Куликов А.А. (сост.) Смешанные задачи для уравнения теплопроводности и уравнения колебаний
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41
> @
³
t
nnn
d)IJ(f)tIJ(a)t(T
0
2
WO
exp , !,, 21
n .
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ:
)tx,(fuau
xxt
2
,
lx
0
, 0
!
t , (7.30)
)
x
(
)
x,(u
M
0 , l
x
d
d
0 , (7.31)
000
11
)t,(u)t,(u
x
E
D
, 0
!
t , (7.32)
0
22
)tl,(u)tl,(u
x
E
D
, 0
!
t . (7.33)
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.30) – (7.33) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
)tx,(u)tx,(u)tx,(u
21
,
ɝɞɟ
1
u
– ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
xxt
uau
1
2
1
,
lx
0
, 0
!
t ,
)x()x,(u
M
0
1
, lx
d
d
0 ,
000
1111
)t,(u)t,(u
x
E
D
, 0
!
t ,
0
1212
)tl,(u)tl,(u
x
E
D
, 0
!
t ,
ɚ
2
u – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
)tx,(fuau
xxt
2
2
2
, l
x
0 , 0
!
t ,
00
2
)x,(u
, l
x
d
d
0 ,
000
2121
)t,(u)t,(u
x
E
D
, 0
!
t ,
0
2222
)tl,(u)tl,(u
x
E
D
, 0
!
t .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɞɚɱɚ (7.30) – (7.33) ɫɜɨɞɢɬɫɹ ɤ ɪɚɫɫɦɨɬɪɟɧɧɵɦ ɜɵɲɟ
ɡɚɞɚɱɚɦ.
3. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞ-
ɧɨɫɬɢ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. ɍɤɚɡɚɧɧɚɹ ɡɚɞɚɱɚ
ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ
)
tx,(uu
, ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɧɟɨɞɧɨ-
ɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
)tx,(fuau
xxt
2
, lx
0 , 0
!
t , (7.34)
ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ
42
)
x
(
)
x,(u
M
0 , l
x
d
d
0 , (7.35)
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
)t()t,(u)t,(u
x 111
00
J
E
D
, 0
!
t , (7.36)
)t()tl,(u)tl,(u
x 222
J
E
D
, 0
!
t . (7.37)
Ɂɞɟɫɶ
)
tx,(
f
,
)
x
(
M
, )t(
1
J
)t(
2
J
– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ, ɚ ɤɨɷɮɮɢɰɢ-
ɟɧɬɵ
2
a ,
1
D
,
1
E
,
2
D
,
2
E
ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɬɟɦ ɠɟ ɭɫɥɨɜɢɹɦ, ɱɬɨ ɢ ɜ ɡɚɞɚɱɟ
(7.1) – (7.4).
ɋɜɟɞɟɦ ɡɚɞɚɱɭ (7.34) – (7.37) ɤ ɡɚɞɚɱɟ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫ-
ɥɨɜɢɹɦɢ.
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.34) – (7.37) ɜ ɜɢɞɟ
)
tx,(w
)
tx,(v
)
tx,(u
,
ɝɞɟ
)
t
x,(
v
– ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɚ
)
t
x,(w – ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤ-
ɰɢɹ, ɞɜɚɠɞɵ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɩɨ
x
ɢɨɞɢɧɪɚɡɩɨt ɢ ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ
ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
)t()t,(w)t,(w
x 111
00
J
E
D
, 0
!
t , (7.38)
)t()tl,(w)tl,(w
x 222
J
E
D
,
0
!
t
. (7.39)
ȿɫɥɢ
0
2
2
2
1
!
E
E
, ɬɨ ɮɭɧɤɰɢɸ
)
t
x,(w
ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
)
t(d
x
)
t(c
)
tx,(w
. (7.40)
ɉɨɞɫɬɚɜɥɹɹ (7.40) ɜ (7.38), (7.39), ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɞɜɭɯ ɥɢɧɟɣɧɵɯ ɚɥ-
ɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɜɭɯ ɧɟɢɡɜɟɫɬɧɵɯ ɮɭɧɤɰɢɣ
)
t(c ɢ
)
t
(d , ɪɟɲɢɜ ɤɨɬɨɪɭɸ, ɧɚɣɞɟɦ ɮɭɧɤɰɢɸ
)
t
x,(w .
Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ
1
21
D
D
, 0
21
E
E
. Ɍɨɝɞɚ, ɟɫɥɢ
)t()t()t(
J
J
J
21
, ɬɨ ɮɭɧɤɰɢɸ
)
tx,(w ɦɨɠɧɨ ɡɚɞɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛ-
ɪɚɡɨɦ:
x
)
t
(
)
tx,(w
J
.
ȿɫɥɢ ɠɟ
)t()t(
21
J
J
z
, ɬɨ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɢɫɤɚɬɶ
)
tx,(w ɜ ɜɢɞɟ
x)t(dx)t(c)tx,(w
2
.
ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɹ
)
t
x,(w
ɢɡɜɟɫɬɧɚ.
ɉɨɞɫɬɚɜɥɹɹ ɮɭɧɤɰɢɸ
wvu
ɜ (7.34) – (7.37) ɢ ɭɱɢɬɵɜɚɹ ɪɚɜɟɧɫɬɜɚ
(7.38), (7.39), ɩɨɥɭɱɢɦ ɡɚɞɚɱɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɨɜɨɣ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ
)
tx,(v :
)tx,(fvav
xxt
~
2
, lx
0 , 0
!
t , (7.41)
43
)
x
(
)
x,(v
M
~
0 , l
x
d
d
0 , (7.42)
000
11
)t,(v)t,(v
x
E
D
, 0
!
t , (7.43)
0
22
)tl,(v)tl,(v
x
E
D
, 0
!
t
, (7.44)
ɝɞɟ
)tx,(w)tx,(wa)tx,(f)tx,(f
txx
2
~
,
)
x,(w
)
x
(
)
x
( 0
M
M
~
.
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜɢɞɚ (7.41) – (7.44) ɪɚɫɫɦɚɬɪɢɜɚɥɨɫɶ ɜ §7, ɩ. 2.
ɉɪɢɦɟɪ 1.
Ɋɟɲɢɬɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ
2
3
2
2
x
txuau
xxt
S
coscos
, 10
x
, 0
!
t , (7.45)
2
2
30
x
)x,(u
S
cos , (7.46)
t)t,(u
x
sin
0 , 21
ttu sin),( . (7.47)
Ɋɟɲɟɧɢɟ
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.45) – (7.47) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
)
t
x,(w
)
t
x,(
v
)
t
x,(u
,
ɝɞɟ
)
tx,(v – ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɚ ɮɭɧɤɰɢɹ
)
tx,(w ɭɞɨɜɥɟɬɜɨɪɹ-
ɟɬ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
t)t,(w
x
sin
0 , 21
t
)
t
,(w sin . (7.48)
Ɏɭɧɤɰɢɸ
)
tx,(w ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
)
t
(dx
)
t
(c
)
t
x,(w
.
ɂɡ (7.48) ɫɥɟɞɭɟɬ, ɱɬɨ
t
)
t(c sin
, 2
)
t(d , ɢ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
2
t
x
)
tx,(w sin .
Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ
)
t
x,(
v
ɹɜɥɹɟɬɫɹ ɪɟ-
ɲɟɧɢɟɦ ɡɚɞɚɱɢ
)tx,(fvav
xxt
~
2
,
10
x
, 0
!
t , (7.49)
)
x
(
)
x,(v
M
~
0 , (7.50)
00
)t,(v
x
, 01
)
t,(v , (7.51)
ɝɞɟ
44
2
3
2
2
x
)tx,(w)tx,(wa)tx,(f)tx,(f
txx
S
cos
~
,
2
30
x
)x,(w)x()x(
S
MM
cos
~
.
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.49) – (7.51) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
)tx,(v)tx,(v)tx,(v
21
,
ɝɞɟ
1
v – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
xxt
vav
1
2
1
, (7.52)
2
30
1
x
)x,(v
S
cos
, (7.53)
00
1
)t,(v
x
, 01
1
)t,(v , (7.54)
ɚ
2
v – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
2
3
2
2
2
2
x
vav
xxt
S
cos
, (7.55)
00
2
)x,(v , (7.56)
00
2
)t,(v
x
, 01
2
)t,(v . (7.57)
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ (7.52) – (7.54). Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚ-
ɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-
Ʌɢɭɜɢɥɥɹ ɹɜɥɹɸɬɫɹ
4
12
22
S
O
)n(
n
,
2
12
x)n(
)x(X
n
S
cos , !,, 21
n .
Ɍɚɤ ɤɚɤ
)x(X
x
)x(
1
3
2
3
S
M
cos
~
,
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ-
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
3
1
M
~
, 0
n
M
~
ɞɥɹ !,, 32
n .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §7, ɩ. 1 ɫɥɟɞɭɟɬ, ɱɬɨ
24
3
22
1
xta
)tx,(v
SS
cosexp
¸
¹
·
¨
©
§
.
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɡɚɞɚɱɭ (7.55) – (7.57).
45
Ɍɚɤ ɤɚɤ
)x(X
x
)tx,(f
2
2
2
3
2
S
cos
~
,
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
)tx,(f
~
ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ-
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
2
2
f
~
, 0
n
f
~
ɞɥɹ 2
z
n .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §7, ɩ. 2 ɫɥɟɞɭɟɬ, ɱɬɨ
2
3
2222
x
)t(T)x(X)t(T)tx,(v
S
cos
,
ɝɞɟ
> @
³
t
df)tIJ(a)t(T
0
22
2
2
WO
~
exp
»
¼
º
«
¬
ª
¸
¹
·
¨
©
§
4
9
1
9
8
22
22
ta
a
S
S
exp
.
ɂɬɚɤ,
2
3
4
9
1
9
8
22
22
2
xta
a
)tx,(v
SS
S
cosexp
»
¼
º
«
¬
ª
¸
¹
·
¨
©
§
,
ɢ
)tx,(w)tx,(v)tx,(v),tx(u
21
¸
¹
·
¨
©
§
24
3
22
xta
SS
cosexp
»
¼
º
«
¬
ª
¸
¹
·
¨
©
§
2
3
4
9
1
9
8
22
22
xta
a
SS
S
cosexp
2
t
x
sin .
ɉɪɢɦɟɪ 2.
Ɋɟɲɢɬɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ
2
2
xuau
xxt
S
cos , 10
x
, 0
!
t , (7.58)
1240
x
)
x,(u
S
cos
, (7.59)
00
)t,(u
x
,
01
)t,(u
x
. (7.60)
Ɋɟɲɟɧɢɟ
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.58) – (7.60) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ (ɫɦ. §7, ɩ. 2)
)tx,(u)tx,(u)tx,(u
21
ɝɞɟ
1
u – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
xxt
uau
1
2
1
, 10
x
, 0
!
t
, (7.61)
46
)x()x,(u
M
0
1
, (7.62)
00
1
)t,(u
x
, 0
1
)tl,(u
x
, (7.63)
124
x
)
x
(
S
M
cos ,
ɚ
2
u – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
)tx,(fuau
xxt
2
2
2
, 10
x
, 0
!
t , (7.64)
00
2
)x,(u , (7.65)
00
2
)t,(u
x
, 0
2
)tl,(u
x
, (7.66)
2
x
)
t
x,(
f
S
cos
.
ɋɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɨɨɬɜɟɬɫɬ-
ɜɭɸɳɟɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɹɜɥɹɸɬɫɹ
2
)ʌn(
n
O
, xn)x(X
n
S
cos
, !,,, 210
n .
Ɍɚɤ ɤɚɤ
)x(X)x(Xx)x(
20
41124
S
M
cos
,
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ-
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
1
0
M
,
4
2
M
,
0
n
M
ɞɥɹ 20,
z
n .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §7, ɩ. 1 ɫɥɟɞɭɟɬ, ɱɬɨ
¦
f
0
2
1
n
nnn
)x(X)tȜa()tx,(u exp
M
x)tʌa(
S
2441
22
cosexp
.
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ (7.64) – (7.66).
Ɍɚɤ ɤɚɤ
)x(X)x(Xx)tx,(f
10
22
S
cos
,
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
)
t
x,(
f
ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ-
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
2
0
f , 1
1
f , 0
n
f ɞɥɹ 2
t
n .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §7, ɩ. 2 ɫɥɟɞɭɟɬ, ɱɬɨ
)x(X)t(T)t(T)x(X)t(T)tx,(u
n
nn
110
0
2
¦
f
,
47
ɝɞɟ
> @
tdfdf)tIJ(a)t(T
tt
2
0
00
0
0
2
0
³³
WWO
exp
,
> @
³
WO
df)tIJ(a)t(T
t
1
0
1
2
1
exp
>
@
)tʌa(
a
22
22
1
1
exp
S
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
t)tx,(u 2
2
>
@
x)tʌa(
a
S
S
cosexp
22
22
1
1
ɢ
t
t
xu 21),(
x)tʌa(
S
244
22
cosexp
+
> @
x)tʌa(
a
S
S
cosexp
22
22
1
1
.
§8. Ɋɟɲɟɧɢɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ
ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ
1. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭ-
ɧɵ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. Ɋɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɡɚɞɚɱɚ
ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ
)
t
x,(uu
, ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɨɞɧɨɪɨɞ-
ɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ
xxtt
uau
2
,
lx
0
, 0
!
t , (8.1)
ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
)
x(
)
x,(u
M
0
,
lx
d
d
0
, (8.2)
)x(xu
t
\
),( 0
, l
x
d
d
0 (8.3)
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
000
11
)t,(u)t,(u
x
E
D
, 0
!
t , (8.4)
0
22
)tl,(u)tl,(u
x
E
D
, 0
!
t . (8.5)
Ɂɞɟɫɶ
0
!
a , 0
!
l ,
1
D
,
1
E
,
2
D
,
2
E
– ɡɚɞɚɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɪɢɱɟɦ
0
2
1
2
1
!
E
D
, 0
2
2
2
2
!
E
D
;
)
x(
M
,
)
x
(
\
– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ.
48
Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (8.1) – (8.5) ɪɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɭɸ ɜɫɩɨɦɨɝɚ-
ɬɟɥɶɧɭɸ ɡɚɞɚɱɭ:
ɇɚɣɬɢ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ
xxtt
UaU
2
, l
x
0 , 0
!
t , (8.6)
ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
000
11
)t,(U)t,(U
x
E
D
, 0
!
t , (8.7)
0
22
)tl,(U)tl,(U
x
E
D
,
0
!
t
(8.8)
ɢ ɩɪɟɞɫɬɚɜɢɦɵɟ ɜ ɜɢɞɟ
)
t
(
T
)
x(
X
)
t
x,U(
(8.9)
ɝɞɟ
)
x
(
X
– ɮɭɧɤɰɢɹ, ɧɟ ɡɚɜɢɫɹɳɚɹ ɨɬ ɩɟɪɟɦɟɧɧɨɣ t ,
)
t(
T
– ɮɭɧɤɰɢɹ, ɧɟ
ɡɚɜɢɫɹɳɚɹ ɨɬ ɩɟɪɟɦɟɧɧɨɣ x .
ɉɨɞɫɬɚɜɥɹɹ (8.9) ɜ (8.6), ɬɚɤ ɠɟ, ɤɚɤ ɢ ɜ §7, ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶ-
ɧɵɟ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɮɭɧɤɰɢɣ
)
t(
T
ɢ
)
x
(
X
:
0
2
c
c
)t(Ta)t(T
O
, (8.10)
0
c
c
)
x(
X
)
x(
X
O
. (8.11)
ɉɨɞɫɬɚɜɥɹɹ (8.9) ɜ (8.7) ɢɜ (8.8), ɩɨɥɭɱɢɦ, ɱɬɨ ɮɭɧɤɰɢɹ
)
x(
X
ɞɨɥɠɧɚ
ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
000
11
c
)(X)(X
E
D
, (8.12)
0
22
c
)l(X)l(X
E
D
. (8.13)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɡɚɞɚɱɟ:
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɜɫɟ ɡɧɚɱɟɧɢɹ ɱɢɫɥɨɜɨɝɨ ɩɚɪɚɦɟɬɪɚ
O
, ɩɪɢ ɤɨɬɨɪɵɯ
ɫɭɳɟɫɬɜɭɸɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
(8.11), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ (8.12), (8.13), ɢ ɧɚɣɬɢ ɷɬɢ
ɪɟɲɟɧɢɹ. ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɜɵɲɟ ɡɚɞɚɱɟ
ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɨ ɧɚɯɨɠɞɟɧɢɢ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɢ ɫɨɛɫɬɜɟɧɧɵɯ
ɮɭɧɤɰɢɣ.
Ɍɚɤ ɠɟ, ɤɚɤ ɢ ɜ §7, ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (8.12),
(8.13) ɨɬɧɨɫɹɬɫɹ ɤ ɨɞɧɨɦɭ ɢɡ ɬɢɩɨɜ I – IV (ɫɦ. §6). Ɍɨɝɞɚ, ɤɚɤ
ɢɡɜɟɫɬɧɨ, ɫɭ-
ɳɟɫɬɜɭɟɬ ɫɱɟɬɧɨɟ ɦɧɨɠɟɫɬɜɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
0
t
n
O
, !,, 21
n , ɢ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ
^ `
)x(X
n
(ɫɦ.
ɉɪɢɥɨɠɟɧɢɟ).
ɉɭɫɬɶ
)t(T
n
– ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (7.11) ɩɪɢ
n
O
O
:
0
2
cc
)t(Ta)t(T
nnn
O
, !,, 21
n . (8.14)
49
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ
0
2
2
2
1
!
E
E
. Ɍɨɝɞɚ ɜɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ
0
!
n
O
,
!,, 21
n
, ɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (8.14) ɢɦɟɟɬ ɜɢɞ
taBtaA)t(T
nnnnn
OO
sincos
, (8.15)
ɝɞɟ
n
A ,
n
B – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ (ɜ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɨɩɭɫɤɚɬɶ
ɫɤɨɛɤɢ ɩɪɢ ɡɚɩɢɫɢ ɚɪɝɭɦɟɧɬɚ ɮɭɧɤɰɢɣ
ta
n
O
cos ɢ ta
n
O
sin ).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɟɲɟɧɢɹɦɢ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɨɣ ɜɵɲɟ ɜɫɩɨɦɨɝɚɬɟɥɶ-
ɧɨɣ ɡɚɞɚɱɢ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɢ
)x(X)t(T)tx,(U
nnn
,
!,, 21
n
.
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.1) – (8.5) ɜ ɜɢɞɟ ɪɹɞɚ
¦¦
f
f
11 n
nn
n
n
)x(X)t(T)tx,(U)tx,(u .
ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɢ
)
x
(
M
ɢ
)
x
(
\
ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɵ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚ-
ɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ:
¦
f
1n
nn
)x(X)x(
MM
,
¦
f
1n
nn
)x(X)x(
\
\
,
ɝɞɟ
xd)x(X)x(
X
n
l
n
n
³
0
2
1
MM
,
xd)x(X)x(
X
n
l
n
n
³
0
2
1
\
\
,
!,, 21
n .
Ɍɨɝɞɚ, ɩɨɬɪɟɛɨɜɚɜ, ɱɬɨɛɵ ɮɭɧɤɰɢɹ
)
t
x,(u ɭɞɨɜɥɟɬɜɨɪɹɥɚ ɧɚɱɚɥɶɧɵɦ
ɭɫɥɨɜɢɹɦ (8.2), (8.3), ɩɨɥɭɱɢɦ ɪɚɜɟɧɫɬɜɚ
¦¦
f
f
11
00
n
nn
n
nn
)x(X)x()x(X)(T)x,(u
MM
,
¦¦
f
f
c
11
00
n
nn
n
nnt
)x(X)x()x(X)(T)x,(u
\\
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɭɸ ɡɚɞɚɱɭ Ʉɨɲɢ ɞɥɹ ɨɩɪɟɞɟɥɟ-
ɧɢɹ ɮɭɧɤɰɢɣ
)t(T
n
:
0
2
c
c
)t(Ta)t(T
nnn
O
, (8.16)
50
nn
)(T
M
0 ,
nn
)(T
\
c
0 ,
!,, 21
n
. (8.17)
ɂɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɟ (8.15) ɞɥɹ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (8.14),
ɧɚɯɨɞɢɦ, ɱɬɨ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (8.16), (8.17) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
ta
a
ta)t(T
n
n
n
nnn
O
O
\
OM
sincos
. (8.18)
ɂɬɚɤ, ɜ ɫɥɭɱɚɟ
0
2
2
2
1
!
E
E
ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.1) – (8.5) ɢɦɟɟɬ ɜɢɞ
¦
f
¸
¸
¹
·
¨
¨
©
§
1n
nn
n
n
nn
)x(Xta
a
ta)tx,(u
O
O
\
OM
sincos . (8.19)
ɉɪɟɞɩɨɥɨɠɢɦ ɬɟɩɟɪɶ, ɱɬɨ
1
21
D
D
, 0
21
E
E
. ɋɨɛɫɬɜɟɧɧɵɦɢ
ɡɧɚɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɹɜɥɹɸɬɫɹ
2
¸
¹
·
¨
©
§
l
n
n
S
O
,
l
xn
xX
n
S
cos)(
, !210 ,,
n ;
ɩɪɢ ɷɬɨɦ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɪɚɜɧɵ
lX
2
0
,
2
2
lX
n
, !,, 21
n .
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.1) – (8.5) ɜ ɜɢɞɟ
¦
f
0n
nn
)x(X)t(T)tx,(u .
Ɇɵ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɢ
)
x
(
M
ɢ
)
x
(
\
ɦɨɝɭɬ ɛɵɬɶ ɪɚɡ-
ɥɨɠɟɧɵ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ
^
`
)x(X
n
:
¦
f
0n
nn
)x(X)x(
MM
,
¦
f
0n
nn
)x(X)x(
\
\
,
ɝɞɟ
³³
ll
xd)x(
l
xd)x(X)x(
X
0
0
0
2
0
0
11
MMM
,
³³
l
n
l
n
n
l
xn
)x(
l
xd)x(X)x(
X
00
2
21
S
MMM
cos ,
!,, 21
n
,
³
l
xd)x(
l
0
0
1
\
\
,
³
l
n
l
xn
)x(
l
0
2
S
\
\
cos , !,, 21
n .