Куликов А.А. (сост.) Смешанные задачи для уравнения теплопроводности и уравнения колебаний
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21
ɑɚɫɬɧɵɦɢ ɫɥɭɱɚɹɦɢ ɡɚɞɚɱɢ (5.1) – (5.3) ɹɜɥɹɸɬɫɹ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɜ
§§3, 4 ɫɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
,)t,x(f
x
)t,x(u
a
t
)t,x(u
w
w
w
w
2
2
2
lx
0 , 0
!
t
,
00
)
,x(u , lx
dd
0 ,
00
0
11
w
w
)t,(u
x
)t,(u
ED
, 0
!
t
, (5.4)
0
22
w
w
)t,l(u
x
)t,l(u
ED
,
0
!
t
ɢ ɫɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ
,)t,x(f
x
)t,x(u
a
t
)t,x(u
w
w
w
w
2
2
2
2
2
l
x
0 , 0
!
t ,
00
)
,x(u ,
0
0
w
w
t
),x(u
, lx
d
d
0 ,
00
0
11
w
w
)t,(u
x
)t,(u
ED
,
0
!
t
, (5.5)
0
22
w
w
)t,l(u
x
)t,l(u
ED
, 0
!
t .
ɉɪɢ ɷɬɨɦ
1
n
,
222
xaA
w
w
,
)
l,( 0
:
, ɝɪɚɧɢɰɚ
*
ɫɨɫɬɨɢɬ ɢɡ
ɞɜɭɯ ɭɱɚɫɬɤɨɜ: ɬɨɱɟɤ
0
x
ɢ l
x
; 2
s
, 1
21
rr . ȼ ɫɥɭɱɚɟ ɡɚɞɚɱɢ
(5.4)
1
m , ɚ ɜ ɫɥɭɱɚɟ ɡɚɞɚɱɢ (5.5) 2
m .
Ɍɟɨɪɟɦɚ (ɩɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ). ɉɭɫɬɶ ɩɪɢ ɤɚɠɞɨɦ ɮɢɤɫɢɪɨɜɚɧɧɨɦ
ɡɧɚɱɟɧɢɢ ɩɚɪɚɦɟɬɪɚ
W
,
T
W
0 , ɫɭɳɟɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɤɥɚɫɫɢɱɟ-
ɫɤɨɟ ɪɟɲɟɧɢɟ
)
;
t
,x(
v
v
W
ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ
0
w
w
vA
t
v
m
m
,
T
Q)t,x(
, (5.6)
00
)
;,
x
(v
W
, 0
0
w
w
t
);,x(v
W
, ! , 0
0
2
2
w
w
m
m
t
);,x(v
W
,
),x(f
t
);,x(v
m
m
W
W
w
w
1
1
0
,
:
x
, (5.7)
0
);t,x(v)D,x(l
xj,i
W
,
i
x
*
,
T
x
0 , (5.8)
22
s
i
,, !1
,
i
rj ,, !1
.
Ɍɨɝɞɚ ɡɚɞɚɱɚ (5.1) – (5.3) ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɤɥɚɫɫɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ
WWW
d);t,x(v)t,x(u
t
³
0
. (5.9)
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ
ɉɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɩɨɦɧɢɦ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ ɨ ɞɢɮɮɟɪɟɧɰɢɪɨ-
ɜɚɧɢɢ ɫɨɛɫɬɜɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ, ɡɚɜɢɫɹɳɟɝɨ ɨɬ ɩɚɪɚɦɟɬɪɚ (ɫɦ., ɧɚɩɪɢɦɟɪ, [1,
§53]):
ɉɭɫɬɶ ɮɭɧɤɰɢɹ
)
,t(
g
W
ɢ ɟɟ ɱɚɫɬɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ),t(g
t
W
ɧɟɩɪɟɪɵɜ-
ɧɵ ɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ
> @
>
@
2211
b,ab,a
u
, ɮɭɧɤɰɢɢ
)
t
(
[
ɢ
)
t
(
K
ɞɢɮɮɟɪɟɧ-
ɰɢɪɭɟɦɵ ɧɚ ɨɬɪɟɡɤɟ
> @
11
b,a ɢ
22
b)t()t(a
d
d
d
K
[
ɞɥɹ ɜɫɟɯ
> @
11
b,at
.
Ɍɨɝɞɚ ɮɭɧɤɰɢɹ
WW
K
[
d),t(g)t(I
)t(
)t(
³
ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɧɚ ɨɬɪɟɡɤɟ
>
@
11
b,a ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɚɹ ɪɚɜɧɚ
)t())t(,t(g)t())t(,t(gd),t(g)t(I
)t(
)t(
t
[[KKWW
K
[
c
c
c
³
.
ɂɫɩɨɥɶɡɭɹ ɭɤɚɡɚɧɧɨɟ ɩɪɚɜɢɥɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɫɨɛɫɬɜɟɧɧɨɝɨ ɢɧ-
ɬɟɝɪɚɥɚ, ɡɚɜɢɫɹɳɟɝɨ ɨɬ ɩɚɪɚɦɟɬɪɚ, ɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (5.7), ɩɨɥɭɱɚɟɦ
ɫɥɟɞɭɸɳɢɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɩɨ ɩɟɪɟɦɟɧɧɨɣ t ɮɭɧɤ-
ɰɢɢ
)
t,
x
(u , ɨɩɪɟɞɟɥɹɟɦɨɣ ɮɨɪɦɭɥɨɣ (5.9):
W
WW
P
P
P
P
d
t
);t,x(v
t
)t,x(u
t
³
w
w
w
w
0
, 11 -m,, !
P
,
w
w
w
w
w
w
³
1
1
0
0
m
m
t
m
m
m
m
t
)t;,x(v
d
t
);t,x(v
t
)t,x(u
W
WW
(5.10)
)t,x(fd
t
);t,x(v
t
m
m
w
w
³
W
WW
0
.
ɂɡ (5.6), (5.9), (5.10) ɫɥɟɞɭɟɬ, ɱɬɨ
w
w
)t,x(uA
t
)t,x(u
m
m
23
)t,x(f)t,x(fd);,x(vA
);,x(v
t
t
m
m
»
¼
º
«
¬
ª
w
w
³
WWT
T
WT
WT
0
,
ɬɨ ɟɫɬɶ ɮɭɧɤɰɢɹ
)
t
,x(u ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ (5.1). Ɉɱɟɜɢɞɧɨ ɬɚɤɠɟ,
ɱɬɨ ɷɬɚ ɮɭɧɤɰɢɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɧɚɱɚɥɶɧɵɦ ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ (5.2),
(5.3).
Ⱦɨɤɚɠɟɦ ɬɟɩɟɪɶ ɟɞɢɧɫɬɜɟɧɧɨɫɬɶ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (5.1) – (5.3). ɉɪɟɞɩɨ-
ɥɨɠɢɦ, ɱɬɨ ɷɬɚ ɡɚɞɚɱɚ ɢɦɟɟɬ ɟɳɟ ɨɞɧɨ ɪɟɲɟɧɢɟ
)t,x(u
1
, ɢ ɪɚɫɫɦɨɬɪɢɦ
ɮɭɧɤɰɢɸ
)t,x(u)t,x(u)t,x(v
~
1
.
Ɏɭɧɤɰɢɹ
)
t,
x
(v
~
ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ
0
w
w
)t,x(v
~
A
t
)t,x(v
~
m
m
,
T
Q)t,x(
, (5.11)
00
)
,x(
v
~
, 0
0
w
w
t
),x(v
~
,
!
, 0
0
1
1
w
w
m
m
t
),x(v
~
, (5.12)
:
x
,
0
)t,x(v
~
)D,x(l
xj,i
,
i
x
*
,
T
x
0 , (5.13)
s
i
,, !1
,
i
rj ,, !1
,
ɤɨɬɨɪɭɸ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɡɚɞɚɱɢ (5.6) – (5.8) ɩɪɢ
0
{
)
t,
x
(
f
. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (5.11) – (5.13) ɹɜɥɹɟɬɫɹ
ɮɭɧɤɰɢɹ
0
{
)
t
,x(
v
~
, ɢ ɷɬɨ ɪɟɲɟɧɢɟ ɟɞɢɧɫɬɜɟɧɧɨ ɜ ɫɢɥɭ ɟɞɢɧɫɬɜɟɧɧɨɫɬɢ
ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (5.6) – (5.8). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
)t,x(u)t,x(u
1
{
, ɱɬɨ ɢ ɬɪɟ-
ɛɨɜɚɥɨɫɶ ɩɨɤɚɡɚɬɶ. Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
Ɂɚɦɟɱɚɧɢɟ. ɂɡ ɩɪɢɜɟɞɟɧɧɨɝɨ ɜɵɲɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɫɥɟɞɭɟɬ, ɱɬɨ
ɟɫɥɢ ɩɪɢ ɤɚɠɞɨɦ ɡɧɚɱɟɧɢɢ ɩɚɪɚɦɟɬɪɚ
W
,
T
W
0
, ɫɭɳɟɫɬɜɭɟɬ ɟɞɢɧɫɬ-
ɜɟɧɧɨɟ ɤɥɚɫɫɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ
)
;
t
,x(
v
v
W
ɡɚɞɚɱɢ Ʉɨɲɢ
0
w
w
wA
t
w
m
m
,
n
R
x
,
T
t
0 , (5.6)
00
)
IJ
;,
x
(w ,
0
0
w
w
t
)IJ;,x(w
, ! , 0
0
2
2
w
w
m
m
t
)IJ;,x(w
,
)IJx,(f
t
)IJ;,x(w
m
m
w
w
1
1
0
,
n
R
x
, (5.7)
ɬɨ ɡɚɞɚɱɚ Ʉɨɲɢ
24
)t,x(f)t,x(uA
t
)t,x(u
m
m
w
w
,
n
R
x
,
T
t
0 ,
00
)
,
x
(u , 0
0
w
w
t
),x(u
, ! , 0
0
1
1
w
w
m
m
t
),x(u
,
n
R
x
,
ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɤɥɚɫɫɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ
IJd)IJ;IJt,x(w)t,x(u
t
³
0
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ ɩɨɡɜɨɥɹɟɬ ɫɜɨɞɢɬɶ ɪɟɲɟɧɢɟ ɢɫɯɨɞ-
ɧɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ
ɨɞɧɨɪɨɞɧɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɤ ɪɟɲɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɡɚɞɚɱɢ
ɞɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ, ɤɨ-
ɬɨɪɚɹ ɨɛɵɱɧɨ ɩɪɨɳɟ, ɱɟɦ ɢɫɯɨɞɧɚɹ ɡɚɞɚɱɚ.
§6. Ɂɚɞɚɱɚ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ
Ɂɚɞɚɱɚ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɜɨɡɧɢɤɚɟɬ ɩɪɢ ɪɟɲɟɧɢɢ ɦɧɨɝɢɯ ɡɚɞɚɱ ɦɚɬɟ-
ɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɜ ɬɨɦ ɱɢɫɥɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨ-
ɩɪɨɜɨɞɧɨɫɬɢ ɢ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ. ȼ ɧɚɫɬɨɹɳɟɦ ɩɚɪɚɝɪɚɮɟ ɦɵ ɨɝɪɚɧɢ-
ɱɢɦɫɹ ɪɚɫɫɦɨɬɪɟɧɢɟɦ ɜɚɠɧɨɝɨ ɱɚɫɬɧɨɝɨ ɫɥɭɱɚɹ ɷɬɨɣ ɡɚɞɚɱɢ, ɩɪɟɞɫɬɚɜɥɹɸ-
ɳɟɝɨ ɧɚɢɛɨɥɶɲɢɣ ɢɧɬɟɪɟɫ ɞɥɹ ɩɪɢɥɨɠɟɧɢɣ. ɋɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɡɚɞɚɱɚ ɮɨɪ-
ɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɜɫɟ ɡɧɚɱɟɧɢɹ ɱɢɫɥɨɜɨɝɨ ɩɚɪɚɦɟɬɪɚ
O
, ɩɪɢ ɤɨɬɨɪɵɯ
ɫɭɳɟɫɬɜɭɸɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ (ɬɨ ɟɫɬɶ ɨɬɥɢɱɧɵɟ ɨɬ ɬɨɠɞɟɫɬɜɟɧɧɨɝɨ ɧɭɥɹ)
ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
0
cc
)
x(
X
)
x(
X
O
,
lx
0
, (6.1)
ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
000
11
c
)(X)(X
E
D
, (6.2)
0
22
c
)l(X)l(X
E
D
, (6.3)
ɢ ɧɚɣɬɢ ɷɬɢ ɪɟɲɟɧɢɹ.
Ɂɞɟɫɶ
l,
1
D
,
1
E
,
2
D
,
2
E
– ɡɚɞɚɧɧɵɟ ɜɟɳɟɫɬɜɟɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɪɢɱɟɦ
0
2
1
2
1
!
E
D
, 0
2
2
2
2
!
E
D
.
Ɍɟ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɚ
O
, ɩɪɢ ɤɨɬɨɪɵɯ ɡɚɞɚɱɚ (6.1) – (6.3) ɢɦɟɟɬ ɧɟ-
ɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ, ɧɚɡɵɜɚɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ, ɚ ɨɬɜɟɱɚɸ-
ɳɢɟ ɢɦ ɪɟɲɟɧɢɹ – ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ.
25
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɢɥɭ ɨɞɧɨɪɨɞɧɨɫɬɢ ɭɪɚɜɧɟɧɢɹ (6.1) ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨ-
ɜɢɣ (6.2), (6.3), ɤɚɠɞɚɹ ɢɡ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɬɨɱɧɨɫɬɶɸ
ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ, ɨɬɥɢɱɧɨɝɨ ɨɬ ɧɭɥɹ (ɬɨ ɟɫɬɶ ɟɫɥɢ
)
x(
X
– ɫɨɛ-
ɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ, ɬɨ ɮɭɧɤɰɢɹ
)
x
(
X
c , ɝɞɟ 0
z
cons
t
c , ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ
ɫɨɛɫɬɜɟɧɧɨɣ).
ɋɜɨɣɫɬɜɚ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɢ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ
1. Ʉɚɠɞɨɦɭ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ (ɫ ɬɨɱɧɨɫɬɶɸ
ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ, ɨɬɥɢɱɧɨɝɨ ɨɬ ɧɭɥɹ) ɬɨɥɶɤɨ ɨɞɧɚ ɫɨɛɫɬɜɟɧɧɚɹ
ɮɭɧɤɰɢɹ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ
ɉɪɟɞɩɨɥɨɠɢɦ ɩɪɨɬɢɜɧɨɟ: ɩɭɫɬɶ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ
O
ɨɬɜɟɱɚɸɬ
ɞɜɚ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵɯ ɪɟɲɟɧɢɹ
)x(X
1
ɢ
)x(X
2
ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ (6.1)
– (6.3). Ɍɨɝɞɚ ɮɭɧɤɰɢɹ
)x(Xc)x(Xc)x(X
2211
,
ɝɞɟ
1
c
ɢ
2
c
– ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɹɜɥɹɟɬɫɹ ɨɛɳɢɦ ɪɟɲɟɧɢɟɦ ɞɢɮ-
ɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (6.1), ɩɪɢɱɟɦ
)
x(
X
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɝɪɚɧɢɱɧɵɦ
ɭɫɥɨɜɢɹɦ (6.2), (6.3). ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɥɸɛɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.1)
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ (6.2), (6.3). Ɉɞɧɚɤɨ ɫɪɟɞɢ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ (6.1)
ɡɚɜɟɞɨɦɨ ɟɫɬɶ ɪɟɲɟɧɢɹ, ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɷɬɢɦ ɭɫɥɨɜɢɹɦ. ɇɚɩɪɢɦɟɪ,
ɪɟɲɟɧɢɟ
)
x
(
Y
ɭɪɚɜɧɟɧɢɹ (6.1), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
11
00
D
E
c
)(Y)(Y , ,
ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɝɪɚɧɢɱɧɨɦɭ ɭɫɥɨɜɢɸ (6.2), ɬɚɤ ɤɚɤ
.000
2
1
2
111
!
c
E
D
E
D
)(Y)(Y
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɢɥɢ ɩɪɨɬɢɜɨɪɟɱɢɟ, ɱɬɨ ɢ ɞɨɤɚɡɵɜɚɟɬ ɫɜɨɣɫɬ-
ɜɨ 1.
2. ɉɭɫɬɶ
P
ɢ
Q
– ɪɚɡɥɢɱɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɢ
)
x(
X
ɢ
)
x(
Y
– ɨɬɜɟɱɚɸɳɢɟ ɢɦ ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ. Ɍɨɝɞɚ
.0
0
³
xd)x(Y)x(X
l
(6.4)
26
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ
ɉɨ ɭɫɥɨɜɢɸ, ɮɭɧɤɰɢɢ
)
x
(
X
ɢ
)
x
(
Y
ɹɜɥɹɸɬɫɹ ɧɟɬɪɢɜɢɚɥɶɧɵɦɢ ɪɟ-
ɲɟɧɢɹɦɢ ɤɪɚɟɜɵɯ ɡɚɞɚɱ
0
cc
)
x
(
X
)
x
(
X
P
, l
x
0 , (6.5)
000
11
c
)(X)(X
E
D
, (6.6)
0
22
c
)l(X)l(X
E
D
, (6.7)
ɢ
0
cc
)
x(
Y
)
x(
Y
Q
, lx
0 , )'6.5(
000
11
c
)(Y)(Y
E
D
, )'6.6(
0
22
c
)l(Y)l(Y
E
D
. )'6.7(
ɍɦɧɨɠɚɹ ɭɪɚɜɧɟɧɢɟ (6.5) ɧɚ ɮɭɧɤɰɢɸ
)
x(
Y
, ɭɪɚɜɧɟɧɢɟ )'6.5( ɧɚ
ɮɭɧɤɰɢɸ
)
x
(
X
ɢ ɜɵɱɢɬɚɹ ɢɡ ɨɞɧɨɝɨ ɩɨɥɭɱɢɜɲɟɝɨɫɹ ɪɚɜɟɧɫɬɜɚ ɞɪɭɝɨɟ,
ɢɦɟɟɦ:
.0
c
c
cc
)
x(
Y
)
x(
X
)
Ȟ
ȝ
(
)
x(
Y
)
x(
X
)
x(
X
)
xY(
(6.8)
Ɋɚɜɟɧɫɬɜɨ (6.8) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
> @
.0
c
c
)x(Y)x(X)Ȟȝ()x(Y)x(X)x(X)x(Y
dx
d
(6.9)
ɂɧɬɟɝɪɢɪɭɹ ɪɚɜɟɧɫɬɜɨ (6.9) ɩɨ ɨɬɪɟɡɤɭ
],[ l0 , ɧɚɯɨɞɢɦ, ɱɬɨ
c
c
³
)l(Y)l(X)x(X)l(Yxd)x(Y)x(X)(
l
0
PQ
)
(
Y
)
(
X
)
(
X
)
(
Y
0000
c
c
. (6.10)
ɍɩɪɚɠɧɟɧɢɟ 1
. ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɹ (6.6), (6.7), )'6.6(, )'6.7(, ɚ ɬɚɤɠɟ
ɭɫɥɨɜɢɹ
0
2
1
2
1
!
E
D
, 0
2
2
2
2
!
E
D
, ɩɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɚɜɚɹ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ
(6.10) ɪɚɜɧɚ ɧɭɥɸ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɢɥɢ, ɱɬɨ
0
0
³
l
xd)x(Y)x(X)(
PQ
.
Ɍɚɤ ɤɚɤ
Q
P
z
, ɬɨ ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ ɪɚɜɟɧɫɬɜɨ (6.4). ɋɜɨɣɫɬɜɨ 2 ɞɨɤɚɡɚ-
ɧɨ.
3.
ȼɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜɟɳɟɫɬɜɟɧɧɵ.
ɍɩɪɚɠɧɟɧɢɟ 2
. Ⱦɨɤɚɡɚɬɶ ɫɜɨɣɫɬɜɨ 3.
27
ɍɤɚɡɚɧɢɟ
. ɉɪɟɞɩɨɥɨɠɢɦ ɩɪɨɬɢɜɧɨɟ: ɩɭɫɬɶ ɫɭɳɟɫɬɜɭɟɬ ɤɨɦɩɥɟɤɫɧɨɟ
ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ
K
[
O
i
, ɝɞɟ
R
K
[
, , 0
z
K
, i – ɦɧɢɦɚɹ ɟɞɢɧɢɰɚ,
ɢ ɟɦɭ ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ
)
x(
X
. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɱɢɫɥɨ
K
[
O
i
ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦ, ɢ ɟɦɭ ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ
ɮɭɧɤɰɢɹ
)x(X (ɮɭɧɤɰɢɹ, ɤɨɦɩɥɟɤɫɧɨ ɫɨɩɪɹɠɟɧɧɚɹ ɤ
)
x(
X
). Ⱦɥɹ ɡɚɜɟɪ-
ɲɟɧɢɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɫɥɟɞɭɟɬ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɫɜɨɣɫɬɜɨɦ 2 (ɩɪɢ ɷɬɨɦ ɭɱɢ-
ɬɵɜɚɟɬɫɹ, ɱɬɨ
O
O
z
).
ɋɥɟɞɫɬɜɢɟ.
Ʌɸɛɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨɫɬɨ-
ɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ, ɨɬɥɢɱɧɨɝɨ ɨɬ ɧɭɥɹ, ɫɨɜɩɚɞɚɟɬ ɫ ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɨɣ
ɮɭɧɤɰɢɟɣ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ
)
x
(
Z
i
)
x
(
Y
)
x
(
X
, ɝɞɟ
)
x
(
Y
ɢ
)
x
(
Z
– ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɵɟ ɮɭɧɤ-
ɰɢɢ, ɨɬɥɢɱɧɵɟ ɨɬ ɬɨɠɞɟɫɬɜɟɧɧɨɝɨ ɧɭɥɹ, ɢ ɩɭɫɬɶ ɮɭɧɤɰɢɹ
)
x(
X
ɨɬɜɟɱɚɟɬ
ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ
O
. ɉɨɞɫɬɚɜɥɹɹ ɜ ɪɚɜɟɧɫɬɜɚ (6.1) – (6.3) ɜɦɟɫɬɨ
)
x
(
X
ɫɭɦɦɭ
)
x
(
Z
i
)
x
(
Y
ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ ɱɢɫɥɨ
O
ɜɟɳɟɫɬɜɟɧɧɨ, ɥɟɝɤɨ
ɜɢɞɟɬɶ, ɱɬɨ
)
x(
Y
ɢ
)
x(
Z
ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ, ɨɬɜɟɱɚɸ-
ɳɢɦɢ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ
O
. ɇɨ ɬɨɝɞɚ, ɜ ɫɢɥɭ ɫɜɨɣɫɬɜɚ 1,
)
x(
Y
c
)
x(
Z
, ɝɞɟ 0
z
c – ɜɟɳɟɫɬɜɟɧɧɚɹ ɩɨɫɬɨɹɧɧɚɹ (ɱɢɫɥɨ c ɜɟɳɟɫɬ-
ɜɟɧɧɨ, ɬɚɤ ɤɚɤ ɮɭɧɤɰɢɢ
)
x
(
Y
ɢ
)
x
(
Z
ɩɪɢɧɢɦɚɸɬ ɜɟɳɟɫɬɜɟɧɧɵɟ ɡɧɚɱɟ-
ɧɢɹ).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
)
x(c)
Y
i
(
)
x(
Z
i
)
x(
Y
)
x(
X
1
, ɬɨ ɟɫɬɶ
ɮɭɧɤɰɢɹ
)
x
(
X
ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ c
i
1 ɫɨɜɩɚɞɚɟɬ ɫ
ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɨɣ ɮɭɧɤɰɢɟɣ
)
x(
Y
. ɋɥɟɞɫɬɜɢɟ ɞɨɤɚɡɚɧɨ.
ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɵɟ
ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ.
ɋɥɟɞɭɸɳɟɟ ɫɜɨɣɫɬɜɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɢ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ
ɩɪɢɜɟɞɟɦ ɛɟɡ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ.
4.
ɋɭɳɟɫɬɜɭɟɬ ɫɱɟɬɧɨɟ ɦɧɨɠɟɫɬɜɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
n
O
,
,,, !21
n
ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɢɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɨɛɫɬɜɟɧɧɵɯ
ɮɭɧɤɰɢɣ
^ `
)x(X
n
. ȼɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɨɠɧɨ ɡɚɧɭɦɟɪɨɜɚɬɶ ɜ ɩɨ-
ɪɹɞɤɟ ɜɨɡɪɚɫɬɚɧɢɹ, ɬɨ ɟɫɬɶ
!!
121 nn
O
O
O
O
,
ɩɪɢɱɟɦ
fo
n
O
ɩɪɢ
fo
n .
ɂɫɩɨɥɶɡɭɹ ɫɜɨɣɫɬɜɨ 4 ɢ ɫɥɟɞɫɬɜɢɟ ɢɡ ɫɜɨɣɫɬɜɚ 3, ɫɜɨɣɫɬɜɨ 2 ɦɨɠɧɨ
ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
28
2'. ɉɭɫɬɶ
n
O
ɢ
m
O
,
mn
z
– ɪɚɡɥɢɱɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɡɚɞɚ-
ɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɢ
)x(X
n
ɢ )x(X
m
– ɨɬɜɟɱɚɸɳɢɟ ɢɦ ɫɨɛɫɬɜɟɧɧɵɟ
ɮɭɧɤɰɢɢ. Ɍɨɝɞɚ ɮɭɧɤɰɢɢ
)x(X
n
ɢ )x(X
m
ɨɪɬɨɝɨɧɚɥɶɧɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ
)l,(L 0
2
(ɫɦ. §1, ɩ. 3), ɬɨ ɟɫɬɶ
³
l
mn
xd)x(X)x(X
0
0 .
Ɇɨɠɧɨ ɞɚɬɶ ɢ ɛɨɥɟɟ ɤɨɪɨɬɤɭɸ ɮɨɪɦɭɥɢɪɨɜɤɭ ɷɬɨɝɨ ɫɜɨɣɫɬɜɚ:
2". ɋɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ, ɨɬɜɟɱɚɸɳɢɟ
ɪɚɡɥɢɱɧɵɦ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ, ɨɪɬɨɝɨɧɚɥɶɧɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ
)l,(L 0
2
.
ɋɜɨɣɫɬɜɨ 2 ɧɚɡɵɜɚɸɬ ɨɛɵɱɧɨ ɫɜɨɣɫɬɜɨɦ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɫɨɛɫɬɜɟɧ-
ɧɵɯ ɮɭɧɤɰɢɣ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ.
ɉɪɢɜɟɞɟɦ ɬɟɩɟɪɶ ɛɟɡ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɨɞɧɨ ɢɡ ɨɫɧɨɜɧɵɯ ɫɜɨɣɫɬɜ ɫɨɛɫɬ-
ɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ.
5. (Ɍɟɨɪɟɦɚ ȼ. Ⱥ. ɋɬɟɤɥɨɜɚ).
ȿɫɥɢ ɮɭɧɤɰɢɹ
)
x
(
f
ɞɜɚɠɞɵ ɧɟɩɪɟɪɵɜɧɨ
ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɧɚ ɨɬɪɟɡɤɟ
],[ l0 ɢ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɞɧɨɪɨɞɧɵɦ ɝɪɚɧɢɱ-
ɧɵɦ ɭɫɥɨɜɢɹɦ
000
11
c
)(f)(f
E
D
,
0
22
c
)l(f)l(f
E
D
,
ɬɨ ɨɧɚ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɚ ɜ ɚɛɫɨɥɸɬɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɫɯɨɞɹɳɢɣɫɹ ɧɚ
ɨɬɪɟɡɤɟ
],[ l0 ɪɹɞ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ:
¦
f
1n
nn
)x(Xf)x(f . (6.11)
Ɋɚɡɥɨɠɟɧɢɟ (6.11) ɧɚɡɵɜɚɟɬɫɹ
ɪɹɞɨɦ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
)
x
(
f
ɩɨ ɫɢɫɬɟ-
ɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ
^ `
)x(X
n
, ɚ ɱɢɫɥɚ
n
f , !,, 21
n – ɤɨɷɮɮɢɰɢɟɧ-
ɬɚɦɢ Ɏɭɪɶɟ
ɷɬɨɣ ɮɭɧɤɰɢɢ.
Ʉɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
)
x
(
f
ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ. ɍɦɧɨɠɢɦ ɪɚɜɟɧɫɬɜɨ (6.11) ɧɚ ɮɭɧɤɰɢɸ
)x(X
m
, ɝɞɟ ɧɨɦɟɪ m
(
1
t
m ) ɮɢɤɫɢɪɨɜɚɧ, ɢ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ ɩɨ ɨɬɪɟɡɤɭ
],[ l0 . Ɍɨɝɞɚ, ɢɫɩɨɥɶɡɭɹ ɫɜɨɣɫɬɜɨ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ,
ɧɚɯɨɞɢɦ, ɱɬɨ
2
0
2
0
mm
l
mm
l
m
Xfxd)x(Xfxd)x(X)x(f
³³
,
!,, 21
m
.
29
Ɂɚɦɟɧɹɹ ɢɧɞɟɤɫ
m ɧɚ n , ɩɨɥɭɱɢɦ, ɱɬɨ
xd)x(X)x(f
X
f
l
n
n
n
³
0
2
1
, !,, 21
n . (6.12)
Ʌɟɦɦɚ.
ɉɭɫɬɶ
)
x(
X
– ɩɪɨɢɡɜɨɥɶɧɨɟ ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɨɟ ɪɟɲɟɧɢɟ
ɭɪɚɜɧɟɧɢɹ (6.1). Ɍɨɝɞɚ ɢɦɟɟɬ ɦɟɫɬɨ ɪɚɜɟɧɫɬɜɨ
³
c
c
c
l
xd)x(X)(X)(X)l(X)l(XX
0
2
2
00
O
. (6.13)
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɹ
)
x(
X
ɨɬɥɢɱɧɚ ɨɬ ɬɨɠɞɟɫɬɜɟɧɧɨɝɨ ɧɭɥɹ
(ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɪɚɜɟɧɫɬɜɨ (6.13) ɨɱɟɜɢɞɧɨ). ɍɦɧɨɠɚɹ ɭɪɚɜɧɟɧɢɟ (6.1)
ɧɚ
)
x(
X
ɢ ɢɧɬɟɝɪɢɪɭɹ ɩɨɥɭɱɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ ɩɨ ɨɬɪɟɡɤɭ ],[ l0 , ɢɦɟɟɦ:
0
0
2
0
cc
³³
xd)x(Xxd)x(X)x(X
ll
O
.
ɉɪɨɢɡɜɨɞɹ ɜ ɩɟɪɜɨɦ ɫɥɚɝɚɟɦɨɦ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɢɧɬɟɝ-
ɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ, ɩɨɥɭɱɢɦ ɪɚɜɟɧɫɬɜɨ (6.13). Ʌɟɦɦɚ ɞɨɤɚɡɚɧɚ.
ɋɥɟɞɫɬɜɢɟ.
ȿɫɥɢ ɥɸɛɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ
)
x(
X
ɡɚɞɚɱɢ
ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɸ
000
t
c
c
)
(
X
)
(
X
)
l(
X
)
l(
X
, (6.14)
ɬɨ ɥɸɛɨɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ
O
ɧɟɨɬɪɢɰɚɬɟɥɶɧɨ.
Ⱦɚɧɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜɵɬɟɤɚɟɬ ɢɡ ɪɚɜɟɧɫɬɜɚ (6.13).
ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
(6.2), (6.3) ɫɩɟɰɢɚɥɶɧɨɝɨ ɜɢɞɚ (ɬɨ ɟɫɬɶ ɧɚɤɥɚɞɵɜɚɬɶ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɨɝɪɚ-
ɧɢɱɟɧɢɹ ɧɚ ɱɢɫɥɚ
1
D
,
1
E
,
2
D
,
2
E
). Ⱥ ɢɦɟɧɧɨ, ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɥɟ-
ɞɭɸɳɢɟ ɬɢɩɵ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ:
I.
00
)
(
X
,
0
)
l(
X
;
II.
000
1
c
)(Xh)(X
, 0
)
l(
X
;
III.
00
)
(
X
, 0
2
c
)l(Xh)l(X ;
IV.
000
1
c
)(Xh)(X , 0
2
c
)l(Xh)l(X ,
ɝɞɟ
0
1
t
h
ɢ
0
2
t
h
– ɡɚɞɚɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ.
Ʉ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɢɦɟɧɧɨ ɬɚɤɨɝɨ
ɜɢɞɚ ɩɪɢɜɨɞɢɬ ɪɟɲɟɧɢɟ ɦɧɨɝɢɯ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɢ, ɜ ɱɚɫɬɧɨ-
ɫɬɢ, ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ, ɨɩɢɫɵɜɚɸɳɢɯ ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ ɜ
30
ɫɬɟɪɠɧɟ, ɢ ɡɚɞɚɱ, ɨɩɢɫɵɜɚɸɳɢɯ ɩɪɨɰɟɫɫɵ ɦɚɥɵɯ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ
ɫɬɪɭɧɵ ɢ ɩɪɨɞɨɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɬɟɪɠɧɹ (ɫɦ. §§3, 4).
Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɜ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɜɢɞɚ I – IV ɧɟɪɚɜɟɧɫɬɜɨ
(6.14) ɜɵɩɨɥɧɟɧɨ (ɩɪɨɜɟɪɢɬɶ ɷɬɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ). ɉɨɷɬɨɦɭ, ɜ ɫɢɥɭ ɫɥɟɞɫɬ-
ɜɢɹ ɢɡ ɩɪɢɜɟɞɟɧɧɨɣ ɜɵɲɟ ɥɟɦɦɵ, ɥɸɛɨɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɡɚɞɚɱɢ
ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɭɤɚɡɚɧɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨ.
Ⱦɚɥɟɟ, ɢɡ (6.13) ɢ
(6.14) ɫɥɟɞɭɟɬ, ɱɬɨ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ
O
ɪɚɜɧɨ ɧɭɥɸ
ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ
000
c
c
)
(
X
)
(
X
)
l(
X
)
l(
X
(6.15)
ɢ
0
0
2
c
³
l
xd)x(X . (6.16)
ɂɡ (6.16) ɜɵɬɟɤɚɟɬ, ɱɬɨ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ
)
x
(
X
, ɨɬɜɟɱɚɸɳɚɹ ɫɨɛ-
ɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ
0
O
, ɹɜɥɹɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ: 0
z{
cons
t
)
x
(
X
; ɩɪɢ
ɷɬɨɦ ɭɫɥɨɜɢɟ (6.15) ɜɵɩɨɥɧɹɟɬɫɹ ɚɜɬɨɦɚɬɢɱɟɫɤɢ. Ɂɚɦɟɬɢɦ, ɧɚɤɨɧɟɰ, ɱɬɨ
ɫɪɟɞɢ ɭɫɥɨɜɢɣ I – IV ɢɦɟɟɬɫɹ ɬɨɥɶɤɨ ɨɞɢɧ ɬɢɩ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ, ɤɨɬɨ-
ɪɵɦ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɮɭɧɤɰɢɹ
0
z
{
cons
t
)
x
(
X
, ɚ ɢɦɟɧɧɨ
00
c
)
(
X
, 0
c
)
l(
X
(6.17)
(ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɭɫɥɨɜɢɣ IV ɩɪɢ
0
21
hh ).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɩɪɚɜɟɞɥɢɜɨ ɫɥɟɞɭɸɳɟɟ ɫɜɨɣɫɬɜɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟ-
ɧɢɣ ɢ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ.
6.
ȿɫɥɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɨɬɧɨɫɹɬɫɹ ɤ
ɨɞɧɨɦɭ ɢɡ ɬɢɩɨɜ I – IV, ɬɨ ɜɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ.
ɇɚɢɦɟɧɶɲɟɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɪɚɜɧɨ ɧɭɥɸ ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨ-
ɝɞɚ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɢɦɟɸɬ ɜɢɞ (6.17). ɇɭɥɟɜɨɦɭ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ
ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ
0
z
{
const
)
x(
X
.
ɋɥɟɞɫɬɜɢɟ.
ȼ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɜɢɞɚ I – IV, ɨɬɥɢɱɧɵɯ ɨɬ
ɭɫɥɨɜɢɣ (6.17), ɜɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɩɨɥɨɠɢɬɟɥɶɧɵ.
ɇɚɣɞɟɦ ɬɟɩɟɪɶ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɡɚɞɚɱɢ
(6.1) – (6.3) ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɜɢɞɚ I – IV. ɉɪɢ ɷɬɨɦ ɦɵ ɛɭɞɟɦ ɪɚɡ-
ɥɢɱɚɬɶ ɫɥɭɱɚɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ II – IV, ɤɨɝɞɚ ɩɨɫɬɨɹɧɧɵɟ
1
h ɢ (ɢɥɢ)
2
h
ɪɚɜɧɵ ɧɭɥɸ, ɢ ɤɨɝɞɚ ɨɧɢ ɩɨɥɨɠɢɬɟɥɶɧɵ. Ⱥ ɢɦɟɧɧɨ, ɪɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸ-
ɳɢɟ 9 ɬɢɩɨɜ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ:
I.
00
)
(
X
, 0
)
l(
X
;
II . ɚ)
00
c
)
(
X
, 0
)
l(
X
( 0
1
h );