Куликов А.А. (сост.) Смешанные задачи для уравнения теплопроводности и уравнения колебаний
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ɎȿȾȿɊȺɅɖɇɈȿ ȺȽȿɇɌɋɌȼɈ ɉɈ ɈȻɊȺɁɈȼȺɇɂɘ
ȽɈɋɍȾȺɊɋɌȼȿɇɇɈȿ ɈȻɊȺɁɈȼȺɌȿɅɖɇɈȿ
ɍɑɊȿɀȾȿɇɂȿ
ȼɕɋɒȿȽɈ ɉɊɈɎȿɋɋɂɈɇȺɅɖɇɈȽɈ ɈȻɊȺɁɈȼȺɇɂə
«ȼɈɊɈɇȿɀɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ
ɍɇɂȼȿɊɋɂɌȿɌ»
ɋɆȿɒȺɇɇɕȿ ɁȺȾȺɑɂ ȾɅə
ɍɊȺȼɇȿɇɂə
ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ
ɂ ɍɊȺȼɇȿɇɂə ɄɈɅȿȻȺɇɂɃ
ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɜɭɡɨɜ
ɋɨɫɬɚɜɢɬɟɥɶ
Ⱥ.Ⱥ. Ʉɭɥɢɤɨɜ
ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ
ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ
2007
2
ɍɬɜɟɪɠɞɟɧɨ ɧɚɭɱɧɨ-ɦɟɬɨɞɢɱɟɫɤɢɦ ɫɨɜɟɬɨɦ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ 28 ɮɟɜɪɚɥɹ
2007 ɝ., ɩɪɨɬɨɤɨɥ ʋ 6
Ɋɟɰɟɧɡɟɧɬ ɞ-ɪɮ.-ɦ. ɧɚɭɤ, ɩɪɨɮ. Ɇ.Ⱥ. Ⱥɪɬɺɦɨɜ
ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɩɨɞɝɨɬɨɜɥɟɧɨ ɧɚ ɤɚɮɟɞɪɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶ-
ɧɵɯ ɭɪɚɜɧɟɧɢɣ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪ-
ɫɢɬɟɬɚ.
Ɋɟɤɨɦɟɧɞɭɟɬɫɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 3–5 ɤɭɪɫɨɜ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ.
Ⱦɥɹ ɫɩɟɰɢɚɥɶɧɨɫɬɢ: 010501 (010200) – ɉɪɢɤɥɚɞɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɢ ɢɧɮɨɪɦɚ-
ɬɢɤɚ
3
ɉɊȿȾɂɋɅɈȼɂȿ
ɇɚɫɬɨɹɳɟɟ ɩɨɫɨɛɢɟ ɫɨɞɟɪɠɢɬ ɜɜɟɞɟɧɢɟ ɜ ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɪɟ-
ɲɟɧɢɹ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ.
ɉɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɩɪɟɠɞɟ ɜɫɟɝɨ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 3 ɤɭɪɫɚ ɮɚɤɭɥɶɬɟ-
ɬɚ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ, ɢɧɮɨɪɦɚɬɢɤɢ ɢ ɦɟɯɚɧɢɤɢ, ɢɡɭɱɚɸɳɢɯ ɞɢɫɰɢɩ-
ɥɢɧɭ «ɍɪɚɜɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ». Ɉɧɨ ɛɭɞɟɬ ɩɨɥɟɡɧɨ ɬɚɤɠɟ ɩɪɢ
ɜɵɩɨɥɧɟɧɢɢ ɤɭɪɫɨɜɵɯ ɢ ɞɢɩɥɨɦɧɵɯ ɪɚɛɨɬ ɢ ɡɚɞɚɧɢɣ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ
ɩɪɚɤɬɢɤɭɦɚ, ɫɨɞɟɪɠɚɳɢɯɫɹ ɜ [7].
ȼ § 1 ɢ § 2 ɩɪɢɜɨɞɢɬɫɹ ɪɹɞ ɜɚɠɧɟɣɲɢɯ ɩɨɧɹɬɢɣ, ɢɫɩɨɥɶɡɭɟɦɵɯ ɜ ɤɭɪɫɟ
«ɍɪɚɜɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ». ȼ § 3 ɢ § 4 ɩɪɢɜɟɞɟɧɚ ɩɨɫɬɚɧɨɜɤɚ
ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɭɪɚɜɧɟ-
ɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ. ȼ § 5 ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɩɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ, ɩɨɡɜɨ-
ɥɹɸɳɢɣ ɫɜɨɞɢɬɶ ɡɚɞɚɱɢ ɨɛɳɟɝɨ ɜɢɞɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɵɯ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ
ɭɪɚɜɧɟɧɢɣ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɡɚɞɚɱɚɦ ɞɥɹ ɨɞ-
ɧɨɪɨɞɧɵɯ ɭɪɚɜɧɟɧɢɣ
. Ɉɬɦɟɬɢɦ, ɱɬɨ ɦɚɬɟɪɢɚɥ § 5 ɨɬɫɭɬɫɬɜɭɟɬ ɜ ɨɛɳɟɞɨɫ-
ɬɭɩɧɨɣ ɭɱɟɛɧɨɣ ɥɢɬɟɪɚɬɭɪɟ ɩɨ ɭɪɚɜɧɟɧɢɹɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ. ȼ § 6
ɢɡɭɱɚɟɬɫɹ ɜɚɠɧɵɣ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ, ɩɪɟɞɫɬɚɜ-
ɥɹɸɳɢɣ ɧɚɢɛɨɥɶɲɢɣ ɢɧɬɟɪɟɫ ɞɥɹ ɩɪɢɥɨɠɟɧɢɣ. ȼ § 7 ɢ § 8 ɪɚɫɫɦɨɬɪɟɧɨ
ɪɟɲɟɧɢɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɭɪɚɜɧɟɧɢɹ
ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ. ȼ ɩɪɢɥɨɠɟɧɢɢ ɩɪɢɜɟ-
ɞɟɧɵ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ
ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɞɥɹ ɡɚɞɚɱɢ ɒɬɭɪɦɚ –
Ʌɢɭɜɢɥɥɹ ɫ ɭɤɚɡɚɧɧɵɦɢ ɜ § 6 ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ.
ɉɨɫɨɛɢɟ ɫɨɞɟɪɠɢɬ ɪɹɞ ɭɩɪɚɠɧɟɧɢɣ ɢ ɡɚɞɚɱ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟ-
ɲɟɧɢɹ ɫɬɭɞɟɧɬɚɦɢ.
§ 1. ɉɨɧɹɬɢɟ ɨɛ ɭɪɚɜɧɟɧɢɹɯ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ
ɢ ɨɛ ɭɪɚɜɧɟɧɢɹɯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ
1. ɍɪɚɜɧɟɧɢɹ, ɢɡɭɱɚɟɦɵɟ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɟ. Ɇɚɬɟɦɚɬɢɱɟ-
ɫɤɚɹ ɮɢɡɢɤɚ – ɧɚɭɤɚ, ɢɡɭɱɚɸɳɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟ-
ɧɢɣ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɢɡɢɤɚ ɧɚɯɨɞɢɬɫɹ ɧɚ ɫɬɵɤɟ ɦɚɬɟɦɚɬɢɤɢ ɢ ɮɢɡɢɤɢ.
ɗɬɚ ɧɚɭɤɚ ɬɟɫɧɨ ɫɜɹɡɚɧɚ ɫ ɮɢɡɢɤɨɣ ɜ ɬɨɣ ɱɚɫɬɢ, ɤɨɬɨɪɚɹ ɤɚɫɚɟɬɫɹ ɩɨɫɬɪɨɟ-
ɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ, ɢ ɜ ɬɨ ɠɟ ɜɪɟɦɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɢɡɢɤɚ – ɷɬɨ
ɪɚɡɞɟɥ ɦɚɬɟɦɚɬɢɤɢ, ɬɚɤ ɤɚɤ
ɦɟɬɨɞɵ ɢɫɫɥɟɞɨɜɚɧɢɹ ɦɨɞɟɥɟɣ ɹɜɥɹɸɬɫɹ ɦɚɬɟ-
ɦɚɬɢɱɟɫɤɢɦɢ.
Ɇɧɨɝɢɟ ɡɚɞɚɱɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɜɨɡɧɢɤɚɸɳɢɟ ɜ ɬɚɤɢɯ ɪɚɡɞɟ-
ɥɚɯ ɮɢɡɢɤɢ, ɤɚɤ ɝɢɞɪɨɞɢɧɚɦɢɤɚ, ɬɟɨɪɢɹ ɭɩɪɭɝɨɫɬɢ, ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɚ, ɤɜɚɧ-
ɬɨɜɚɹ ɦɟɯɚɧɢɤɚ ɢ ɞɪɭɝɢɯ, ɩɪɢɜɨɞɹɬ ɤ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɹɦ ɫ ɱɚ-
ɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ. ȼ ɨɬɥɢɱɢɟ ɨɬ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ
ɭɪɚɜɧɟɧɢɣ, ɜ ɤɨɬɨɪɵɯ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɣ ɩɟɪɟ-
4
ɦɟɧɧɨɣ, ɜ ɭɪɚɜɧɟɧɢɹɯ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ ɡɚ-
ɜɢɫɢɬ ɨɬ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ, ɚ ɫɚɦɨ ɭɪɚɜɧɟɧɢɟ ɫɨɞɟɪɠɢɬ ɱɚɫɬɧɵɟ
ɩɪɨɢɡɜɨɞɧɵɟ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ.
ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɧɚɪɹɞɭ ɫ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ
ɞɥɹ ɨɩɢɫɚɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ ɢɫɩɨɥɶɡɭɸɬɫɹ ɢɧɬɟɝɪɚɥɶɧɵɟ ɢ ɢɧɬɟɝɪɨ-
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ. Ɉɞɧɚɤɨ ɜ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬ-
ɪɢɜɚɬɶ ɬɨɥɶɤɨ ɬɟ ɡɚɞɚɱɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ,
ɤɨɬɨɪɵɟ ɩɪɢɜɨɞɹɬ ɤ
ɭɪɚɜɧɟɧɢɹɦ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ.
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞ-
ɧɵɦɢ ɢɦɟɟɬ ɜɢɞ:
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x(uu
– ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, F –
ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɫɜɨɢɯ ɚɪɝɭɦɟɧɬɨɜ. Ɉɛɵɱɧɨ ɜɟɤɬɨɪ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟ-
ɦɟɧɧɵɯ ɜ ɭɪɚɜɧɟɧɢɢ (1.1) ɢɡɦɟɧɹɟɬɫɹ ɧɚ ɧɟɤɨɬɨɪɨɦ ɨɬɤɪɵɬɨɦ ɦɧɨɠɟɫɬɜɟ
,
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ɤɨɬɨɪɨɟ ɧɚɡɵɜɚɟɬɫɹ ɨɛɥɚɫɬɶɸ ɡɚɞɚɧɢɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ.
Ʉɥɚɫɫɢɱɟɫɤɢɦ (ɪɟɝɭɥɹɪɧɵɦ) ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (1.1) ɧɚ ɨɬɤɪɵɬɨɦ
ɦɧɨɠɟɫɬɜɟ
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ɜɦɟɫɬɟ ɫɨ ɜɫɟɦɢ ɫɜɨɢɦɢ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ, ɜɯɨ-
ɞɹɳɢɦɢ ɜ ɭɪɚɜɧɟɧɢɟ (1.1), ɢ ɨɛɪɚɳɚɸɳɚɹ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ
0
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x ɜ ɬɨɠɞɟɫɬɜɨ.
ɉɨɪɹɞɤɨɦ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɧɚɡɵɜɚɟɬɫɹ ɦɚɤɫɢ-
ɦɚɥɶɧɵɣ ɢɡ ɩɨɪɹɞɤɨɜ ɜɯɨɞɹɳɢɯ ɜ ɧɟɝɨ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ.
ɍɪɚɜɧɟɧɢɟ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɵɦ, ɟɫɥɢ ɨɧɨ
ɥɢɧɟɣɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ ɢ ɟɺ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ.
ɇɚɩɪɢɦɟɪ, ɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞ-
ɤɚ ɢɦɟɟɬ ɜɢɞ:
,)x(fu)x(c
x
u
)x(b
xx
u
)x(a
i
n
i
i
ji
n
j,i
j,i
w
w
ww
w
¦¦
1
2
1
(1.2)
ɝɞɟ
,
j,i
a ,
i
b ɫ, f – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. Ɏɭɧɤɰɢɢ ,
j,i
a ,
i
b ɫ ɧɚɡɵɜɚɸɬɫɹ ɤɨ-
ɷɮɮɢɰɢɟɧɬɚɦɢ ɭɪɚɜɧɟɧɢɹ (1.2), ɚ ɮɭɧɤɰɢɹ f – ɫɜɨɛɨɞɧɵɦ ɱɥɟɧɨɦ (ɩɪɚɜɨɣ
ɱɚɫɬɶɸ). ȿɫɥɢ
,0
{
)
x(
f
ɬɨ ɭɪɚɜɧɟɧɢɟ (1.2) ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦ, ɜ
ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ – ɧɟɨɞɧɨɪɨɞɧɵɦ.
ɍɪɚɜɧɟɧɢɟ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɧɚɡɵɜɚɟɬɫɹ ɤɜɚɡɢɥɢɧɟɣɧɵɦ, ɟɫ-
ɥɢ ɨɧɨ ɥɢɧɟɣɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɫɟɯ ɫɬɚɪɲɢɯ ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ ɧɟɢɡɜɟɫɬɧɨɣ
ɮɭɧɤɰɢɢ. Ʉɜɚɡɢɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɢɦɟɟɬ ɜɢɞ:
5
,0
1
2
1
1
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
ww
w
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
¦
nji
n
j,i
n
j,i
x
u
,,
x
u
,u,xB
xx
u
x
u
,,
x
u
,u,xA !!
ɝɞɟ
j,i
A
ɢ ȼ – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ.
ɇɚɢɛɨɥɟɟ ɱɚɫɬɨ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɟ ɜɫɬɪɟɱɚɸɬɫɹ ɞɢɮɮɟɪɟɧɰɢ-
ɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ. ɇɢɠɟ ɩɪɢ-
ɜɨɞɹɬɫɹ ɧɟɤɨɬɨɪɵɟ ɩɪɢɦɟɪɵ ɬɚɤɢɯ ɭɪɚɜɧɟɧɢɣ.
ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
,F
x
u
p
xt
u
n
i
ii
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
w
w
¦
1
U
(1.3)
ɨɩɢɫɵɜɚɟɬ, ɜ ɱɚɫɬɧɨɫɬɢ, ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ, ɚ ɬɚɤɠɟ ɩɪɨɰɟɫɫ
ɞɢɮɮɭɡɢɢ ɱɚɫɬɢɰ ɜ ɫɪɟɞɟ. ȼ ɭɪɚɜɧɟɧɢɢ (1.3) ɢ ɞɚɥɟɟ ɜ ɧɚɫɬɨɹɳɟɦ ɩɚɪɚɝɪɚ-
ɮɟ
,
U
p ɢ F ɨɛɨɡɧɚɱɚɸɬ ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ, ɡɚɜɢɫɹɳɢɟ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɨɬ
ɜɟɤɬɨɪɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ
)x,,x(x
n
!
1
(ɱɢɫɥɨ n ɦɨɠɟɬ
ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ 1, 2, 3), ɨɬ ɜɪɟɦɟɧɢ t ,
ɚ ɬɚɤɠɟ ɨɬ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ
)
t,
x
(u ɢ ɟɟ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. ɉɪɢ ɷɬɨɦ ɮɭɧɤɰɢɢ
U
ɢ p ɭɞɨɜɥɟɬɜɨ-
ɪɹɸɬ ɭɫɥɨɜɢɹɦ
,const 0
0
! t
U
U
.const 0
0
!
t
pp ȿɫɥɢ ,
0
U
U
{
,
0
pp
{
ɬɨ ɭɪɚɜɧɟɧɢɟ (1.3) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
,ǻ fua
t
u
w
w
2
ɝɞɟ ,
00
U
pa
0
U
Ff
ɢ
¦
w
w
n
i
i
x
1
2
2
ǻ
— ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ, ɧɚɡɵɜɚɟɦɵɣ ɨɩɟɪɚɬɨɪɨɦ Ʌɚɩɥɚɫɚ.
ɍɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɣ
F
x
u
p
xt
u
n
i
ii
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
w
w
¦
1
2
2
U
(1.4)
ɨɩɢɫɵɜɚɟɬ, ɜ ɱɚɫɬɧɨɫɬɢ, ɦɚɥɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧ ɢ ɫɬɟɪɠɧɟɣ ɩɪɢ
,1
n
ɦɟɦɛɪɚɧ ɩɪɢ
,2
n ɤɨɥɟɛɚɧɢɹ ɜ ɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɟ ɩɪɢ ,3
n
ɚ ɬɚɤɠɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɤɨɥɟɛɚɧɢɹ. ɉɪɢ
,
0
U
U
{
0
pp
{
ɭɪɚɜɧɟɧɢɟ (1.4)
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
.ǻ fua
t
u
w
w
2
2
2
(1.5)
ɍɪɚɜɧɟɧɢɟ (1.5) ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ.
6
ȼ ɫɥɭɱɚɟ ɫɬɚɰɢɨɧɚɪɧɵɯ (ɬɨ ɟɫɬɶ ɧɟ ɡɚɜɢɫɹɳɢɯ ɨɬ ɜɪɟɦɟɧɢ) ɹɜɥɟɧɢɣ
ɮɭɧɤɰɢɢ
,
U
p, F , u ɧɟ ɡɚɜɢɫɹɬ ɨɬ t , ɢ ɭɪɚɜɧɟɧɢɹ (1.3), (1.4) ɩɟɪɟɯɨɞɹɬ ɜ
ɭɪɚɜɧɟɧɢɟ
.F
x
u
p
x
n
i
ii
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
¦
1
(1.6)
ɉɪɢ
0
pp
{
ɭɪɚɜɧɟɧɢɟ (1.6) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
,ǻ
f
u
(1.7)
ɝɞɟ
.
0
pFf
ɍɪɚɜɧɟɧɢɟ (1.7) ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɉɭɚɫɫɨɧɚ, ɚ ɩɪɢ
0
{
f
– ɭɪɚɜɧɟɧɢɟɦ Ʌɚɩɥɚɫɚ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɭɪɚɜɧɟɧɢɸ (1.7) ɭɞɨɜɥɟɬɜɨɪɹ-
ɸɬ ɩɨɬɟɧɰɢɚɥɵ ɷɥɟɤɬɪɨɫɬɚɬɢɱɟɫɤɨɝɨ ɩɨɥɹ ɢ ɩɨɥɹ ɬɹɝɨɬɟɧɢɹ.
2. ɇɟɤɨɬɨɪɵɟ ɨɛɨɡɧɚɱɟɧɢɹ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɞɚɥɶɧɟɣɲɟɦ. ɉɭɫɬɶ
)ss(s
n
,,!
1
– ɜɟɤɬɨɪ, ɤɨɦɩɨɧɟɧɬɵ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ-
ɦɢ ɰɟɥɵɦɢ ɱɢɫɥɚɦɢ. Ɍɚɤɨɣ ɜɟɤɬɨɪ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɦɭɥɶɬɢɢɧɞɟɤɫɚ. ɑɢɫɥɨ
n
ss !
1
ɧɚɡɵɜɚɟɬɫɹ ɞɥɢɧɨɣ ɦɭɥɶɬɢɢɧɞɟɤɫɚ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ ɱɟɪɟɡ s .
ɑɟɪɟɡ
x
D ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɮɨɪɦɚɥɶɧɵɣ ɜɟɤɬɨɪ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ
ɨɩɟɪɚɬɨɪɨɜ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɩɨ ɩɟɪɟɦɟɧɧɵɦ
n
x,,x !
1
:
,)D,,D(D
n
xxx
!
1
,
ix
xD
i
w
w
.n,,
i
!1
ɋɢɦɜɨɥɨɦ
s
x
D ɨɛɨɡɧɚɱɢɦ
ɫɦɟɲɚɧɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨɪɹɞɤɚ
s : ,
n
n
s
x
s
x
s
x
DDD "
1
1
ɬɚɤ ɱɬɨ
;)(
n
s
n
s
s
s
x
xx
u
xuD
ww
w
"
1
1
ɩɪɢ ɷɬɨɦ, ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ,
)x(u)x(uD
0
. ɂɧɨɝɞɚ ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ
ɬɚɤɠɟ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ ɞɥɹ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ:
,
i
x
x
u
u
i
w
w
,
2
2
i
xx
x
u
u
ii
w
w
,
ji
xx
xx
u
u
ji
ww
w
2
n,,
j
,
i
!1
ɢɬ. ɩ.
3. ɉɪɨɫɬɪɚɧɫɬɜɨ ɮɭɧɤɰɢɣ
)l,(L 0
2
. ɑɟɪɟɡ )l,(L 0
2
ɛɭɞɟɦ ɨɛɨɡɧɚ-
ɱɚɬɶ ɫɨɜɨɤɭɩɧɨɫɬɶ (ɩɪɨɫɬɪɚɧɫɬɜɨ) ɜɫɟɯ ɨɩɪɟɞɟɥɟɧɧɵɯ ɧɚ ɢɧɬɟɪɜɚɥɟ
)
l,( 0
ɮɭɧɤɰɢɣ
)
x
(
f
(ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɤɨɦɩɥɟɤɫɧɨɡɧɚɱɧɵɯ), ɞɥɹ ɤɨɬɨɪɵɯ
.
f
³
xd)x(f
l
0
2
(1.8)
Ɉɬɦɟɬɢɦ, ɱɬɨ ɷɥɟɦɟɧɬɚɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ
)l,(L 0
2
ɹɜɥɹɸɬɫɹ ɜɫɟ ɤɭɫɨɱ-
ɧɨ-ɧɟɩɪɟɪɵɜɧɵɟ ɧɚ ɢɧɬɟɪɜɚɥɟ
)
l,( 0 ɮɭɧɤɰɢɢ
)
x(
f
, ɞɥɹ ɤɨɬɨɪɵɯ ɫɯɨɞɢɬ-
7
ɫɹ ɢɧɬɟɝɪɚɥ (1.8), ɢ, ɜ ɱɚɫɬɧɨɫɬɢ, ɜɫɟ ɮɭɧɤɰɢɢ, ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɵɟ ɧɚ
ɨɬɪɟɡɤɟ
][ l,0 .
Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɮɭɧɤɰɢɣ
)l,(L 0
2
ɥɢɧɟɣɧɨ, ɬɨ ɟɫɬɶ ɟɫɥɢ
ɮɭɧɤɰɢɢ
)l,(Lgf 0
2
,
, ɬɨ
)l,(Lgf 0
2
, ɢ ɟɫɥɢ c – ɩɪɨɢɡɜɨɥɶɧɚɹ
ɤɨɦɩɥɟɤɫɧɚɹ ɩɨɫɬɨɹɧɧɚɹ, ɬɨ
)l,(Lfc 0
2
(ɩɨɤɚɡɚɬɶ ɷɬɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ).
ȼ ɩɪɨɫɬɪɚɧɫɬɜɟ
)l,(L 0
2
ɦɨɠɧɨ ɜɜɟɫɬɢ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ
)
g
,
f
(
ɮɭɧɤɰɢɣ
)
x
(
f
ɢ
)
x
(
g
:
³
l
xd)x(g)x(f)g,f(
0
(ɱɟɪɟɡ
f ɨɛɨɡɧɚɱɚɟɬɫɹ ɮɭɧɤɰɢɹ, ɤɨɦɩɥɟɤɫɧɨ ɫɨɩɪɹɠɟɧɧɚɹ ɤ ɮɭɧɤɰɢɢ
f
),
ɢ ɧɨɪɦɭ
f ɮɭɧɤɰɢɢ
f
:
21
0
2
21
¸
¸
¹
·
¨
¨
©
§
³
l
xd)x(f)ff(f , . (1.9)
ɇɨɪɦɚ (1.9) ɨɛɥɚɞɚɟɬ ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣɫɬɜɚɦɢ:
1)
0
t
f , ɩɪɢɱɟɦ 0
f ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ ɮɭɧɤɰɢɹ
)
x(
f
ɪɚɜɧɚ ɧɭɥɸ ɩɨɱɬɢ ɜɟɡɞɟ ɧɚ ɢɧɬɟɪɜɚɥɟ
)
l,( 0 (ɬɨ ɟɫɬɶ ɪɚɜɧɚ ɧɭɥɸ ɜɫɸɞɭ ɧɚ
ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ, ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ, ɜɨɡɦɨɠɧɨ, ɦɧɨɠɟɫɬɜɚ ɦɟɪɵ ɧɭɥɶ
1
);
2)
fcfc
ɞɥɹ ɥɸɛɨɝɨ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ c;
3) ɂɦɟɟɬ ɦɟɫɬɨ ɧɟɪɚɜɟɧɫɬɜɨ ɬɪɟɭɝɨɥɶɧɢɤɚ
gfgf
d
. (1.10)
ɋɜɨɣɫɬɜɚ 1) ɢ 2) ɜɵɬɟɤɚɸɬ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɢɡ ɨɩɪɟɞɟɥɟɧɢɹ ɧɨɪɦɵ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɧɟɪɚɜɟɧɫɬɜɚ (1.10) ɩɪɢɜɟɞɟɧɨ, ɧɚɩɪɢɦɟɪ, ɜ [4, ɝɥ. I, §1].
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɧɨɠɟɫɬɜɨ ɮɭɧɤɰɢɣ
)l,(L 0
2
, ɧɚɞɟɥɟɧɧɨɟ ɧɨɪɦɨɣ
(1.9),
ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɧɨɪɦɢɪɨɜɚɧɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ.
Ɏɭɧɤɰɢɢ
g
f
,
ɢɡ )l,(L 0
2
ɧɚɡɵɜɚɸɬɫɹ ɨɪɬɨɝɨɧɚɥɶɧɵɦɢ ɜ ɷɬɨɦ ɩɪɨ-
ɫɬɪɚɧɫɬɜɟ, ɟɫɥɢ ɢɯ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɪɚɜɧɨ ɧɭɥɸ, ɬɨ ɟɫɬɶ
0
0
³
l
xd)x(g)x(f .
1
Ƚɨɜɨɪɹɬ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɧɚ ɱɢɫɥɨɜɨɣ ɩɪɹɦɨɣ ɢɦɟɟɬ ɦɟɪɭ ɧɭɥɶ, ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ
0
!
H
ɨɧɨ ɦɨɠɟɬ ɛɵɬɶ ɩɨɤɪɵɬɨ ɢɧɬɟɪɜɚɥɚɦɢ, ɫɭɦɦɚ ɞɥɢɧ ɤɨɬɨɪɵɯ ɦɟɧɶɲɟ
H
(ɫɦ., ɧɚ-
ɩɪɢɦɟɪ, [4, ɝɥ. I, §1] ).
8
§2. Ʉɪɚɟɜɵɟ ɡɚɞɚɱɢ ɞɥɹ ɭɪɚɜɧɟɧɢɣ ɫ ɱɚɫɬɧɵɦɢ
ɩɪɨɢɡɜɨɞɧɵɦɢ
1. Ʉɨɪɪɟɤɬɧɵɟ ɤɪɚɟɜɵɟ ɡɚɞɚɱɢ. ɉɪɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ ɦɨɞɟɥɢɪɨɜɚ-
ɧɢɢ ɤɚɤɨɝɨ-ɥɢɛɨ ɮɢɡɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɧɟɞɨɫɬɚɬɨɱɧɨ ɡɚɞɚɜɚɬɶ ɬɨɥɶɤɨ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɦɭ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɢɫɤɨɦɚɹ ɮɭɧɤɰɢɹ,
ɨɩɢɫɵɜɚɸɳɚɹ ɞɚɧɧɵɣ ɩɪɨɰɟɫɫ (ɩɨɞ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɟɣ ɦɨɠɟɬ ɩɨɧɢɦɚɬɶɫɹ
ɤɚɤ ɫɤɚɥɹɪɧɚɹ, ɬɚɤ ɢ ɜɟɤɬɨɪɧɚɹ ɮɭɧɤɰɢɹ, ɚ ɩɨɞ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟ-
ɧɢɟɦ – ɥɢɛɨ ɨɞɧɨ ɭɪɚɜɧɟɧɢɟ, ɥɢɛɨ ɡɚɩɢɫɚɧɧɚɹ ɜ ɜɟɤɬɨɪɧɨɣ ɮɨɪɦɟ ɫɢɫɬɟɦɚ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ
ɭɪɚɜɧɟɧɢɣ, ɤɨɬɨɪɨɣ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ
ɜɟɤɬɨɪɧɚɹ ɮɭɧɤɰɢɹ). ɗɬɨ ɫɜɹɡɚɧɨ ɫ ɬɟɦ, ɱɬɨ ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ, ɜɨɨɛɳɟ
ɝɨɜɨɪɹ, ɢɦɟɸɬ ɨɞɧɨɡɧɚɱɧɵɣ ɯɚɪɚɤɬɟɪ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɨɩɢɫɵɜɚɸɳɢɟ ɢɯ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢɦɟɸɬ ɦɧɨɠɟɫɬɜɨ ɪɟɲɟɧɢɣ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ,
ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɛɵɤɧɨɜɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɡɚɜɢɫɢɬ ɨɬ
ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɫɬɨɹɧɧɵɯ, ɱɢɫɥɨ ɤɨɬɨɪɵɯ ɪɚɜɧɨ ɩɨɪɹɞɤɭ ɭɪɚɜɧɟɧɢɹ, ɚɪɟ-
ɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ
, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɡɚɜɢɫɢɬ ɨɬ ɩɪɨ-
ɢɡɜɨɥɶɧɵɯ ɮɭɧɤɰɢɣ. ɇɚɩɪɢɦɟɪ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ
0
2
ww
w
yx
u
(2.1)
ɢɦɟɟɬ ɜɢɞ
,
)
y
(
g
)
x
(
f
)
y
,
x
(u
ɝɞɟ f ɢ g – ɩɪɨɢɡɜɨɥɶɧɵɟ ɞɢɮɮɟɪɟɧɰɢ-
ɪɭɟɦɵɟ ɮɭɧɤɰɢɢ (ɩɨɤɚɡɚɬɶ ɷɬɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ, ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɜ ɭɪɚɜɧɟ-
ɧɢɟ (2.1) ɩɨ x, ɚ ɩɨɥɭɱɟɧɧɨɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɭɪɚɜɧɟɧɢɟ – ɩɨ y).
Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɧɚɣɬɢ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɭɪɚɜɧɟɧɢɹ, ɨɩɢɫɵɜɚɸɳɟɟ ɤɨɧɤɪɟɬɧɵɣ ɮɢɡɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ, ɧɟɨɛɯɨɞɢɦɨ
ɡɚɞɚɬɶ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɤɨɬɨɪɵɦ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɢɫɤɨɦɚɹ ɮɭɧɤɰɢɹ.
Ɍɚɤɢɦɢ ɭɫɥɨɜɢɹɦɢ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɦɨɝɭɬ ɹɜɥɹɬɶɫɹ
ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɤɨ-
ɬɨɪɵɟ ɡɚɞɚɸɬ ɫɨɫɬɨɹɧɢɟ ɞɚɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɢ
ɤɪɚɟɜɵɟ (ɝɪɚɧɢɱɧɵɟ) ɭɫɥɨɜɢɹ, ɤɨɬɨɪɵɟ ɡɚɞɚɸɬ ɪɟɠɢɦ ɧɚ ɝɪɚɧɢɰɟ ɬɨɣ ɨɛɥɚɫ-
ɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɝɞɟ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɰɟɫɫ. ɇɚɱɚɥɶɧɵɟ ɢ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ
ɞɨɥɠɧɵ ɛɵɬɶ ɩɨɫɬɚɜɥɟɧɵ ɬɚɤ, ɱɬɨɛɵ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸ-
ɳɟɟ ɷɬɢɦ ɭɫɥɨɜɢɹɦ, ɫɭɳɟɫɬɜɨɜɚɥɨ ɢ ɛɵɥɨ ɟɞɢɧɫɬɜɟɧɧɵɦ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ
ɡɚɞɚɱɚ, ɡɚɤɥɸɱɚɸɳɚɹɫɹ ɜ ɧɚɯɨɠɞɟɧɢɢ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜ-
ɧɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨ ɡɚɞɚɧɧɵɦ ɧɚɱɚɥɶɧɵɦ ɢ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ, ɧɚ-
ɡɵɜɚɟɬɫɹ ɤɪɚɟɜɨɣ ɡɚɞɚɱɟɣ ɞɥɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ.
ȼ ɡɚɞɚɱɚɯ, ɨɩɢɫɵɜɚɸɳɢɯ ɪɟɚɥɶɧɵɟ ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ, ɢɫɯɨɞɧɵɟ
ɞɚɧɧɵɟ, ɧɚɩɪɢɦɟɪ, ɧɚɱɚɥɶɧɵɟ ɢ ɝɪɚɧɢɱɧɵɟ ɡɧɚɱɟɧɢɹ ɢɫɤɨɦɨɝɨ ɪɟɲɟɧɢɹ,
ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɪɚɜɧɟɧɢɣ, ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ ɢ ɬ. ɞ., ɨɛɵɱɧɨ ɩɨ-
ɥɭɱɚɸɬɫɹ ɩɭɬɟɦ ɢɡɦɟɪɟɧɢɣ
, ɢ ɩɨɷɬɨɦɭ ɹɜɥɹɸɬɫɹ ɩɪɢɛɥɢɠɟɧɧɵɦɢ. Ʉɚɤ ɩɪɚ-
ɜɢɥɨ, ɷɬɢ ɩɪɢɛɥɢɠɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɚɥɨ ɨɬɥɢɱɚɸɬɫɹ ɨɬ ɬɨɱɧɵɯ ɡɧɚɱɟɧɢɣ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɜɟɥɢɱɢɧ, ɢ ɩɨɝɪɟɲɧɨɫɬɶ ɢɦɟɟɬ ɫɥɭɱɚɣɧɵɣ ɯɚɪɚɤɬɟɪ. Ⱦɥɹ
ɨɞɧɨɡɧɚɱɧɨɣ ɮɢɡɢɱɟɫɤɨɣ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɧɭɠɧɨ ɛɵɬɶ ɭɜɟ-
ɪɟɧɧɵɦ ɜ ɬɨɦ, ɱɬɨ ɷɬɨ ɪɟɲɟɧɢɟ ɧɟ ɛɭɞɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɟɬɶ ɨɬ ɩɨɝɪɟɲ-
9
ɧɨɫɬɟɣ ɢɡɦɟɪɟɧɢɹ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ, ɬɨ ɟɫɬɶ «ɦɚɥɵɦ» (ɜ ɨɩɪɟɞɟɥɟɧɧɨɦ
ɫɦɵɫɥɟ) ɢɡɦɟɧɟɧɢɹɦ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ ɞɨɥɠɧɵ ɨɬɜɟɱɚɬɶ «ɦɚɥɵɟ» ɢɡɦɟɧɟ-
ɧɢɹ ɪɟɲɟɧɢɹ. Ɍɨɱɧɟɟ, ɪɟɲɟɧɢɟ ɞɨɥɠɧɨ ɧɟɩɪɟɪɵɜɧɨ ɡɚɜɢɫɟɬɶ (ɜ ɡɚɪɚɧɟɟ ɨɩ-
ɪɟɞɟɥɟɧɧɨɦ ɫɦɵɫɥɟ) ɨɬ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ. ɗɬɨ ɫɜɨɣɫɬɜɨ ɧɚɡɵɜɚɸɬ ɭɫɬɨɣ-
ɱɢɜɨɫɬɶɸ ɪɟɲɟɧɢɹ ɩɨ ɢɫɯɨɞɧɵɦ ɞɚɧɧɵɦ.
Ʉɪɚɟɜɚɹ ɡɚɞɚɱɚ ɧɚɡɵɜɚɟɬɫɹ ɤɨɪɪɟɤɬɧɨɣ, ɟɫɥɢ ɟɟ ɪɟɲɟɧɢɟ ɫɭɳɟɫɬɜɭɟɬ ɢ
ɟɞɢɧɫɬɜɟɧɧɨ ɜ ɧɟɤɨɬɨɪɨɦ ɤɥɚɫɫɟ ɮɭɧɤɰɢɣ, ɚ ɬɚɤɠɟ ɭɫɬɨɣɱɢɜɨ ɩɨ ɢɫɯɨɞɧɵɦ
ɞɚɧɧɵɦ.
ɇɚɯɨɠɞɟɧɢɟ ɤɨɪɪɟɤɬɧɵɯ ɩɨɫɬɚɧɨɜɨɤ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɣ ɜ
ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɨɩɢɫɵɜɚɸɳɢɯ ɬɟ ɢɥɢ ɢɧɵɟ ɮɢɡɢɱɟɫɤɢɟ ɹɜɥɟɧɢɹ
(ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ), ɢ ɪɚɡɪɚɛɨɬɤɚ ɦɟɬɨɞɨɜ ɢɯ ɪɟɲɟɧɢɹ ɫɨɫɬɚɜɥɹɸɬ ɨɫ-
ɧɨɜɧɨɟ ɫɨɞɟɪɠɚɧɢɟ ɩɪɟɞɦɟɬɚ ɭɪɚɜɧɟɧɢɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ.
2. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɤɪɚɟɜɵɯ ɡɚɞɚɱ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟ-
ɦɵɣ ɮɢɡɢɱɟɫɤɢɣ
ɩɪɨɰɟɫɫ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɮɭɧɤɰɢɟɣ u, ɡɚɜɢɫɹɳɟɣ, ɜɨɨɛɳɟ
ɝɨɜɨɪɹ, ɨɬ ɜɟɤɬɨɪɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ
)x,,x(x
n
!
1
(ɱɢɫɥɨ
n ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ 1, 2, 3) ɢ ɨɬ ɜɪɟɦɟɧɢ t. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ
ɜɟɤɬɨɪ x ɢɡɦɟɧɹɟɬɫɹ ɜ ɧɟɤɨɬɨɪɨɣ ɨɛɥɚɫɬɢ
.
n
R
:
ȿɫɥɢ ɮɭɧɤɰɢɹ u ɧɟ ɡɚ-
ɜɢɫɢɬ ɨɬ t, ɬɨ ɟɫɬɶ
)
x
(uu
, ɬɨ ɜ ɞɚɥɶɧɟɣɲɟɦ ɜɦɟɫɬɨ ɬɟɪɦɢɧɚ «ɮɢɡɢɱɟ-
ɫɤɢɣ ɩɪɨɰɟɫɫ» ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɟɪɦɢɧ «ɫɬɚɰɢɨɧɚɪɧɨɟ ɮɢɡɢɱɟɫɤɨɟ ɹɜ-
ɥɟɧɢɟ» ɢ ɧɚɡɵɜɚɬɶ ɫɬɚɰɢɨɧɚɪɧɵɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɦɭ
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɷɬɚ ɮɭɧɤɰɢɹ. ȿɫɥɢ ɠɟ ɮɭɧɤɰɢɹ u ɡɚɜɢɫɢɬ ɨɬ t, ɬɨ ɭɤɚɡɚɧɧɨɟ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɧɟɫɬɚɰɢɨɧɚɪɧɵɦ.
ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɨɝɪɚɧɢɱɢɦɫɹ ɪɚɫɫɦɨɬɪɟɧɢɟɦ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫ-
ɫɨɜ, ɤɨɬɨɪɵɟ ɨɩɢɫɵɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ ɜɢɞɚ
)t,x(f)t,x(u)D,D,t,x(A
t
)t,x(u
tx
m
m
w
w
, (2.2)
ɢ ɫɬɚɰɢɨɧɚɪɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ, ɨɩɢɫɵɜɚɟɦɵɯ ɭɪɚɜɧɟɧɢɹɦɢ ɜɢɞɚ
)x(g)t,x(u)D,x(L
x
. (2.3)
Ɂɞɟɫɶ
1
t
m – ɡɚɞɚɧɧɨɟ ɰɟɥɨɟ,
,)D,,D(D
n
xxx
!
1
,
ix
xD
i
w
w
,n,,
i
!1
tD
t
w
w
(ɫɦ. §1, ɩ. 2),
P
P
P
t
s
x
rs
m
stx
DD)tx(a)DDtx(A ,,,,
,
¦¦
d
1
0
(2.4)
— ɥɢɧɟɣɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ, ɩɨɪɹɞɨɤ ɤɨɬɨɪɨɝɨ ɧɟ ɩɪɟɜɨɫɯɨ-
ɞɢɬ
1
mr (ɡɞɟɫɶ 1
t
r – ɰɟɥɨɟ), ɫɨɞɟɪɠɚɳɢɣ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɩɟɪɟɦɟɧ-
ɧɵɦ
n
xx ,,!
1
, t (ɫɭɦɦɢɪɨɜɚɧɢɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (2.4) ɩɪɨɢɫɯɨ-
ɞɢɬ ɩɨ ɜɫɟɦ ɦɭɥɶɬɢɢɧɞɟɤɫɚɦ
)ss(s
n
,,!
1
, ɞɥɢɧɵ ɤɨɬɨɪɵɯ
n
sss !
1
ɧɟ ɩɪɟɜɨɫɯɨɞɹɬ r , ɢ ɩɨ ɜɫɟɦ 110
m,,, !
P
);
10
s
x
rs
sx
D)x(a)Dx(L
¦
d
,
– ɥɢɧɟɣɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ ɩɨɪɹɞɤɚ r
;
)tx(a
s
,
,
P
, )x(a
s
,
)
t
,x(
f
,
)
x(
g
– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. Ɏɭɧɤɰɢɢ )tx(a
s
,
,
P
(ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
)x(a
s
) ɧɚɡɵɜɚɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɨɩɟɪɚɬɨɪɚ )D,D,t,x(A
tx
(ɫɨɨɬɜɟɬ-
ɫɬɜɟɧɧɨ ɨɩɟɪɚɬɨɪɚ
)D,x(L
x
).
Ɉɛɥɚɫɬɶɸ ɡɚɞɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (2.2) (ɫɦ. §1) ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɰɢɥɢɧɞɪ
^
`
Tt,xR)t,x(Q
n
T
0
1
:
: , ɝɞɟ T – ɧɟɤɨɬɨɪɚɹ ɩɨɥɨɠɢɬɟɥɶ-
ɧɚɹ ɩɨɫɬɨɹɧɧɚɹ ɢɥɢ
f
. Ɉɛɥɚɫɬɶɸ ɡɚɞɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (2.3) ɹɜɥɹɟɬɫɹ
:
.
Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɱɚɫɬɧɵɦɢ ɫɥɭɱɚɹɦɢ ɭɪɚɜɧɟɧɢɹ (2.2) ɹɜɥɹɸɬɫɹ ɥɢ-
ɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɣ ɢ ɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɚ
ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɹ (2.3) ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɉɭɚɫɫɨɧɚ.
Ɋɚɡɥɢɱɚɸɬ ɫɥɟɞɭɸɳɢɟ ɬɪɢ ɨɫɧɨɜɧɵɯ ɬɢɩɚ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟ-
ɧɢɣ (2.2) ɢ (2.3).
Ⱥ) Ɂɚɞɚɱɚ Ʉɨɲɢ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (2.2)
ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ
n
R
:
. ɋɬɚɜɹɬɫɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
)x(u),x(u
0
0
, )x(u
t
),x(u
1
0
w
w
, ! , )x(u
t
),x(u
m
m
m
1
1
1
0
w
w
,
(2.5)
ɝɞɟ
n
R
x
ɢ
)x(u
0
, … ,
)x(u
m 1
– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨ-
ɜɢɹ ɨɬɫɭɬɫɬɜɭɸɬ.
Ȼ) ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (2.3)
ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ
n
R
z
:
. Ɂɚɞɚɸɬɫɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (2.5) ɜ
ɬɨɱɤɚɯ
:
x
ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɝɪɚɧɢɰɟ ɨɛɥɚɫɬɢ
:
.
ȼ) Ʉɪɚɟɜɚɹ ɡɚɞɚɱɚ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (2.3)
ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ
n
R
z
:
. Ɂɚɞɚɸɬɫɹ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɝɪɚɧɢ-
ɰɟ ɨɛɥɚɫɬɢ
:
, ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɟɫɬɟɫɬɜɟɧɧɨ, ɨɬɫɭɬɫɬɜɭɸɬ.
3. Ʉɥɚɫɫɢɱɟɫɤɢɟ ɢ ɨɛɨɛɳɟɧɧɵɟ ɩɨɫɬɚɧɨɜɤɢ ɤɪɚɟɜɵɯ ɡɚɞɚɱ. ȼɞɚɥɶ-
ɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɨɫɬɚɧɨɜɤɢ ɤɪɚɟɜɵɯ ɡɚɞɚɱ, ɯɚɪɚɤɬɟɪɢ-
ɡɭɸɳɢɟɫɹ ɬɟɦ, ɱɬɨ ɢɯ ɪɟɲɟɧɢɹ ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɝɥɚɞɤɢɦɢ (ɬɨ
ɟɫɬɶ ɧɟɩɪɟɪɵɜɧɵɦɢ ɢ ɢɦɟɸɳɢɦɢ ɧɟɩɪɟɪɵɜɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ, ɢɫɩɨɥɶɡɭɟ-
ɦɵɟ ɜ ɩɨɫɬɚɧɨɜɤɟ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ), ɢ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ
ɭɪɚɜɧɟɧɢɸ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚɱɚɥɶɧɵɦ ɢ (ɢɥɢ) ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ) ɜ ɤɚɠ-
ɞɨɣ ɬɨɱɤɟ ɨɛɥɚɫɬɢ ɡɚɞɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ
ɦɧɨɠɟɫɬɜɚ, ɧɚ ɤɨɬɨɪɨɦ ɫɬɚɜɹɬɫɹ ɧɚɱɚɥɶɧɵɟ ɢ (ɢɥɢ) ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ). Ɍɚ-