ISSN 1812-3368. ȼɟɫɬɧɢɤ ɆȽɌɍ ɢɦ. ɇ.ɗ. Ȼɚɭɦɚɧɚ. ɋɟɪ. «ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ». 2012
115
ɡɚɞɚɱ ɞɥɹ ɭɩɪɭɝɨɣ ɩɨɥɭɩɥɨɫɤɨɫɬɢ. Ɂɚɞɚɱɚ ɨ ɞɟɣɫɬɜɢɢ ɜɟɪɬɢɤɚɥɶɧɨɝɨ
ɲɬɚɦɩɚ ɜ ɪɚɦɤɚɯ ɷɬɨɣ ɦɨɞɟɥɢ ɪɚɧɟɟ ɪɚɫɫɦɚɬɪɢɜɚɥɚɫɶ ɜ ɪɚɛɨɬɟ [10].
ȼ ɪɚɡɜɢɬɢɟ [10] ɨɛɫɭɠɞɚɟɬɫɹ ɩɪɢɛɥɢɠɟɧɧɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɞɥɹ
ɭɩɪɭɝɨɣ ɩɨɥɭɩɥɨɫɤɨɫɬɢ, ɱɚɫɬɶ ɩɨɜɟɪɯɧɨɫɬɢ ɤɨɬɨɪɨɣ ɩɨɤɪɵɬɚ ɝɢɛɤɨɣ
ɧɟɪɚɫɬɹɠɢɦɨɣ ɦɟɦɛɪɚɧɨɣ, ɚ ɬɚɤɠɟ ɞɥɹ ɛɨɥɟɟ ɨɛɳɟɣ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚ-
ɱɢ, ɞɥɹ ɤɨɬɨɪɨɣ ɧɚ ɱɚɫɬɢ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɞɚɧɵ ɨɛɚ ɩɟɪɟɦɟɳɟɧɢɹ. ɉɨɤɚ-
ɡɚɧɨ, ɱɬɨ ɜ ɩɟɪɜɨɣ ɢɡ ɭɩɨɦɹɧɭɬɵɯ ɡɚɞɚɱ, ɤɚɤ ɢ ɜ ɪɚɛɨɬɟ [10], ɩɨɥɟ ɜɨɥ-
ɧɵ Ɋɷɥɟɹ ɦɨɠɟɬ ɛɵɬɶ ɜɵɪɚɠɟɧɨ ɜ ɬɟɪɦɢɧɚɯ ɨɞɧɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ
ɮɭɧɤɰɢɢ. ɗɬɨ ɹɜɥɹɟɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦ ɭɩɪɨɳɟɧɢɟɦ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ
ɬɪɚɞɢɰɢɨɧɧɨɣ ɮɨɪɦɭɥɢɪɨɜɤɨɣ ɡɚɞɚɱɢ. Ɋɚɫɫɦɨɬɪɟɧ ɩɪɢɦɟɪ, ɢɥɥɸɫɬɪɢ-
ɪɭɸɳɢɣ ɷɮɮɟɤɬɢɜɧɨɫɬɶ ɩɪɟɞɥɚɝɚɟɦɨɝɨ ɩɨɞɯɨɞɚ. ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ
ɚɧɚɥɨɝɢɱɧɚɹ ɫɤɚɥɹɪɧɚɹ ɩɨɫɬɚɧɨɜɤɚ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɜɨɡɦɨɠɧɨɣ; ɞɥɹ ɧɟɝɨ
ɪɟɲɟɧɢɟ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɞɜɟ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ. ɉɨɥɭɱɟɧɧɵɟ
ɪɟɡɭɥɶɬɚɬɵ ɦɨɝɭɬ ɧɚɣɬɢ ɪɹɞ ɢɧɬɟɪɟɫɧɵɯ ɩɪɢɥɨɠɟɧɢɣ, ɜ ɱɚɫɬɧɨɫɬɢ ɩɪɢ
ɢɫɫɥɟɞɨɜɚɧɢɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɪɟɳɢɧ ɜ ɭɩɪɭɝɢɯ ɬɟɥɚɯ [11].
Ⱥɫɢɦɩɬɨɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɞɥɹ ɜɨɥɧɵ Ɋɷɥɟɹ ɜ ɫɥɭɱɚɟ ɭɩɪɭɝɨɣ
ɩɨɥɭɩɥɨɫɤɨɫɬɢ.
ɉɪɢɜɟɞɟɦ ɤɪɚɬɤɨɟ ɨɩɢɫɚɧɢɟ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɣ ɦɨɞɟ-
ɥɢ ɞɥɹ ɜɨɥɧɵ Ɋɷɥɟɹ [3]. Ɋɚɫɫɦɨɬɪɢɦ ɭɩɪɭɝɭɸ ɩɨɥɭɩɥɨɫɤɨɫɬɶ
, 0
.
x
y
f f d f
ȼ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ, ɡɚɞɚɧɧɵɯ ɜ ɬɟɪɦɢ-
ɧɚɯ ɧɚɩɪɹɠɟɧɢɣ,
0
0
( , );
( , ),
yy y
xy y
P x t
Q x t
V
V
(1)
ɡɚɬɭɯɚɧɢɟ ɜ ɝɥɭɛɶ ɫɪɟɞɵ ɨɩɢɫɵɜɚɟɬɫɹ ɩɫɟɜɞɨɫɬɚɬɢɱɟɫɤɢɦɢ ɷɥɥɢɩɬɢ-
ɱɟɫɤɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ
2
2
2
1 2
2
0;
k
y
x
M
M
w
w
w
w
(2)
2
2
2
2 2
2
0,
k
y
x
\
\
w
w
w
w
(3)
ɝɞɟ
M
ɢ
\
ɭɩɪɭɝɢɟ ɩɨɬɟɧɰɢɚɥɵ Ʌɚɦɟ;
1/ 2 2
2
1
R
m
m
c
k
c
§
·
¨
¸
©
¹
ɩɨɫɬɨɹɧ-
ɧɵɟ,
1, 2
m
(
1
,
c
2
,
c
R
c
ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɩɪɨɞɨɥɶɧɨɣ, ɩɨ-
ɩɟɪɟɱɧɨɣ ɢ ɪɷɥɟɟɜɫɤɨɣ ɜɨɥɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ).
ȼɨɡɦɨɠɧɨɫɬɶ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɩɨɥɹ ɜɨɥɧɵ Ɋɷɥɟɹ ɜ ɬɟɪɦɢɧɚɯ ɨɞɧɨɣ
ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ, ɭɫɬɚɧɨɜɥɟɧɧɚɹ ɜ ɪɚɛɨɬɟ [4], ɩɨɡɜɨɥɹɟɬ ɫɜɹ-
ɡɚɬɶ ɩɨɬɟɧɰɢɚɥɵ ɫɨɨɬɧɨɲɟɧɢɹɦɢ
*
*
1
2
2
2
2
2
2
2
2
2
2
2
,
,
,
, ;
,
,
,
, ,
1
1
k
k
x k y t
x k y t
x k y t
x k y t
k
k
\
M
M
\
(4)