1 2,u x x x x x c (2) 0,0 0 0u c
1
2
1 2 23xu x x x
2
3
1 1xu x x
0 0 0u ,
31 2 1 1 2,u x x x x x . 2 2u C
1 2 1 1 2 2, , , ,x x x x x xu u u u u .
) 1 2,u x x
1
2
1 2 23 001xu , x x x
2
3
1 1xu x x
0 0 0u ,
, 1 2,u x x 2 2C ,
1 2 2 1
2
1 2, ,x x x xu u x x .
1 2
2
13 001 1x xu , x 2 12
13 1x xu x .
1 2 2 1x x x x
u u u
2 2u C .
-
6
) ()
2 2C , , , (), .
5. ) :
1 1 2 2 2 3 3 33 4 4 0x x x x x x x xu u u u .
)
1 1 2 2
2
1 2 0x x x xu x x u , 1 0x , 2 0x , .
) 1 1 2 2 2 3 3 3
3 4 4 0x x x x x x x xu u u u
, ,
, 1 1i j i
n n
i j x x i j x
i j i
a u b u cu d
ijA a (. . 3.1, 73,
), 11 3,a 22 1a , 33 4a , 12 21 13 21 31 0a a a a a
23 32 2a a
,
3 0 0
0 1 2
0 2 4
A
3,0,5.
, 3 .
)
1 1 2 2
2
1 2 0x x x xu x x u , 1 0x , 2 0x (1) , .
(1) 1 1 1 2 2 2
2 0x x x x x xAu Bu u 1, 0A B 2
1 2x x .
, 2
21 2
1
dx B B Ax x
dx A
, 2 0x .
1
222 1 1
1
dxx x x
dx
11
22
1
11
12
xdx
dx
221
2 1 22 42
xx c x x c
c . 2
1 24x x 2
1 24x x .
-
7
1
2 1
21
2
22
80
22
xx x
Jx
xx
, 1 0x ,
. 1 1 1
2x x x , 2 22
2x x
x
1 1 1 1 1
2x x n x x nu u u n u x u u
2 2 2 2
2
2x x n x x nu u u n u u u
x
1 1 1 1 1 1 1 1 1 1
2 2
... 2x x x nn x n x x x x n x xu u u n u n u u n
1 1
2
12 2 2x x nn n nu u u u x u u
2 2 2 2 2 2 2 2 2 2
2 2
2x x x nn x n x x x x n x xu u u n u n u u n
2 2
2
32 2
2 12x x nn n nu u u u u u
x x
(1)
1 1 2 2
2 22 2 1 11 2 1
2 2
4 4 2 2 0x x x x n nx x
u x x u u x u ux x
2
12
x
28
x
1 1 2 2
2
1 2
6 2 2 60 8 0x x x xu x x u u u u
6. ) :
1 1 1 2 2 2 10x x x x x x xu u u u .
) .
1 1 1 2 2 2 1
0x x x x x x xu u u u (1)
1 1 1 2 2 2 1 2
2 , ,x x x x x xAu Bu u x x u
1
1,2
A B 1 .
23
04
B A ,
2 .
-
8
) (1)
,
11
2
11
2
A
1
2
3
2
1 1, 1 2 1,1 .
1
1 1 1
2 2 2
1 1 1 1 1 11
1 1 1 1 1 12
x x x
x x x
1 2
1 2
1
2
1
2
x x
x x
21 2,x x , 1 0J .
, 1
1
2xu u u
1 1
1 1 1
4 4 2x xu u u u
1 2
1 1
4 4x xu u u
2 2
1 1 1
4 4 2x xu u u u
(1)
1 1 1 2 2 2 10x x x x x x xu u u u
3 1 10
4 4 2u u u u
3 10
2 2u u u u
, , .
7. ) 1 1 2 2
2
1 0x x x xu x u u , 2
1 2x ,x
, .
) 0t x xxxu a u u bu a, ,b , Korteweg-de Vries. :
1
22
12 3
A Au x,t Asech x a t
b
,
.
( 1 2
cosh x xsechx
x e e
).
) 1 1 2 2
2
1 0x x x xu x u u , 2
1 2x ,x (1)
-
9
1 1 1 2 1 2
2 , ,8x x x xAu Bu u x x u
211, 0,A B x ,
2 2
1 1 20, 0,B A x x x
(1) 21 2x ,x 1 0x .
222 1
1 2 2 1
1
22
dx xx x c x x c
dx c .
1 0x , 2x , (1)
2x c 1x .
)
1
22
12 3
A Au x,t Asech x a t
b
2secA h cx dt
1
2
12
Ac
b
1
2
12 3
A Ad a
b
.
2
1 1sec sinh
cosh cosh
d dhu u
du du u u
2sec sinhh u u
sinh coshd
u udu
cosh sinh
du u
du
2 2cosh sinh 1u u
,
22sec sec sinhxu Ac h cx dt h cx dt cx dt
32 sec sinhAc h cx dt cx dt
2 2 22 3sec sinh sec sinhxxu Ac h cx dt cx dt h cx dt cx dt 2 32 sec coshAc h cx dt cx dt
2 4 2 22 3sec sinh secAc h cx dt cx dt h cx dt
2 2 2 22 sec 3sec cosh 1 1Ac h cx dt h cx dt cx dt
2 2 22 sec 2 3secAc h cx dt h cx dt
2 2 2 44 sec 6 secAc h cx dt c h cx dt
2 24 2sec sec sinhxxxu Ac h cx dt h cx dt cx dt
3 3 224 sec sec sinhAc h cx dt h cx dt cx dt
3 38 sec sinhAc h cx dt cx dt 3 524 sec sinhAc h cx dt cx dt
, 2 sec sinhtu Ad h cx dt cx dt ,
-
10
2 52 sec sinhxuu A c h cx dt cx dt
1
22 52 sec sinh
12
AA h cx dt cx dt
b
(1)
... 0t x xxxu a u u bu
8.
t xRu x,t Vu x,t Ku x,t 0x , 0t (1) V,R,K>0, . ) (1).
) Kt
Ru x,t f x ct e
1 0 1K K
V VR R
f u e u u e HV V
Heaviside, 1 0
0 0
,H
,
(1) 00u x, u x 10u ,t u t .
) t xRu Vu Ku 0, 0x t (1)
V,R,K>0, 0t xV K
u u uR R
(2)
.
dx V Vx t c
dt R R
1
Vx t
R
x
(2)
0V V V Ku u u u
R R R R 0
V Ku u
R R
,K
Vu e f
,
,Kx
VV
u x t e f x tR
(3)
f 1C .
2
Vx t
R
t
(2)
0 0V V Ku u u u
R R R
-
11
0K
u uR
0K
Re u
,
K
Ru e f
,Kt
RV
u x t e f x tR
(4)
f 1C .
) 1
f (4), u
0,0u x u 10,u t u 3,
,x t V
x tR
0t 0 0,0 , 0x x .
0
Vx t x
R
(4) 0 0 0 0,0 , 0u x f x u x x
, (1) 0V V
x t x tR R
0,Kt
RV
u x t e u x tR
(5)
V
x tR
0x 00, t
0V
x t tR
. , (4) 0x
0
0 0 1 00,Kt
RV
u t e f t u tR
0
0 1 0
Kt
RV
f t e u tR
1K
VR
f e uV
(4) V
x tR
1,K K VKx tt
V V RRR R V
u x t e e u x tV V R
1,Kx
VR
u x t e u x tV
, V
x tR
u 0 0
lim ,0 lim 0,x t
u x u t
0 10 0
lim lim 1x t
u x u t
(4) V
x tR
, 0Kt
Ru x t e f
, 0,0lim , 0
x tu x t f L
,
Kt
Ru x t e L
-
12
,u x t , 0
1
,
,
,
Kt
R
Kx
V
V Ve u x t x t
R Ru x t
V V Ve u x t x t
R R R
.
2
:
0
1
,
,
,
Kt
R
Kx
V
V Vu x t e x t
R Ru x t
V V Vu x t e x t
R R R
V
x tR
0,Kt
Rxu x t u x e
0 0, 0K Kt t
R Rt
V Ku x t u x e u x e
R R
(1), (2),
.
, V
x tR
1 1, 'K Kx x
V Vx
R Ku x t u t e u t e
V V
1,Kx
Vtu x t u t e
(1)
t xRu Vu Ku 1 1 1 , 0K K Kx x x
V V VRu t e Ru t e Ku t e Ku x t
.
, 00
lim 0Kt
R
Vx t
R
u u e
1 1 10
lim 0 0 0K K V Kt t t
V V R R
Vx t
R
u u e u e u e
. u V
x tR
0 10 0u u .
u 0V
x tR
0t , 10,u t u t
0 , 0R
t x xV
0,0u x u x
.
-
13
9. ) :
0u u
c x x K x Q x,tt x x
0 x L 0t (1)
0 00 0u
K ,t tx
0t (2)
0 Lu
K L L,t tx
0t (3)
0u x, f x (4)
u x,t ,
Q x,t , 0K x , x ,c x . ) - . , , . ,
1t xxu x,t u x,t 0 x L 0t (5)
0 1xu ,t 0t (6)
xu L,t b 0t (7)
0u x, f x
0 x L (8)
) b (5)-(8); ) .
) L L ,
. Ox
, 0x x .
x , c x ,
0K x . ,Q x t , x ,
0 t L t 0t .
f x .
, (1) u x x ,
,Q x t . . (2) (3) . (4)
0t .
-
14
) (4) .
2
1 2,2
xu x t c t x c t
1,xu x t x c t
1 2,c t c t 1C t .
(5) 10, 1 1xu t c t .
0u
t
2 20c t c t c , u
(4)-(6)
2
2,2
xu x t x c , 0 1x .
) (6) , 1xu L t b b L 1b L . , u , , .
) (1) 0,1x
1 1
0
0 0 0
, ,
L
x x
uc x x u x t dx K x dx Q x t dx
t x x
0 00 0
, 0,, 0 ,
L L
x
u L t u tc x x u x t dx K x K Q x t dx
t x x
(),
1
0 0
, 1 1 1
L
u x t dx b dx b Lt
() ,
0
, 0 1
L
u x t dx b Lt
.