Куликов А.А. (сост.) Смешанные задачи для уравнения теплопроводности и уравнения колебаний
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11
ɤɢɟ ɪɟɲɟɧɢɹ ɧɚɡɵɜɚɸɬɫɹ ɤɥɚɫɫɢɱɟɫɤɢɦɢ (ɪɟɝɭɥɹɪɧɵɦɢ), ɚ ɩɨɫɬɚɧɨɜɤɢ ɫɨɨɬ-
ɜɟɬɫɬɜɭɸɳɢɯ ɤɪɚɟɜɵɯ ɡɚɞɚɱ – ɤɥɚɫɫɢɱɟɫɤɢɦɢ ɩɨɫɬɚɧɨɜɤɚɦɢ. Ɍɚɤɢɦ ɨɛɪɚ-
ɡɨɦ, ɤɥɚɫɫɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɩɪɟɞɩɨɥɚɝɚɟɬ, ɧɚɩɪɢɦɟɪ, ɧɟɩɪɟɪɵɜ-
ɧɨɫɬɶ ɩɪɚɜɵɯ ɱɚɫɬɟɣ ɜ ɭɪɚɜɧɟɧɢɢ, ɧɚɱɚɥɶɧɵɯ ɢ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɹɯ. Ɉɞɧɚɤɨ
ɜɨ ɦɧɨɝɢɯ ɡɚɞɚɱɚɯ, ɜɫɬɪɟɱɚɸɳɢɯɫɹ ɜ ɩɪɢɥɨɠɟɧɢɹɯ, ɷɬɢ ɮɭɧɤɰɢɢ ɦɨɝɭɬ
ɢɦɟɬɶ ɫɢɥɶɧɵɟ ɨɫɨɛɟɧɧɨɫɬɢ. Ⱦɥɹ ɬɚɤɢɯ ɡɚɞɚɱ ɤɥɚɫɫɢɱɟɫɤɢɟ ɩɨɫɬɚɧɨɜɤɢ ɭɠɟ
ɨɤɚɡɵɜɚɸɬɫɹ ɧɟɞɨɫɬɚɬɨɱɧɵɦɢ, ɢ ɩɪɢɯɨɞɢɬɫɹ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ
ɨɛɨɛɳɟɧɧɵɟ ɩɨɫɬɚɧɨɜɤɢ, ɜ ɤɨɬɨɪɵɯ ɡɚɪɚɧɟɟ ɧɟ ɬɪɟɛɭɟɬɫɹ, ɱɬɨɛɵ ɪɟɲɟɧɢɟ
ɛɵɥɨ ɞɨɫɬɚɬɨɱɧɨɟ ɱɢɫɥɨ ɪɚɡ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɦ ɢ ɞɚɠɟ ɧɟɩɪɟɪɵɜɧɵɦ.
ɍɤɚɡɚɧɧɵɟ ɪɟɲɟɧɢɹ ɧɨɫɹɬ ɧɚɡɜɚɧɢɟ ɨɛɨɛɳɟɧɧɵɯ ɪɟɲɟɧɢɣ.
ɉɟɪɟɣɬɢ ɤ ɨɛɨɛɳɟɧɧɵɦ ɩɨɫɬɚɧɨɜɤɚɦ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɦɨɠɧɨ ɪɚɡɥɢɱɧɵ-
ɦɢ ɫɩɨɫɨɛɚɦɢ. Ɉɞɢɧ ɢɡ ɧɢɯ ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɬɚɤ ɧɚɡɵɜɚɟ-
ɦɵɯ ɨɛɨɛɳɟɧɧɵɯ
ɮɭɧɤɰɢɣ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɯ ɫɨɛɨɣ ɥɢɧɟɣɧɵɟ ɧɟɩɪɟɪɵɜɧɵɟ
ɮɭɧɤɰɢɨɧɚɥɵ ɧɚɞ ɩɪɨɫɬɪɚɧɫɬɜɚɦɢ ɛɟɫɤɨɧɟɱɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɯ ɮɭɧɤ-
ɰɢɣ [4]. Ⱦɪɭɝɢɟ ɩɨɞɯɨɞɵ ɤ ɨɛɨɛɳɟɧɧɵɦ ɩɨɫɬɚɧɨɜɤɚɦ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɢ ɩɨ-
ɧɹɬɢɹɦ ɨɛɨɛɳɟɧɧɵɯ ɪɟɲɟɧɢɣ ɫɨɞɟɪɠɚɬɫɹ, ɧɚɩɪɢɦɟɪ, ɜ ɭɱɟɛɧɨɦ ɩɨɫɨɛɢɢ
[5].
§3. ɉɨɫɬɚɧɨɜɤɚ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ
ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Ɋɚɫɫɦɨɬɪɢɦ ɫɬɟɪɠɟɧɶ ɞɥɢɧɵ
l
, ɜ ɤɨɬɨɪɨɦ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɰɟɫɫ ɪɚɫ-
ɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ. Ȼɨɤɨɜɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɫɬɟɪɠɧɹ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɬɟɩɥɨ-
ɢɡɨɥɢɪɨɜɚɧɧɨɣ (ɷɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɬɟɩɥɨ ɱɟɪɟɡ ɛɨɤɨɜɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɧɟ
ɩɪɨɯɨɞɢɬ). Ɉɫɶ ɫɬɟɪɠɧɹ ɩɪɢɦɟɦ ɡɚ ɨɫɶ ɚɛɫɰɢɫɫ, ɚ ɥɟɜɵɣ ɤɨɧɟɰ – ɡɚ ɧɚɱɚɥɨ
ɨɬɫɱɟɬɚ. ɋɬɟɪɠɟɧɶ ɫɱɢɬɚɟɬɫɹ ɧɚɫɬɨɥɶɤɨ ɬɨɧɤɢɦ, ɱɬɨ ɜ ɤɚɠɞɵɣ ɦɨɦɟɧɬ ɜɪɟ-
ɦɟɧɢ t ɬɟɦɩɟɪɚɬɭɪɭ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ
ɥɸɛɨɝɨ ɟɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɦɨɠɧɨ
ɫɱɢɬɚɬɶ ɨɞɢɧɚɤɨɜɨɣ.
ɉɭɫɬɶ
)
t
,x(uu
– ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɜ ɩɨɩɟɪɟɱɧɨɦ
ɫɟɱɟɧɢɢ ɫɬɟɪɠɧɹ, ɨɬɜɟɱɚɸɳɟɦ ɬɨɱɤɟ ɫ ɚɛɫɰɢɫɫɨɣ
x , ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɧɚ ɨɫɢ
ɫɬɟɪɠɧɹ (ɜ ɞɚɥɶɧɟɣɲɟɦ ɞɥɹ ɤɪɚɬɤɨɫɬɢ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɮɭɧɤɰɢɸ
)
t
,x(u
ɬɟɦɩɟɪɚɬɭɪɨɣ ɜ ɬɨɱɤɟ
x
ɫɬɟɪɠɧɹ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ). Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ
[ɝɥ. III, §1], ɱɬɨ ɮɭɧɤɰɢɹ
)
t,
x
(u ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɞɧɨɦɟɪɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
,F
x
u
k
xt
u
c
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
w
w
U
l
x
0 , 0
!
t , (3.1)
ɝɞɟ
0
!
c , 0
!
U
, 0
!
k – ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɭɞɟɥɶɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ, ɩɥɨɬ-
ɧɨɫɬɶ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɬɟɪɠɧɹ, F – ɩɥɨɬɧɨɫɬɶ ɜɧɭɬɪɟɧ-
ɧɢɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ ɜ ɫɬɟɪɠɧɟ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɟɥɢɱɢɧɚ c ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ
ɤɨɥɢɱɟɫɬɜɭ ɬɟɩɥɚ, ɤɨɬɨɪɨɟ ɧɭɠɧɨ ɫɨɨɛɳɢɬɶ ɟɞɢɧɢɰɟ ɦɚɫɫɵ ɫɬɟɪɠɧɹ, ɱɬɨɛɵ
ɩɨɜɵɫɢɬɶ ɟɟ ɬɟɦɩɟɪɚɬɭɪɭ ɧɚ ɨɞɢɧ ɝɪɚɞɭɫ; ɜɟɥɢɱɢɧɚ
U
ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ ɦɚɫ-
12
ɫɟ ɫɬɟɪɠɧɹ, ɩɪɢɯɨɞɹɳɟɣɫɹ ɧɚ ɟɞɢɧɢɰɭ ɟɝɨ ɨɛɴɟɦɚ; ɜɟɥɢɱɢɧɚ
k ɹɜɥɹɟɬɫɹ
ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ ɜ ɡɚɤɨɧɟ Ɏɭɪɶɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟ-
ɩɥɚ (ɫɦ. ɧɢɠɟ); ɧɚɤɨɧɟɰ, ɜɟɥɢɱɢɧɚ F ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ ɤɨɥɢɱɟɫɬɜɭ ɬɟɩɥɚ, ɜɵ-
ɞɟɥɹɟɦɨɝɨ ɢɥɢ ɩɨɝɥɨɳɚɟɦɨɝɨ ɜ ɟɞɢɧɢɰɟ ɨɛɴɟɦɚ ɫɬɟɪɠɧɹ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟ-
ɧɢ
ȼɨɨɛɳɟ ɝɨɜɨɪɹ, ɜɟɥɢɱɢɧɵ
c ,
U
, k , F ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɨɬ x , t ɢ
ɬɟɦɩɟɪɚɬɭɪɵ
u . Ɉɞɧɚɤɨ ɜɨ ɦɧɨɝɢɯ ɜɚɠɧɵɯ ɞɥɹ ɩɪɢɥɨɠɟɧɢɣ ɡɚɞɚɱɚɯ ɮɭɧɤ-
ɰɢɢ
c ,
U
, k , F ɦɟɧɹɸɬɫɹ ɫ ɢɡɦɟɧɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɞɨɫɬɚɬɨɱɧɨ ɦɟɞɥɟɧɧɨ
ɢ ɮɭɧɤɰɢɢ
c
,
U
,
k
ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɜɪɟɦɟɧɢ. ɉɨɷɬɨɦɭ ɫ ɞɨɫɬɚɬɨɱɧɨɣ ɞɥɹ
ɦɧɨɝɢɯ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɬɨɱɧɨɫɬɶɸ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ
c ,
U
, k ɡɚɜɢ-
ɫɹɬ ɬɨɥɶɤɨ ɨɬ
x
, ɚ ɮɭɧɤɰɢɹ F ɡɚɜɢɫɢɬ ɨɬ
x
, t .
ȼ ɫɥɭɱɚɟ ɨɞɧɨɪɨɞɧɨɝɨ ɫɬɟɪɠɧɹ ɮɭɧɤɰɢɢ
c ,
U
, k ɹɜɥɹɸɬɫɹ ɩɨɫɬɨɹɧ-
ɧɵɦɢ, ɢ ɭɪɚɜɧɟɧɢɟ (3.1) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
,f
x
u
a
t
u
w
w
w
w
2
2
2
lx
0
,
0
!
t
,
ɝɞɟ
)c(ka
U
, )c(Ff
U
.
ɋɨɝɥɚɫɧɨ ɡɚɤɨɧɭ Ɏɭɪɶɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ, ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ,
ɩɪɨɯɨɞɹɳɟɟ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ox ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɟɞɢɧɢɰɭ
ɩɥɨɳɚɞɢ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɹ, ɨɬɜɟɱɚɸɳɟɝɨ ɬɨɱɤɟ ɫ ɚɛɫɰɢɫɫɨɣ x
ɪɚɜɧɨ
x
)t,x(u
)x(k)t,x(q
w
w
. (3.2)
Ɏɭɧɤɰɢɹ
)
t
,x(q ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ. Ɂɧɚ-
ɱɟɧɢɟ ɮɭɧɤɰɢɢ
q
ɜ ɬɨɱɤɟ )t,x(
00
ɫɱɢɬɚɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɵɦ, ɟɫɥɢ ɜ ɧɟɤɨ-
ɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ
0
x ɫɬɟɪɠɧɹ ɜ ɬɟɱɟɧɢɟ ɧɟɤɨɬɨɪɨɝɨ ɩɪɨɦɟɠɭɬɤɚ
ɜɪɟɦɟɧɢ
> @
tdt,t
00
ɬɟɩɥɨ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɬɟɱɟɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ox, ɢ
ɨɬɪɢɰɚɬɟɥɶɧɵɦ ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ. ɗɬɢɦ ɨɛɴɹɫɧɹɟɬɫɹ ɜɵɛɨɪ ɡɧɚɤɚ « – » ɜ
ɮɨɪɦɭɥɟ (3.2). Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɟɫɥɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ
0
x ɡɚ ɩɪɨɦɟɠɭɬɨɤ
ɜɪɟɦɟɧɢ
> @
tdt,t
00
ɬɟɩɥɨ ɩɪɨɬɟɤɚɟɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ox, ɬɨ ɬɟɦɩɟɪɚ-
ɬɭɪɚ
)t,x(u
0
ɭɛɵɜɚɟɬ ɜ ɭɤɚɡɚɧɧɨɣ ɨɤɪɟɫɬɧɨɫɬɢ, ɬɚɤ ɤɚɤ ɬɟɩɥɨ ɪɚɫɩɪɨɫɬɪɚ-
ɧɹɟɬɫɹ ɢɡ ɦɟɫɬ ɫ ɛɨɥɟɟ ɜɵɫɨɤɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ ɤ ɦɟɫɬɚɦ ɫ ɛɨɥɟɟ ɧɢɡɤɨɣ. ɉɨ-
ɷɬɨɦɭ
0
00
w
w
x
)t,x(u
, ɢ, ɜ ɫɢɥɭ (3.2),
0
00
!
)t,x(q
, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɨ-
ɝɥɚɲɟɧɢɸ ɨ ɡɧɚɤɟ ɜɟɥɢɱɢɧɵ
q . Ⱥɧɚɥɨɝɢɱɧɨ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɢ ɫɥɭɱɚɣ, ɤɨ-
ɝɞɚ ɬɟɩɥɨ ɩɪɨɬɟɤɚɟɬ ɩɪɨɬɢɜ ɧɚɩɪɚɜɥɟɧɢɹ ɨɫɢ Ox.
Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɩɨɥɧɨɫɬɶɸ ɨɩɢɫɚɬɶ ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ ɜ
ɫɬɟɪɠɧɟ, ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɧɚɱɚɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɬɨɱɟɤ ɫɬɟɪɠɧɹ ɢ ɬɟɩ-
13
ɥɨɜɨɣ ɪɟɠɢɦ ɧɚ ɟɝɨ ɤɨɧɰɚɯ, ɬɨ ɟɫɬɶ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (3.1) ɧɭɠɧɨ ɡɚɞɚɬɶ ɧɚ-
ɱɚɥɶɧɵɟ ɢ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ.
ɉɭɫɬɶ
)
x(
M
– ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɬɨɱɤɟ x ɫɬɟɪɠɧɹ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
0
t
. Ɍɨɝɞɚ ɫɬɚɜɢɬɫɹ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ
)
x
(
)
,
x
(u
M
0 , l
x
d
d
0 . (3.3)
ɇɚ ɤɨɧɰɚɯ ɫɬɟɪɠɧɹ ɦɨɝɭɬ ɛɵɬɶ ɢɡɜɟɫɬɧɵ ɥɢɛɨ ɬɟɦɩɟɪɚɬɭɪɚ, ɥɢɛɨ
ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɥɢɛɨ ɦɨɝɭɬ ɛɵɬɶ ɡɚɞɚɧɵ ɭɫɥɨɜɢɹ ɬɟɩɥɨɨɛɦɟɧɚ
ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ. ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɧɚ ɥɟɜɨɦ ɤɨɧɰɟ ɫɬɟɪɠɧɹ ɡɚɞɚɧɚ
ɬɟɦɩɟɪɚɬɭɪɚ
)
t
(
s
, ɚ ɧɚ ɩɪɚɜɨɦ ɤɨɧɰɟ ɡɚɞɚɧɚ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ
)
t(w , ɬɨ ɫɬɚɜɹɬɫɹ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ
)
t(
s
)
t,(u
0 , )t(w
x
)t,l(u
)l(k
w
w
. (3.4)
Ʉɨɧɟɰ
0
x
ɢɥɢ l
x
ɫɬɟɪɠɧɹ ɧɚɡɵɜɚɟɬɫɹ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɵɦ, ɟɫ-
ɥɢ ɱɟɪɟɡ ɧɟɝɨ ɧɟ ɩɪɨɬɟɤɚɟɬ ɬɟɩɥɨ (ɬɨ ɟɫɬɶ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɧɚ
ɷɬɨɦ ɤɨɧɰɟ ɪɚɜɧɚ ɧɭɥɸ). ȿɫɥɢ, ɧɚɩɪɢɦɟɪ, ɤɨɧɟɰ
0
x
ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧ, ɬɨ
ɩɨɥɭɱɚɟɦ ɤɪɚɟɜɨɟ ɭɫɥɨɜɢɟ
0
0
w
w
x
)t,(u
.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɧɚ ɤɨɧɰɚɯ ɫɬɟɪɠɧɹ ɩɪɨɢɫɯɨɞɢɬ ɬɟɩɥɨɨɛɦɟɧ ɩɨ ɡɚ-
ɤɨɧɭ ɇɶɸɬɨɧɚ. ɗɬɨɬ ɡɚɤɨɧ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɩɟɪɟɞɚɜɚɟ-
ɦɨɟ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɫ ɟɞɢɧɢɰɵ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɜ ɨɤɪɭɠɚɸ-
ɳɭɸ ɫɪɟɞɭ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɢ ɨɤ-
ɪɭɠɚɸɳɟɣ ɫɪɟɞɵ, ɬɨ ɟɫɬɶ ɪɚɜɧɨ
)
u
~
U(
H
,
ɝɞɟ
U – ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ, u
~
– ɬɟɦɩɟɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɣ
ɫɪɟɞɵ,
H
– ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ, ɧɚɡɵɜɚɟɦɵɣ ɤɨɷɮɮɢɰɢɟɧ-
ɬɨɦ ɬɟɩɥɨɨɛɦɟɧɚ.
ɂɫɩɨɥɶɡɭɹ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɢ ɡɚɤɨɧ
ɇɶɸɬɨɧɚ, ɩɨɥɭɱɚɟɦ ɭɫɥɨɜɢɹ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɤɨɧɰɚɯ ɫɬɟɪɠɧɹ:
> @
)t(p)t,(uH
x
t),(u
)(k
11
0
0
0
w
w
, (3.5)
> @
)t(p)t,l(uH
x
)t,l(u
)l(k
22
w
w
, (3.6)
ɝɞɟ
0
1
!
H
(ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
0
2
!
H
) – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɥɟɜɨɦ
(ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ ɩɪɚɜɨɦ) ɤɨɧɰɟ,
)t(p
1
(ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ )t(p
2
) – ɬɟɦɩɟ-
ɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɨɤɨɥɨ ɥɟɜɨɝɨ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɤɨɥɨ ɩɪɚɜɨɝɨ)
14
ɤɨɧɰɚ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ
1
H ɢ
2
H
ɩɨɫɬɨɹɧɧɵ.
ɍɫɥɨɜɢɹ (3.5), (3.6) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
)t()t,(uh
x
)t,(u
11
0
0
J
w
w
, (3.7)
)t()t,l(uh
x
)t,l(u
22
J
w
w
, (3.8)
ɝɞɟ
0
0
1
1
!
)(k
H
h
, 0
2
2
!
)l(k
H
h
,
)(k
)t(pH
)t(
0
11
1
J
,
)l(k
)t(pH
)t(
22
2
J
.
ȼɫɟ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɜɵɲɟ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (3.1) ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
)t()t,(u
x
)t,(u
111
0
0
JED
w
w
, 0
!
t , (3.9)
)t()t,l(u
x
)t,l(u
222
JED
w
w
, 0
!
t
, (3.10)
ɝɞɟ
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
;
)t(
1
J
,
)t(
2
J
– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ.
ȿɫɥɢ
0
1
D
, 1
1
E
, ɬɨ ɭɫɥɨɜɢɟ (3.9) ɧɚɡɵɜɚɟɬɫɹ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɟɦ
ɩɟɪɜɨɝɨ ɪɨɞɚ (ɧɚ ɥɟɜɨɦ ɤɨɧɰɟ); ɟɫɥɢ
0
1
z
D
, 0
1
E
, ɬɨ ɭɤɚɡɚɧɧɨɟ ɭɫɥɨɜɢɟ
ɧɚɡɵɜɚɟɬɫɹ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɟɦ ɜɬɨɪɨɝɨ ɪɨɞɚ; ɟɫɥɢ ɠɟ
0
1
z
D
,
0
1
z
E
, ɬɨ
ɨɧɨ ɧɚɡɵɜɚɟɬɫɹ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɟɦ ɬɪɟɬɶɟɝɨ ɪɨɞɚ. Ⱥɧɚɥɨɝɢɱɧɨ ɜɜɨɞɹɬɫɹ ɢ
ɩɨɧɹɬɢɹ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ ɩɟɪɜɨɝɨ, ɜɬɨɪɨɝɨ ɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɧɚ ɩɪɚɜɨɦ ɤɨɧ-
ɰɟ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɞɚɱɚ ɨ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɢ ɬɟɩɥɚ ɜ ɬɨɧɤɨɦ ɫɬɟɪɠɧɟ ɫ
ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɨɣ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɫɬɚɜɢɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (3.1), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ
ɭɫɥɨɜɢɸ (3.3) ɢ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ (3.9), (3.10), ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
1
D
,
1
E
,
2
D
,
2
E
ɢ ɮɭɧɤɰɢɢ )t(
1
J
, )t(
2
J
ɡɚɞɚɸɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɚɦɢ ɜɢ-
ɞɚ (3.4) – (3.6). ɉɪɢ ɷɬɨɦ ɜɨɡɦɨɠɧɚ ɩɨɫɬɚɧɨɜɤɚ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ ɪɚɡɥɢɱɧɨ-
ɝɨ ɪɨɞɚ ɧɚ ɪɚɡɥɢɱɧɵɯ ɤɨɧɰɚɯ ɫɬɟɪɠɧɹ.
15
§4. ɉɨɫɬɚɧɨɜɤɚ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ
ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ
Ʉɚɤ ɛɵɥɨ ɨɬɦɟɱɟɧɨ ɜ §1, ɤ ɭɪɚɜɧɟɧɢɸ ɤɨɥɟɛɚɧɢɣ ɩɪɢɜɨɞɢɬ ɦɚɬɟɦɚɬɢ-
ɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɤɨɥɟɛɚɬɟɥɶɧɵɯ ɩɪɨɰɟɫɫɨɜ ɪɚɡɥɢɱɧɨɣ ɮɢɡɢɱɟɫɤɨɣ ɩɪɢɪɨ-
ɞɵ. ɉɪɢɦɟɪɚɦɢ ɬɚɤɢɯ ɩɪɨɰɟɫɫɨɜ ɹɜɥɹɸɬɫɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɧɢɠɟ ɦɚɥɵɟ
ɩɨɩɟɪɟɱɧɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ ɢ ɦɚɥɵɟ ɩɪɨɞɨɥɶɧɵɟ ɤɨɥɟɛɚɧɢɹ ɭɩɪɭɝɨɝɨ
ɫɬɟɪɠɧɹ.
ɉɨɞ ɫɬɪɭɧɨɣ ɩɨɧɢɦɚɸɬ ɬɨɧɤɭɸ ɧɢɬɶ, ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɫɜɨɛɨɞɧɨ ɢɡɝɢ-
ɛɚɬɶɫɹ, ɬɨ ɟɫɬɶ ɧɟ ɨɤɚɡɵɜɚɟɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢɡɦɟɧɟɧɢɸ ɟɺ ɮɨɪɦɵ, ɧɟ ɫɜɹ-
ɡɚɧɧɨɦɭ ɫ ɢɡɦɟɧɟɧɢɟɦ ɞɥɢɧɵ.
Ɋɚɫɫɦɨɬɪɢɦ ɧɚɬɹɧɭɬɭɸ ɫɬɪɭɧɭ ɞɥɢɧɵ l, ɡɚɤɪɟɩɥɟɧɧɭɸ ɧɚ ɤɨɧɰɚɯ.
ɉɭɫɬɶ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɫɬɪɭɧɚ ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɨɫɢ ɚɛɫɰɢɫɫ Ox ɢɟɟ
ɥɟɜɵɣ ɤɨɧɟɰ ɧɚɯɨɞɢɬɫɹ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ. Ɍɨɝɞɚ ɤɚɠɞɭɸ ɬɨɱɤɭ ɫɬɪɭɧɵ
ɦɨɠɧɨ ɨɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɡɧɚɱɟɧɢɟɦ ɟɟ ɚɛɫɰɢɫɫɵ x.
ȿɫɥɢ ɜɵɜɟɫɬɢ ɫɬɪɭɧɭ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ (ɧɚɩɪɢɦɟɪ
, ɨɬɬɹɧɭɬɶ ɟɟ
ɢ ɨɬɩɭɫɬɢɬɶ ɢɥɢ ɭɞɚɪɢɬɶ ɩɨ ɧɟɣ), ɬɨ ɫɬɪɭɧɚ ɧɚɱɧɟɬ ɤɨɥɟɛɚɬɶɫɹ. Ɇɵ ɛɭɞɟɦ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɩɨɩɟɪɟɱɧɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ, ɬɨ ɟɫɬɶ ɬɚɤɢɟ ɤɨɥɟɛɚ-
ɧɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɞɜɢɠɟɧɢɟ ɬɨɱɟɤ ɫɬɪɭɧɵ ɩɪɨɢɫɯɨɞɢɬ ɜ ɨɞɧɨɣ ɩɥɨɫɤɨɫɬɢ
xOu ɢ ɜɫɟ ɬɨɱɤɢ ɞɜɢɠɭɬɫɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɨɫɢ Ox.
Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ
)
t,
x
(uu
ɜɟɥɢɱɢɧɭ ɨɬɤɥɨɧɟɧɢɹ ɫɬɪɭɧɵ ɨɬ ɩɨɥɨ-
ɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜ ɬɨɱɤɟ ɫ ɚɛɫɰɢɫɫɨɣ x,
l
x
d
d
0 ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ,
ɨɬɫɱɢɬɵɜɚɟɦɭɸ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ou.
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɪɢ ɤɚɠɞɨɦ ɮɢɤɫɢɪɨɜɚɧɧɨɦ
0
tt
ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ
)t,x(u)x(
0
M
ɞɚɟɬ ɮɨɪɦɭ ɫɬɪɭɧɵ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
0
t . ɉɪɢ ɮɢɤɫɢɪɨ-
ɜɚɧɧɨɦ ɡɧɚɱɟɧɢɢ
0
xx
ɮɭɧɤɰɢɹ
)t,x(u)t(s
0
ɞɚɟɬ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ
ɬɨɱɤɢ ɫɬɪɭɧɵ ɫ ɚɛɫɰɢɫɫɨɣ
0
x ɜɞɨɥɶ ɩɪɹɦɨɣ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɨɫɢ Ou, ɩɪɨɢɡ-
ɜɨɞɧɚɹ
)t,x(u)t(s
t 0
c
– ɫɤɨɪɨɫɬɶ ɷɬɨɝɨ ɞɜɢɠɟɧɢɹ, ɚ ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ
)t,x(u)t(s
tt 0
cc
– ɭɫɤɨɪɟɧɢɟ.
Ɇɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɦɚɥɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ, ɬɨ ɟɫɬɶ ɬɚ-
ɤɢɟ ɤɨɥɟɛɚɧɢɹ, ɤɨɝɞɚ ɜɟɥɢɱɢɧɚ
u
ɩɪɢɧɢɦɚɟɬ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɟ ɡɧɚɱɟɧɢɹ ɢ
ɤɜɚɞɪɚɬɨɦ ɜɟɥɢɱɢɧɵ
x
u ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɟɞɢɧɢɰɟɣ.
Ʌɢɧɟɣɧɨɣ ɩɥɨɬɧɨɫɬɶɸ ɫɬɪɭɧɵ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɮɭɧɤɰɢɸ
)
x(
U
, ɨɩ-
ɪɟɞɟɥɟɧɧɭɸ ɧɚ ɨɬɪɟɡɤɟ
> @
l,0 ɢ ɨɛɥɚɞɚɸɳɭɸ ɬɟɦ ɫɜɨɣɫɬɜɨɦ, ɱɬɨ ɦɚɫɫɚ
ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɚɥɨɝɨ ɭɱɚɫɬɤɚ
>
@
xdx,x
ɫɬɟɪɠɧɹ ɪɚɜɧɚ
x
d
)
x
(
U
(ɢɧɵ-
ɦɢ ɫɥɨɜɚɦɢ, ɜɟɥɢɱɢɧɚ
U
ɪɚɜɧɚ ɦɚɫɫɟ ɫɬɪɭɧɵ, ɩɪɢɯɨɞɹɳɟɣɫɹ ɧɚ ɟɞɢɧɢɰɭ ɟɟ
ɞɥɢɧɵ).
ȼ ɩɪɨɰɟɫɫɟ ɤɨɥɟɛɚɧɢɣ ɧɚ ɤɚɠɞɵɣ ɭɱɚɫɬɨɤ ɫɬɪɭɧɵ ɞɟɣɫɬɜɭɸɬ ɜɧɟɲɧɢɟ
ɫɢɥɵ, ɧɚɩɪɚɜɥɟɧɧɵɟ ɩɚɪɚɥɥɟɥɶɧɨ ɨɫɢ Ou, ɢ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ.
16
ɉɭɫɬɶ
))t,x(u,x(P
t
x
– ɬɨɱɤɚ ɫɬɪɭɧɵ ɫ ɚɛɫɰɢɫɫɨɣ x ɜ ɦɨɦɟɧɬ ɜɪɟ-
ɦɟɧɢ t . ɑɟɪɟɡ
)t,x(T
G
ɨɛɨɡɧɚɱɢɦ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɭɱɚ-
ɫɬɨɤ
t
x
t
PP
0
ɫɨ ɫɬɨɪɨɧɵ ɭɱɚɫɬɤɚ
t
l
t
x
PP . Ɉɬɦɟɬɢɦ, ɱɬɨ ɫɨɝɥɚɫɧɨ ɬɪɟɬɶɟɦɭ ɡɚ-
ɤɨɧɭ ɇɶɸɬɨɧɚ, ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɭɱɚɫɬɨɤ
t
l
t
x
PP ɫɨ ɫɬɨɪɨɧɵ
ɭɱɚɫɬɤɚ
t
x
t
PP
0
ɪɚɜɧɚ
)t,x(T
G
.
Ɍɚɤ ɤɚɤ ɫɬɪɭɧɚ ɧɟ ɫɨɩɪɨɬɢɜɥɹɟɬɫɹ ɢɡɝɢɛɭ, ɬɨ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɫɬɪɭɧɵ ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɫɬɪɭɧɟ
ɜ ɷɬɨɣ ɬɨɱɤɟ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ ɬɚɤɠɟ [3, ɝɥ. 1, §1], ɱɬɨ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ
ɫɥɭɱɚɟ ɦɚɥɵɯ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɜɟɥɢɱɢɧɚ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɩɨɫɬɨɹɧɧɚ:
const
{
0
T)t,x(T
G
ɞɥɹ ɜɫɟɯ l
x
d
d
0 , 0
t
t .
ɉɥɨɬɧɨɫɬɶɸ ɜɧɟɲɧɢɯ ɫɢɥ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɮɭɧɤɰɢɸ
)
t
,x(F , ɨɛɥɚ-
ɞɚɸɳɭɸ ɬɟɦ ɫɜɨɣɫɬɜɨɦ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ
0
t
t
ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ Ou ɪɟɡɭɥɶ-
ɬɢɪɭɸɳɟɝɨ ɜɟɤɬɨɪɚ ɜɧɟɲɧɢɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɧɚ
ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɚɥɵɣ ɭɱɚɫɬɨɤ
t
dxx
t
x
PP
ɪɚɜɧɚ
x
d
)
t,
x
(F .
ɂɫɩɨɥɶɡɭɹ ɜɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ, ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ [3, ɝɥ. 1, §1], ɱɬɨ
ɮɭɧɤɰɢɹ
)
t,
x
(u ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ
,F
x
u
T
t
u
w
w
w
w
2
2
0
2
2
U
lx
0
,
0
!
t
. (4.1)
ɍɪɚɜɧɟɧɢɟ (4.1) ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɭɪɚɜɧɟɧɢɹ ɦɚɥɵɯ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚ-
ɧɢɣ ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɬɪɭɧɵ. ȿɫɥɢ ɜɧɟɲɧɢɟ ɫɢɥɵ ɨɬɫɭɬɫɬɜɭɸɬ (
0
{
F ), ɬɨ
ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ ɧɚɡɵɜɚɸɬɫɹ ɫɜɨɛɨɞɧɵɦɢ, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ – ɜɵɧɭɠ-
ɞɟɧɧɵɦɢ.
ȿɫɥɢ ɫɬɪɭɧɚ ɨɞɧɨɪɨɞɧɚ, ɬɨ ɟɟ ɩɥɨɬɧɨɫɬɶ
U
ɩɨɫɬɨɹɧɧɚ, ɢ ɭɪɚɜɧɟɧɢɟ
ɤɨɥɟɛɚɧɢɣ (4.1) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
,f
x
u
a
t
u
w
w
w
w
2
2
2
2
2
l
x
0 , 0
!
t , (4.2)
ɝɞɟ
U
0
Ta
,
U
Ff
.
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ (4.1) ɹɜɥɹɟɬɫɹ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɹ
ɤɨɥɟɛɚɧɢɣ (1.4) ɩɪɢ
1
n ,
0
Tp
, ɚ ɭɪɚɜɧɟɧɢɟ (4.2) – ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ
ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ (1.5).
Ʉ ɭɪɚɜɧɟɧɢɸ (1.4) ɩɪɢɜɨɞɢɬ ɬɚɤɠɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɩɪɨɰɟɫɫɚ
ɤɨɥɟɛɚɧɢɣ ɫɬɟɪɠɧɹ (ɢɥɢ ɩɪɭɠɢɧɵ).
Ɋɚɫɫɦɨɬɪɢɦ ɭɩɪɭɝɢɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɵ l, ɡɚɧɢɦɚɸɳɢɣ ɜ ɩɨɥɨɠɟɧɢɢ
ɪɚɜɧɨɜɟɫɢɹ ɨɬɪɟɡɨɤ
> @
l,0 ɨɫɢ Ox. ɉɪɨɰɟɫɫ ɩɪɨɞɨɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɬɟɪɠɧɹ
ɦɨɠɟɬ ɛɵɬɶ ɨɩɢɫɚɧ ɮɭɧɤɰɢɟɣ
)
t,
x
(u , ɩɪɟɞɫɬɚɜɥɹɸɳɟɣ ɫɨɛɨɣ ɫɦɟɳɟɧɢɟ ɤ
ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ
t ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɫɬɟɪɠɧɹ, ɢɦɟɜɲɟɣ ɜ ɩɨɥɨɠɟɧɢɢ
17
ɪɚɜɧɨɜɟɫɢɹ ɚɛɫɰɢɫɫɭ
x
(ɫɦɟɳɟɧɢɹ ɨɬɫɱɢɬɵɜɚɸɬɫɹ ɜ ɩɨɥɨɠɢɬɟɥɶɧɨɦ ɧɚ-
ɩɪɚɜɥɟɧɢɢ ɨɫɢ Ox). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
t ɤɨɨɪɞɢɧɚɬɚ ɭɤɚ-
ɡɚɧɧɨɣ ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɫɬɟɪɠɧɹ ɪɚɜɧɚ
)
t
,x(ux
.
ɉɭɫɬɶ
> @
xx,x ǻ
– ɧɟɤɨɬɨɪɵɣ ɭɱɚɫɬɨɤ ɫɬɟɪɠɧɹ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨ-
ɜɟɫɢɹ. Ɉɬɧɨɫɢɬɟɥɶɧɵɦ ɭɞɥɢɧɟɧɢɟɦ ɷɬɨɝɨ ɭɱɚɫɬɤɚ ɤ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ t ɧɚ-
ɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ
112
l)ll(
, ɝɞɟ
2
l – ɞɥɢɧɚ ɭɱɚɫɬɤɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
t
,
xl ǻ
1
– ɞɥɢɧɚ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ. ɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɢ
)
t
,x(u ɫɥɟɞɭɟɬ, ɱɬɨ
))t,x(ux()t,xx(uxxl ǻǻ
2
)
t,
x
(u
)
t,
x
x
(u
x
ǻǻ ,
ɩɨɷɬɨɦɭ
x
)t,x(u)t,xx(u
l
ll
ǻ
ǻ
1
12
.
ȼ ɩɪɟɞɟɥɟ ɩɪɢ
0
o
x
ǻ (ɬɨ ɟɫɬɶ ɫɱɢɬɚɹ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɭɱɚɫɬɨɤ
ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɭɸ ɞɥɢɧɭ) ɩɨɥɭɱɢɦ, ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɭɞɥɢɧɟɧɢɟ
ɫɬɟɪɠɧɹ ɜ ɬɨɱɤɟ x ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ
)t,x(u
x
.
ɉɭɫɬɶ
)t,x(T
G
– ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɜ ɩɨɩɟɪɟɱɧɨɦ ɫɟɱɟɧɢɢ ɫɬɟɪɠɧɹ, ɨɬ-
ɜɟɱɚɸɳɟɦ ɬɨɱɤɟ
x
, ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t , ɢ
)
t
,x(
T
– ɩɪɨɟɤɰɢɹ ɜɟɤɬɨɪɚ
)t,x(T
G
ɧɚ ɨɫɶ Ox. ɋɨɝɥɚɫɧɨ ɡɚɤɨɧɭ Ƚɭɤɚ, ɭɤɚɡɚɧɧɚɹ ɩɪɨɟɤɰɢɹ ɩɪɨɩɨɪɰɢɨ-
ɧɚɥɶɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨɦɭ ɭɞɥɢɧɟɧɢɸ ɫɬɟɪɠɧɹ ɜ ɷɬɨɣ ɬɨɱɤɟ ɢ ɩɥɨɳɚɞɢ ɟɝɨ
ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ, ɬɨ ɟɫɬɶ ɢɦɟɟɬ ɦɟɫɬɨ ɫɨɨɬɧɨɲɟɧɢɟ
,
x
)t,x(u
SE)t,x(T
w
w
(4.3)
ɝɞɟ
)
x
(
E
E
– ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ, ɧɚ-
ɡɵɜɚɟɦɵɣ ɦɨɞɭɥɟɦ ɘɧɝɚ ɜ ɬɨɱɤɟ
x
,
S
– ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ
ɫɬɟɪɠɧɹ.
ɂɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɟ (4.3), ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ [2, ɝɥ. II, §1], ɱɬɨ ɮɭɧɤ-
ɰɢɹ
)
t,
x
(u ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ
,F
x
u
SE
xt
u
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
w
w
2
2
U
lx
0 , 0
!
t
, (4.4)
ɝɞɟ
)
x
(
U
U
– ɥɢɧɟɣɧɚɹ ɩɥɨɬɧɨɫɬɶ ɫɬɟɪɠɧɹ,
)
t,
x
(FF
– ɩɥɨɬɧɨɫɬɶ
ɜɧɟɲɧɢɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɫɬɟɪɠɟɧɶ ɜɞɨɥɶ ɟɝɨ ɨɫɢ.
ȿɫɥɢ ɫɬɟɪɠɟɧɶ ɨɞɧɨɪɨɞɟɧ, ɬɨ ɮɭɧɤɰɢɢ
U
ɢ
E
ɩɪɢɧɢɦɚɸɬ ɩɨɫɬɨɹɧɧɵɟ
ɡɧɚɱɟɧɢɹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɟ (4.4) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ (4.2), ɝɞɟ
18
U
SEa
,
U
Ff
. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɞɧɨ ɢ ɬɨ ɠɟ ɭɪɚɜɧɟɧɢɟ (4.2) ɨɩɢ-
ɫɵɜɚɟɬ ɞɜɚ ɪɚɡɥɢɱɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚ: ɦɚɥɵɟ ɩɨɩɟɪɟɱɧɵɟ ɤɨɥɟɛɚɧɢɹ
ɨɞɧɨɪɨɞɧɨɣ ɫɬɪɭɧɵ ɢ ɩɪɨɞɨɥɶɧɵɟ ɤɨɥɟɛɚɧɢɹ ɨɞɧɨɪɨɞɧɨɝɨ ɫɬɟɪɠɧɹ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɫɬɚɧɨɜɤɭ ɧɚɱɚɥɶɧɵɯ ɢ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ ɞɥɹ ɭɪɚɜɧɟɧɢɹ
ɤɨɥɟɛɚɧɢɣ (4.2). ȼɧɚɱɚɥɟ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɥɭɱɚɣ ɦɚɥɵɯ ɩɨɩɟɪɟɱɧɵɯ
ɤɨɥɟɛɚɧɢɣ ɨɞɧɨɪɨɞɧɨɣ ɫɬɪɭɧɵ.
Ɍɚɤ ɤɚɤ ɩɪɨɰɟɫɫ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɡɚɜɢɫɢɬ ɨɬ ɟɟ ɧɚɱɚɥɶɧɨɣ
ɮɨɪɦɵ,
ɡɚɞɚɜɚɟɦɨɣ ɮɭɧɤɰɢɟɣ
)
,x(u 0
, ɢ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ),x(u
t
0 , ɬɨ ɞɥɹ
ɨɩɢɫɚɧɢɹ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ ɧɭɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
)
x
(
)
,
x
(u
M
0 , )x(
t
),x(u
\
w
w
0
, l
x
d
d
0 , (4.5)
ɝɞɟ
)
x
(
M
ɢ
)
x
(
\
– ɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ.
ȿɫɥɢ ɤɨɧɰɵ
0
x
ɢɥɢ
lx
ɫɬɪɭɧɵ ɡɚɤɪɟɩɥɟɧɵ, ɬɨ ɫɬɚɜɹɬɫɹ ɤɪɚɟɜɵɟ
ɭɫɥɨɜɢɹ
00
)
t,(u , 0
)
t,l(u , 0
!
t .
ȿɫɥɢ ɤɨɧɰɵ ɫɬɪɭɧɵ ɞɜɢɠɭɬɫɹ ɩɨ ɢɡɜɟɫɬɧɨɦɭ ɡɚɤɨɧɭ, ɬɨ ɫɬɚɜɹɬɫɹ ɤɪɚɟ-
ɜɵɟ ɭɫɥɨɜɢɹ
)t()t,(u
1
0
P
, )t()t,l(u
2
P
, 0
!
t , (4.6)
ɝɞɟ
)t(
1
P
ɢ )t(
2
P
– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ.
ȼɨɡɦɨɠɧɵ ɢ ɞɪɭɝɢɟ ɬɢɩɵ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ. ɉɨɫɬɚɧɨɜɤɭ ɬɚɤɢɯ ɭɫ-
ɥɨɜɢɣ ɪɚɫɫɦɨɬɪɢɦ ɧɚ ɩɪɢɦɟɪɟ ɩɪɨɰɟɫɫɚ ɩɪɨɞɨɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɨɞɧɨɪɨɞ-
ɧɨɝɨ ɫɬɟɪɠɧɹ.
ȿɫɥɢ ɨɞɢɧ ɢɡ ɤɨɧɰɨɜ ɫɬɟɪɠɧɹ, ɧɚɩɪɢɦɟɪ,
0
x
, ɞɜɢɠɟɬɫɹ ɩɨ ɢɡɜɟɫɬ-
ɧɨɦɭ ɡɚɤɨɧɭ, ɨɩɪɟɞɟɥɹɟɦɨɦɭ ɮɭɧɤɰɢɟɣ
)
t(
P
, ɚ ɧɚ ɤɨɧɰɟ l
x
ɡɚɞɚɧɚ
ɜɧɟɲɧɹɹ ɫɢɥɚ, ɩɪɨɟɤɰɢɹ ɤɨɬɨɪɨɣ ɧɚ ɨɫɶ Ox ɪɚɜɧɚ
)
t(
Z
, ɬɨ ɩɨɥɭɱɚɟɦ ɤɪɚɟ-
ɜɵɟ ɭɫɥɨɜɢɹ
)t()t,(u
1
0
P
, (4.7)
)t(
x
)t,l(u
SE
Z
w
w
, 0
!
t . (4.8)
ɍɫɥɨɜɢɟ (4.8) ɫɥɟɞɭɟɬ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ (4.3) ɢ ɬɨɝɨ ɮɚɤɬɚ, ɱɬɨ ɫɢɥɚ ɧɚ-
ɬɹɠɟɧɢɹ ɧɚ ɤɨɧɰɟ
lx
ɪɚɜɧɚ ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɷɬɨɦɭ ɤɨɧɰɭ ɜɧɟɲɧɟɣ ɫɢɥɟ.
ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɤɨɧɟɰ
0
x
ɡɚɤɪɟɩɥɟɧ, ɚ ɤɨɧɟɰ l
x
ɫɜɨɛɨɞɟɧ (ɬɨ
ɟɫɬɶ ɧɚ ɧɟɝɨ ɧɟ ɞɟɣɫɬɜɭɸɬ ɜɧɟɲɧɢɟ ɫɢɥɵ), ɬɨ ɭɫɥɨɜɢɹ (4.7), (4.8) ɩɪɢɧɢɦɚ-
ɸɬ ɜɢɞ
00
)
t,(u , 0
w
w
x
)t,l(u
.
19
Ɍɢɩɢɱɧɵɦ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɭɫɥɨɜɢɟ ɭɩɪɭɝɨɝɨ ɡɚɤɪɟɩɥɟɧɢɹ, ɧɚɩɪɢɦɟɪ,
ɞɥɹ ɤɨɧɰɚ
lx
:
)t,l(u
x
)t,l(u
SE
V
w
w
.
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɤɨɧɟɰ
l
x
ɦɨɠɟɬ ɩɟɪɟɦɟɳɚɬɶɫɹ, ɧɨ ɭɩɪɭɝɚɹ ɫɢɥɚ
ɡɚɤɪɟɩɥɟɧɢɹ ɜɵɡɵɜɚɟɬ ɧɚ ɷɬɨɦ ɤɨɧɰɟ ɧɚɬɹɠɟɧɢɟ, ɫɬɪɟɦɹɳɟɟɫɹ ɜɟɪɧɭɬɶ ɫɦɟ-
ɫɬɢɜɲɢɣɫɹ ɤɨɧɟɰ ɜ ɩɪɟɠɧɟɟ ɩɨɥɨɠɟɧɢɟ. ɗɬɚ ɫɢɥɚ, ɫɨɝɥɚɫɧɨ ɡɚɤɨɧɭ Ƚɭɤɚ,
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɫɦɟɳɟɧɢɸ
)
t,l(u ; ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ
0
!
V
ɧɚɡɵɜɚɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɠɟɫɬɤɨɫɬɢ ɡɚɤɪɟɩɥɟɧɢɹ.
ȿɫɥɢ ɬɨɱɤɚ (ɢɥɢ ɫɢɫɬɟɦɚ), ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɢɦɟɟɬ ɦɟɫɬɨ ɭɩɪɭɝɨɟ
ɡɚɤɪɟɩɥɟɧɢɟ, ɩɟɪɟɦɟɳɚɟɬɫɹ ɢ ɟɟ ɨɬɤɥɨɧɟɧɢɟ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɪɚɜɧɨ
)
t(
T
, ɬɨ ɤɪɚɟɜɨɟ ɭɫɥɨɜɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
> @
)t()t,l(u
x
)t,l(u
SE
TV
w
w
. (4.9)
ɍɫɥɨɜɢɟ ɭɩɪɭɝɨɝɨ ɡɚɤɪɟɩɥɟɧɢɹ ɧɚ ɥɟɜɨɦ ɤɨɧɰɟ ɢɦɟɟɬ ɜɢɞ
> @
)t()t,(u
~
x
)t,(u
SE
KV
w
w
0
0
, (4.10)
ɝɞɟ
0
!
V
~
– ɤɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɡɚɤɪɟɩɥɟɧɢɹ ɧɚ ɷɬɨɦ ɤɨɧɰɟ; ɮɭɧɤɰɢɹ
)
t(
K
ɨɩɪɟɞɟɥɹɟɬɫɹ ɚɧɚɥɨɝɢɱɧɨ
)
t(
T
.
ɍɫɥɨɜɢɹ (4.9), (4.10) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
)t()t,(uh
x
)t,(u
11
0
0
J
w
w
,
)t()t,l(uh
x
)t,l(u
22
J
w
w
,
ɝɞɟ
0
1
!
E
S
~
h
V
, 0
2
!
E
S
h
V
,
E
S
)t(
~
)t(
K
V
J
1
,
E
S
)t(
)t(
T
V
J
2
(ɫɪ. ɫ ɭɫɥɨɜɢɹɦɢ (3.7), (3.8)).
ȼɫɟ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɜɵɲɟ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (4.2) ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ ɬɚɤ ɠɟ, ɤɚɤ ɢ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨ-
ɩɪɨɜɨɞɧɨɫɬɢ:
)t()t,(u
x
)t,(u
111
0
0
JED
w
w
, 0
!
t , (4.11)
20
)t()t,l(u
x
)t,l(u
222
JED
w
w
, 0
!
t , (4.12)
ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
1
D
,
1
E
,
2
D
,
2
E
ɢ ɮɭɧɤɰɢɢ )t(
1
J
, )t(
2
J
ɡɚɞɚɸɬɫɹ ɜ ɫɨ-
ɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɚɦɢ ɜɢɞɚ (4.6) – (4.10) (ɩɪɢ ɷɬɨɦ, ɨɱɟɜɢɞɧɨ, ɜɵɩɨɥɧɹ-
ɸɬɫɹ ɭɫɥɨɜɢɹ
0
2
1
2
1
!
E
D
, 0
2
2
2
2
!
E
D
).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ (4.2) ɡɚ-
ɤɥɸɱɚɟɬɫɹ ɜ ɧɚɯɨɠɞɟɧɢɢ ɪɟɲɟɧɢɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨ ɧɚ-
ɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ (4.5) ɢ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ (4.11), (4.12).
§5. ɇɟɨɞɧɨɪɨɞɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ
ɩɪɨɢɡɜɨɞɧɵɦɢ. ɉɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ
ɉɭɫɬɶ
:
– ɨɛɥɚɫɬɶ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ
n
R
ɫ ɝɪɚɧɢɰɟɣ
*
, ɫɨɫɬɨɹɳɟɣ ɢɡ
ɭɱɚɫɬɤɨɜ
Į
ī,,ī !
1
. ɉɭɫɬɶ T – ɡɚɞɚɧɧɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ ɢɥɢ
f
ɢ
^
`
Tt,xR)t,x(Q
n
T
0
1
:
:
– ɰɢɥɢɧɞɪ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ
1
n
R
, ɨɫɧɨɜɚɧɢɟɦ ɤɨɬɨɪɨɝɨ ɹɜɥɹɟɬɫɹ
:
. ȼɨɛ-
ɥɚɫɬɢ
T
Q
ɪɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ
)t,x(f)t,x(uA
t
)t,x(u
m
m
w
w
,
T
Q)t,x(
, (5.1)
ɝɞɟ
1
t
m – ɡɚɞɚɧɧɨɟ ɰɟɥɨɟ,
ȝ
t
s
x
rs
m
ȝ
ȝ,stx
DD)x(a)D,D,x(A
¦¦
d
1
0
ɥɢɧɟɣɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ, ɩɨɪɹɞɨɤ ɤɨɬɨɪɨɝɨ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ
1
mr (ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ ɫɦ. ɜ §1), )x(a
s
P
,
,
)
t
,x(
f
– ɡɚ-
ɞɚɧɧɵɟ ɮɭɧɤɰɢɢ.
Ⱦɥɹ ɭɪɚɜɧɟɧɢɹ (5.1) ɫɬɚɜɹɬɫɹ ɨɞɧɨɪɨɞɧɵɟ ɧɚɱɚɥɶɧɵɟ ɢ ɤɪɚɟɜɵɟ ɭɫɥɨ-
ɜɢɹ:
00
)
,x(u
, 0
0
w
w
t
),x(u
,
!
, 0
0
1
1
w
w
m
m
t
),x(u
,
:
x
, (5.2)
0
)t,x(u)D,x(l
xj,i
,
i
x
*
,
T
x
0 , (5.3)
D
,, !1
i
,
i
rj ,, !1
,
ɝɞɟ
)D,x(l
xj,i
– ɥɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɨɩɟɪɚɬɨɪɵ, 1
t
i
r – ɡɚɞɚɧ-
ɧɵɟ ɰɟɥɵɟ (ɬɨ ɟɫɬɶ ɧɚ ɤɚɠɞɨɦ ɭɱɚɫɬɤɟ
i
*
,
D
,, !1
i
ɝɪɚɧɢɰɵ
*
ɦɨɠɟɬ
ɫɬɚɜɢɬɶɫɹ ɧɟɫɤɨɥɶɤɨ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ, ɫɜɨɢɯ ɞɥɹ ɤɚɠɞɨɝɨ ɭɱɚɫɬɤɚ).