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
ɍɬɜɟɪɠɞɟɧɨ ɧɚɭɱɧɨ-ɦɟɬɨɞɢɱɟɫɤɢɦ ɫɨɜɟɬɨɦ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ 28 ɮɟɜɪɚɥɹ 2007 ɝ., ɩɪɨɬɨɤɨɥ ʋ 6
Ɋɟɰɟɧɡɟɧɬ ɞ-ɪ ɮ.-ɦ. ɧɚɭɤ, ɩɪɨɮ. Ɇ.Ⱥ. Ⱥɪɬɺɦɨɜ
ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɩɨɞɝɨɬɨɜɥɟɧɨ ɧɚ ɤɚɮɟɞɪɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. Ɋɟɤɨɦɟɧɞɭɟɬɫɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 3–5 ɤɭɪɫɨɜ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ.
Ⱦɥɹ ɫɩɟɰɢɚɥɶɧɨɫɬɢ: 010501 (010200) – ɉɪɢɤɥɚɞɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɢ ɢɧɮɨɪɦɚɬɢɤɚ 2
ɉɊȿȾɂɋɅɈȼɂȿ ɇɚɫɬɨɹɳɟɟ ɩɨɫɨɛɢɟ ɫɨɞɟɪɠɢɬ ɜɜɟɞɟɧɢɟ ɜ ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ. ɉɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɩɪɟɠɞɟ ɜɫɟɝɨ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 3 ɤɭɪɫɚ ɮɚɤɭɥɶɬɟɬɚ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ, ɢɧɮɨɪɦɚɬɢɤɢ ɢ ɦɟɯɚɧɢɤɢ, ɢɡɭɱɚɸɳɢɯ ɞɢɫɰɢɩɥɢɧɭ «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 ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ. ȼ § 5 ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɩɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ, ɩɨɡɜɨɥɹɸɳɢɣ ɫɜɨɞɢɬɶ ɡɚɞɚɱɢ ɨɛɳɟɝɨ ɜɢɞɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɵɯ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɡɚɞɚɱɚɦ ɞɥɹ ɨɞɧɨɪɨɞɧɵɯ ɭɪɚɜɧɟɧɢɣ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɦɚɬɟɪɢɚɥ § 5 ɨɬɫɭɬɫɬɜɭɟɬ ɜ ɨɛɳɟɞɨɫɬɭɩɧɨɣ ɭɱɟɛɧɨɣ ɥɢɬɟɪɚɬɭɪɟ ɩɨ ɭɪɚɜɧɟɧɢɹɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ. ȼ § 6 ɢɡɭɱɚɟɬɫɹ ɜɚɠɧɵɣ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɣ ɧɚɢɛɨɥɶɲɢɣ ɢɧɬɟɪɟɫ ɞɥɹ ɩɪɢɥɨɠɟɧɢɣ. ȼ § 7 ɢ § 8 ɪɚɫɫɦɨɬɪɟɧɨ ɪɟɲɟɧɢɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ. ȼ ɩɪɢɥɨɠɟɧɢɢ ɩɪɢɜɟɞɟɧɵ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɞɥɹ ɡɚɞɚɱɢ ɒɬɭɪɦɚ – Ʌɢɭɜɢɥɥɹ ɫ ɭɤɚɡɚɧɧɵɦɢ ɜ § 6 ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. ɉɨɫɨɛɢɟ ɫɨɞɟɪɠɢɬ ɪɹɞ ɭɩɪɚɠɧɟɧɢɣ ɢ ɡɚɞɚɱ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ ɫɬɭɞɟɧɬɚɦɢ.
§ 1. ɉɨɧɹɬɢɟ ɨɛ ɭɪɚɜɧɟɧɢɹɯ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɢ ɨɛ ɭɪɚɜɧɟɧɢɹɯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ 1. ɍɪɚɜɧɟɧɢɹ, ɢɡɭɱɚɟɦɵɟ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɟ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɢɡɢɤɚ – ɧɚɭɤɚ, ɢɡɭɱɚɸɳɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɢɡɢɤɚ ɧɚɯɨɞɢɬɫɹ ɧɚ ɫɬɵɤɟ ɦɚɬɟɦɚɬɢɤɢ ɢ ɮɢɡɢɤɢ. ɗɬɚ ɧɚɭɤɚ ɬɟɫɧɨ ɫɜɹɡɚɧɚ ɫ ɮɢɡɢɤɨɣ ɜ ɬɨɣ ɱɚɫɬɢ, ɤɨɬɨɪɚɹ ɤɚɫɚɟɬɫɹ ɩɨɫɬɪɨɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ, ɢ ɜ ɬɨ ɠɟ ɜɪɟɦɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɢɡɢɤɚ – ɷɬɨ ɪɚɡɞɟɥ ɦɚɬɟɦɚɬɢɤɢ, ɬɚɤ ɤɚɤ ɦɟɬɨɞɵ ɢɫɫɥɟɞɨɜɚɧɢɹ ɦɨɞɟɥɟɣ ɹɜɥɹɸɬɫɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ. Ɇɧɨɝɢɟ ɡɚɞɚɱɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɜɨɡɧɢɤɚɸɳɢɟ ɜ ɬɚɤɢɯ ɪɚɡɞɟɥɚɯ ɮɢɡɢɤɢ, ɤɚɤ ɝɢɞɪɨɞɢɧɚɦɢɤɚ, ɬɟɨɪɢɹ ɭɩɪɭɝɨɫɬɢ, ɷɥɟɤɬɪɨɞɢɧɚɦɢɤɚ, ɤɜɚɧɬɨɜɚɹ ɦɟɯɚɧɢɤɚ ɢ ɞɪɭɝɢɯ, ɩɪɢɜɨɞɹɬ ɤ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɹɦ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ. ȼ ɨɬɥɢɱɢɟ ɨɬ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɜ ɤɨɬɨɪɵɯ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɣ ɩɟɪɟ3
ɦɟɧɧɨɣ, ɜ ɭɪɚɜɧɟɧɢɹɯ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ ɡɚɜɢɫɢɬ 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 ɢɧɬɟɝɪɨɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ. Ɉɞɧɚɤɨ ɜ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɬɟ ɡɚɞɚɱɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɤɨɬɨɪɵɟ ɩɪɢɜɨɞɹɬ ɤ ɭɪɚɜɧɟɧɢɹɦ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɢɦɟɟɬ ɜɢɞ:
§ · wu wu w ku ¸ 0, (1.1) F ¨¨ x, u, ,! , ,! , k k ¸ w x1 w xn w x1 " w x n ¹ © ɝɞɟ x ( x1 ,! , x n ) R n – ɜɟɤɬɨɪ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ (ɡɞɟɫɶ R n – n-ɦɟɪɧɨɟ ɟɜɤɥɢɞɨɜɨ ɩɪɨɫɬɪɚɧɫɬɜɨ), u u ( x ) – ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, F – 1
n
ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɫɜɨɢɯ ɚɪɝɭɦɟɧɬɨɜ. Ɉɛɵɱɧɨ ɜɟɤɬɨɪ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ ɜ ɭɪɚɜɧɟɧɢɢ (1.1) ɢɡɦɟɧɹɟɬɫɹ ɧɚ ɧɟɤɨɬɨɪɨɦ ɨɬɤɪɵɬɨɦ ɦɧɨɠɟɫɬɜɟ : R n , ɤɨɬɨɪɨɟ ɧɚɡɵɜɚɟɬɫɹ ɨɛɥɚɫɬɶɸ ɡɚɞɚɧɢɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ. Ʉɥɚɫɫɢɱɟɫɤɢɦ (ɪɟɝɭɥɹɪɧɵɦ) ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (1.1) ɧɚ ɨɬɤɪɵɬɨɦ ɦɧɨɠɟɫɬɜɟ : 0 : ɧɚɡɵɜɚɟɬɫɹ ɜɫɹɤɚɹ ɮɭɧɤɰɢɹ u ( x ), ɨɩɪɟɞɟɥɟɧɧɚɹ ɢ ɧɟɩɪɟɪɵɜɧɚɹ ɜ : 0 ɜɦɟɫɬɟ ɫɨ ɜɫɟɦɢ ɫɜɨɢɦɢ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ, ɜɯɨɞɹɳɢɦɢ ɜ ɭɪɚɜɧɟɧɢɟ (1.1), ɢ ɨɛɪɚɳɚɸɳɚɹ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ x : 0 ɜ ɬɨɠɞɟɫɬɜɨ. ɉɨɪɹɞɤɨɦ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɧɚɡɵɜɚɟɬɫɹ ɦɚɤɫɢɦɚɥɶɧɵɣ 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 ɩɨɪɹɞɤɚ ɢɦɟɟɬ ɜɢɞ: n
¦ ai , j ( x )
i,j 1
w 2u w xi w x j
n
wu
¦b ( x )wx i
i 1
c( x ) u
f ( x ),
(1.2)
i
ɝɞɟ a i , j , bi , ɫ, f – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. Ɏɭɧɤɰɢɢ a i , j , bi , ɫ ɧɚɡɵɜɚɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɭɪɚɜɧɟɧɢɹ (1.2), ɚ ɮɭɧɤɰɢɹ f – ɫɜɨɛɨɞɧɵɦ ɱɥɟɧɨɦ (ɩɪɚɜɨɣ ɱɚɫɬɶɸ). ȿɫɥɢ f ( x ) { 0, ɬɨ ɭɪɚɜɧɟɧɢɟ (1.2) ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦ, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ – ɧɟɨɞɧɨɪɨɞɧɵɦ. ɍɪɚɜɧɟɧɢɟ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɧɚɡɵɜɚɟɬɫɹ ɤɜɚɡɢɥɢɧɟɣɧɵɦ, ɟɫɥɢ ɨɧɨ ɥɢɧɟɣɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɫɟɯ ɫɬɚɪɲɢɯ ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ. Ʉɜɚɡɢɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɢɦɟɟɬ ɜɢɞ: 4
n
¦A
i,j
i,j 1
§ § wu w u · w 2u wu wu · ¨¨ x , u , ¸¸ ¸ ,! , B ¨¨ x , u , ,! , w x1 w xn ¹ w xi w x j w x1 w x n ¸¹ © ©
0,
ɝɞɟ A i , j ɢ ȼ – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. ɇɚɢɛɨɥɟɟ ɱɚɫɬɨ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɟ ɜɫɬɪɟɱɚɸɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ. ɇɢɠɟ ɩɪɢɜɨɞɹɬɫɹ ɧɟɤɨɬɨɪɵɟ ɩɪɢɦɟɪɵ ɬɚɤɢɯ ɭɪɚɜɧɟɧɢɣ. ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
U
w § wu · ¨¨ p ¸ F, w x i ¸¹ i ©
n
wu wt
¦wx i 1
(1.3)
ɨɩɢɫɵɜɚɟɬ, ɜ ɱɚɫɬɧɨɫɬɢ, ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ, ɚ ɬɚɤɠɟ ɩɪɨɰɟɫɫ ɞɢɮɮɭɡɢɢ ɱɚɫɬɢɰ ɜ ɫɪɟɞɟ. ȼ ɭɪɚɜɧɟɧɢɢ (1.3) ɢ ɞɚɥɟɟ ɜ ɧɚɫɬɨɹɳɟɦ ɩɚɪɚɝɪɚɮɟ U , p ɢ F ɨɛɨɡɧɚɱɚɸɬ ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ, ɡɚɜɢɫɹɳɢɟ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɨɬ ɜɟɤɬɨɪɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ x ( x1 ,! , x n ) (ɱɢɫɥɨ n ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ 1, 2, 3), ɨɬ ɜɪɟɦɟɧɢ t , ɚ ɬɚɤɠɟ ɨɬ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ u ( x , t ) ɢ ɟɟ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. ɉɪɢ ɷɬɨɦ ɮɭɧɤɰɢɢ U ɢ p ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɹɦ U t U 0 const ! 0, p t p0 const ! 0. ȿɫɥɢ U { U 0 , p { p0 , ɬɨ ɭɪɚɜɧɟɧɢɟ (1.3) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
wu wt ɝɞɟ a
p0 U 0 , f
a 2ǻ u f ,
F U0 ɢ
w2
n
¦ wx
ǻ
i 1
2 i
— ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ, ɧɚɡɵɜɚɟɦɵɣ ɨɩɟɪɚɬɨɪɨɦ Ʌɚɩɥɚɫɚ. ɍɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɣ
U
w2u wt2
n
w § wu · ¨¨ p ¸F w x i ¸¹ i ©
¦wx i 1
(1.4)
ɨɩɢɫɵɜɚɟɬ, ɜ ɱɚɫɬɧɨɫɬɢ, ɦɚɥɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧ ɢ ɫɬɟɪɠɧɟɣ ɩɪɢ n 1, ɦɟɦɛɪɚɧ ɩɪɢ n 2, ɤɨɥɟɛɚɧɢɹ ɜ ɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɟ ɩɪɢ n 3, ɚ ɬɚɤɠɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɤɨɥɟɛɚɧɢɹ. ɉɪɢ U { U 0 , p { p0 ɭɪɚɜɧɟɧɢɟ (1.4) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
w2 u wt2
a 2ǻ u f .
ɍɪɚɜɧɟɧɢɟ (1.5) ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ. 5
(1.5)
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 n
w § wu · ¨¨ p ¸ w x i ¸¹ i ©
¦wx i 1
F .
(1.6)
ɉɪɢ p { p0 ɭɪɚɜɧɟɧɢɟ (1.6) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
f,
ǻu ɝɞɟ f
(1.7)
F p0 . ɍɪɚɜɧɟɧɢɟ (1.7) ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɉɭɚɫɫɨɧɚ, ɚ ɩɪɢ
f { 0 – ɭɪɚɜɧɟɧɢɟɦ Ʌɚɩɥɚɫɚ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɭɪɚɜɧɟɧɢɸ (1.7) ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɩɨɬɟɧɰɢɚɥɵ ɷɥɟɤɬɪɨɫɬɚɬɢɱɟɫɤɨɝɨ ɩɨɥɹ ɢ ɩɨɥɹ ɬɹɝɨɬɟɧɢɹ. 2. ɇɟɤɨɬɨɪɵɟ ɨɛɨɡɧɚɱɟɧɢɹ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɞɚɥɶɧɟɣɲɟɦ. ɉɭɫɬɶ
s
( s1 , ! , sn ) – ɜɟɤɬɨɪ, ɤɨɦɩɨɧɟɧɬɵ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ-
ɦɢ ɰɟɥɵɦɢ ɱɢɫɥɚɦɢ. Ɍɚɤɨɣ ɜɟɤɬɨɪ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɦɭɥɶɬɢɢɧɞɟɤɫɚ. ɑɢɫɥɨ s1 ! sn ɧɚɡɵɜɚɟɬɫɹ ɞɥɢɧɨɣ ɦɭɥɶɬɢɢɧɞɟɤɫɚ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ ɱɟɪɟɡ s . ɑɟɪɟɡ D x ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɮɨɪɦɚɥɶɧɵɣ ɜɟɤɬɨɪ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ ɨɩɟɪɚɬɨɪɨɜ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɩɨ ɩɟɪɟɦɟɧɧɵɦ x1 ,! , x n :
Dx
( D x1 ,! , D xn ), D xi
w w xi , i
1 ,! , n. ɋɢɦɜɨɥɨɦ D xs ɨɛɨɡɧɚɱɢɦ
ɫɦɟɲɚɧɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨɪɹɞɤɚ s : D xs
D xs11 " D xsnn , ɬɚɤ ɱɬɨ s
D xs u ( x )
w u ; s1 s w x1 " w x n n
ɩɪɢ ɷɬɨɦ, ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ, D 0 u ( x ) u ( x ) . ɂɧɨɝɞɚ ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɚɤɠɟ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ ɞɥɹ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ:
u xi
wu , u xi xi w xi
w 2u , u xi x j w x i2
w 2u , i, j w xi w x j
1 ,! , n
ɢ ɬ. ɩ. 3. ɉɪɨɫɬɪɚɧɫɬɜɨ ɮɭɧɤɰɢɣ L 2 ( 0 , l ) . ɑɟɪɟɡ L 2 ( 0 , l ) ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɫɨɜɨɤɭɩɧɨɫɬɶ (ɩɪɨɫɬɪɚɧɫɬɜɨ) ɜɫɟɯ ɨɩɪɟɞɟɥɟɧɧɵɯ ɧɚ ɢɧɬɟɪɜɚɥɟ ( 0 , l ) ɮɭɧɤɰɢɣ f ( x ) (ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɤɨɦɩɥɟɤɫɧɨɡɧɚɱɧɵɯ), ɞɥɹ ɤɨɬɨɪɵɯ l
³
2
f ( x ) d x f.
(1.8)
0
Ɉɬɦɟɬɢɦ, ɱɬɨ ɷɥɟɦɟɧɬɚɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ L 2 ( 0 , l ) ɹɜɥɹɸɬɫɹ ɜɫɟ ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɵɟ ɧɚ ɢɧɬɟɪɜɚɥɟ ( 0 , l ) ɮɭɧɤɰɢɢ f ( x ) , ɞɥɹ ɤɨɬɨɪɵɯ ɫɯɨɞɢɬ6
ɫɹ ɢɧɬɟɝɪɚɥ (1.8), ɢ, ɜ ɱɚɫɬɧɨɫɬɢ, ɜɫɟ ɮɭɧɤɰɢɢ, ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɵɟ ɧɚ ɨɬɪɟɡɤɟ [ 0 , l ] . Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɮɭɧɤɰɢɣ L 2 ( 0 , l ) ɥɢɧɟɣɧɨ, ɬɨ ɟɫɬɶ ɟɫɥɢ ɮɭɧɤɰɢɢ f , g L 2 ( 0 , l ) , ɬɨ f g L 2 ( 0 , l ) , ɢ ɟɫɥɢ c – ɩɪɨɢɡɜɨɥɶɧɚɹ ɤɨɦɩɥɟɤɫɧɚɹ ɩɨɫɬɨɹɧɧɚɹ, ɬɨ c f L 2 ( 0 , l ) (ɩɨɤɚɡɚɬɶ ɷɬɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ). ȼ ɩɪɨɫɬɪɚɧɫɬɜɟ L 2 ( 0 , l ) ɦɨɠɧɨ ɜɜɟɫɬɢ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ( f , g ) ɮɭɧɤɰɢɣ f ( x ) ɢ g ( x ) : l
( f ,g )
³ f ( x ) g( x ) d x 0
(ɱɟɪɟɡ f ɨɛɨɡɧɚɱɚɟɬɫɹ ɮɭɧɤɰɢɹ, ɤɨɦɩɥɟɤɫɧɨ ɫɨɩɪɹɠɟɧɧɚɹ ɤ ɮɭɧɤɰɢɢ f ), ɢ ɧɨɪɦɭ f ɮɭɧɤɰɢɢ f : 12
12
( f, f )
f
· §l 2 ¨ f ( x ) d x¸ . ¸ ¨ ¹ ©0
³
(1.9)
ɇɨɪɦɚ (1.9) ɨɛɥɚɞɚɟɬ ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣɫɬɜɚɦɢ: 1) f t 0 , ɩɪɢɱɟɦ f 0 ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ ɮɭɧɤɰɢɹ f ( x ) ɪɚɜɧɚ ɧɭɥɸ ɩɨɱɬɢ ɜɟɡɞɟ ɧɚ ɢɧɬɟɪɜɚɥɟ ( 0 , l ) (ɬɨ ɟɫɬɶ ɪɚɜɧɚ ɧɭɥɸ ɜɫɸɞɭ ɧɚ ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ, ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ, ɜɨɡɦɨɠɧɨ, ɦɧɨɠɟɫɬɜɚ ɦɟɪɵ ɧɭɥɶ 1); 2) c f
c f ɞɥɹ ɥɸɛɨɝɨ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ c;
3) ɂɦɟɟɬ ɦɟɫɬɨ ɧɟɪɚɜɟɧɫɬɜɨ ɬɪɟɭɝɨɥɶɧɢɤɚ
f g d f g .
(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 2 ( 0 , l ) , ɧɚɞɟɥɟɧɧɨɟ ɧɨɪɦɨɣ (1.9), ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɧɨɪɦɢɪɨɜɚɧɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ. Ɏɭɧɤɰɢɢ f , g ɢɡ L 2 ( 0 , l ) ɧɚɡɵɜɚɸɬɫɹ ɨɪɬɨɝɨɧɚɥɶɧɵɦɢ ɜ ɷɬɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɟɫɥɢ ɢɯ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɪɚɜɧɨ ɧɭɥɸ, ɬɨ ɟɫɬɶ l
³ f ( x ) g( x ) d x
0.
0
1
Ƚɨɜɨɪɹɬ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɧɚ ɱɢɫɥɨɜɨɣ ɩɪɹɦɨɣ ɢɦɟɟɬ ɦɟɪɭ ɧɭɥɶ, ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ
H ! 0 ɨɧɨ ɦɨɠɟɬ ɛɵɬɶ ɩɨɤɪɵɬɨ ɢɧɬɟɪɜɚɥɚɦɢ, ɫɭɦɦɚ ɞɥɢɧ ɤɨɬɨɪɵɯ ɦɟɧɶɲɟ H (ɫɦ., ɧɚ-
ɩɪɢɦɟɪ, [4, ɝɥ. I, §1] ).
7
§2. Ʉɪɚɟɜɵɟ ɡɚɞɚɱɢ ɞɥɹ ɭɪɚɜɧɟɧɢɣ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ 1. Ʉɨɪɪɟɤɬɧɵɟ ɤɪɚɟɜɵɟ ɡɚɞɚɱɢ. ɉɪɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ ɦɨɞɟɥɢɪɨɜɚɧɢɢ 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 ɭɪɚɜɧɟɧɢɟɦ – 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, ɚ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɡɚɜɢɫɢɬ ɨɬ ɩɪɨɢɡɜɨɥɶɧɵɯ ɮɭɧɤɰɢɣ. ɇɚɩɪɢɦɟɪ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ
w 2u w xw y
0
(2.1)
ɢɦɟɟɬ ɜɢɞ u ( x , y ) f ( x ) g ( y ), ɝɞɟ f ɢ g – ɩɪɨɢɡɜɨɥɶɧɵɟ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɟ ɮɭɧɤɰɢɢ (ɩɨɤɚɡɚɬɶ ɷɬɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ, ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɜ ɭɪɚɜɧɟɧɢɟ (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, ɤɨɬɨɪɵɟ ɡɚɞɚɸɬ ɫɨɫɬɨɹɧɢɟ ɞɚɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɢ ɤɪɚɟɜɵɟ (ɝɪɚɧɢɱɧɵɟ) ɭɫɥɨɜɢɹ, ɤɨɬɨɪɵɟ ɡɚɞɚɸɬ ɪɟɠɢɦ ɧɚ ɝɪɚɧɢɰɟ ɬɨɣ ɨɛɥɚɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɝɞɟ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɰɟɫɫ. ɇɚɱɚɥɶɧɵɟ ɢ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ ɞɨɥɠɧɵ ɛɵɬɶ ɩɨɫɬɚɜɥɟɧɵ ɬɚɤ, ɱɬɨɛɵ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɷɬɢɦ ɭɫɥɨɜɢɹɦ, ɫɭɳɟɫɬɜɨɜɚɥɨ ɢ ɛɵɥɨ ɟɞɢɧɫɬɜɟɧɧɵɦ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɡɚɞɚɱɚ, ɡɚɤɥɸɱɚɸɳɚɹɫɹ ɜ ɧɚɯɨɠɞɟɧɢɢ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨ ɡɚɞɚɧɧɵɦ ɧɚɱɚɥɶɧɵɦ ɢ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ, ɧɚɡɵɜɚɟɬɫɹ 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 ɩɨɥɭɱɚɸɬɫɹ ɩɭɬɟɦ ɢɡɦɟɪɟɧɢɣ, ɢ ɩɨɷɬɨɦɭ ɹɜɥɹɸɬɫɹ ɩɪɢɛɥɢɠɟɧɧɵɦɢ. Ʉɚɤ ɩɪɚɜɢɥɨ, ɷɬɢ ɩɪɢɛɥɢɠɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɚɥɨ ɨɬɥɢɱɚɸɬɫɹ ɨɬ ɬɨɱɧɵɯ ɡɧɚɱɟɧɢɣ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɜɟɥɢɱɢɧ, ɢ ɩɨɝɪɟɲɧɨɫɬɶ ɢɦɟɟɬ ɫɥɭɱɚɣɧɵɣ ɯɚɪɚɤɬɟɪ. Ⱦɥɹ ɨɞɧɨɡɧɚɱɧɨɣ ɮɢɡɢɱɟɫɤɨɣ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɧɭɠɧɨ ɛɵɬɶ ɭɜɟɪɟɧɧɵɦ ɜ ɬɨɦ, ɱɬɨ ɷɬɨ ɪɟɲɟɧɢɟ ɧɟ ɛɭɞɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɟɬɶ ɨɬ ɩɨɝɪɟɲ8
ɧɨɫɬɟɣ ɢɡɦɟɪɟɧɢɹ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ, ɬɨ ɟɫɬɶ «ɦɚɥɵɦ» (ɜ ɨɩɪɟɞɟɥɟɧɧɨɦ ɫɦɵɫɥɟ) ɢɡɦɟɧɟɧɢɹɦ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ ɞɨɥɠɧɵ ɨɬɜɟɱɚɬɶ «ɦɚɥɵɟ» ɢɡɦɟɧɟɧɢɹ ɪɟɲɟɧɢɹ. Ɍɨɱɧɟɟ, ɪɟɲɟɧɢɟ ɞɨɥɠɧɨ ɧɟɩɪɟɪɵɜɧɨ ɡɚɜɢɫɟɬɶ (ɜ ɡɚɪɚɧɟɟ ɨɩɪɟɞɟɥɟɧɧɨɦ ɫɦɵɫɥɟ) ɨɬ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ. ɗɬɨ ɫɜɨɣɫɬɜɨ ɧɚɡɵɜɚɸɬ ɭɫɬɨɣɱɢɜɨɫɬɶɸ 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 ɨɫɧɨɜɧɨɟ ɫɨɞɟɪɠɚɧɢɟ ɩɪɟɞɦɟɬɚ ɭɪɚɜɧɟɧɢɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ. 2. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɤɪɚɟɜɵɯ ɡɚɞɚɱ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɮɢɡɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɮɭɧɤɰɢɟɣ u, ɡɚɜɢɫɹɳɟɣ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɨɬ ɜɟɤɬɨɪɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ x ( x1 ,! , x n ) (ɱɢɫɥɨ n ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ 1, 2, 3) ɢ ɨɬ ɜɪɟɦɟɧɢ t. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɜɟɤɬɨɪ x ɢɡɦɟɧɹɟɬɫɹ ɜ ɧɟɤɨɬɨɪɨɣ ɨɛɥɚɫɬɢ : R n . ȿɫɥɢ ɮɭɧɤɰɢɹ u ɧɟ ɡɚɜɢɫɢɬ ɨɬ t, ɬɨ ɟɫɬɶ u u ( x ) , ɬɨ ɜ ɞɚɥɶɧɟɣɲɟɦ ɜɦɟɫɬɨ ɬɟɪɦɢɧɚ «ɮɢɡɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ» ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɟɪɦɢɧ «ɫɬɚɰɢɨɧɚɪɧɨɟ ɮɢɡɢɱɟɫɤɨɟ ɹɜɥɟɧɢɟ» 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 ɩɪɨɰɟɫɫɨɜ, ɤɨɬɨɪɵɟ ɨɩɢɫɵɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ ɜɢɞɚ
w m u( x,t ) A ( x , t , D x , Dt ) u ( x , t ) wt m
f ( x ,t ),
(2.2)
ɢ ɫɬɚɰɢɨɧɚɪɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ, ɨɩɢɫɵɜɚɟɦɵɯ ɭɪɚɜɧɟɧɢɹɦɢ ɜɢɞɚ
g( x ).
L ( x , Dx ) u( x , t )
i
( D x1 ,! , D xn ), D xi
Ɂɞɟɫɶ m t 1 – ɡɚɞɚɧɧɨɟ ɰɟɥɨɟ, D x 1 ,! , n, Dt w w t (ɫɦ. §1, ɩ. 2),
A ( x , t , D x , Dt )
m 1
¦¦ a s dr P
s, P
( x , t ) D xs DtP
(2.3)
w w xi ,
(2.4)
0
— ɥɢɧɟɣɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ, ɩɨɪɹɞɨɤ ɤɨɬɨɪɨɝɨ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ r m 1 (ɡɞɟɫɶ r t 1 – ɰɟɥɨɟ), ɫɨɞɟɪɠɚɳɢɣ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɩɟɪɟɦɟɧɧɵɦ x1 , ! , x n , t (ɫɭɦɦɢɪɨɜɚɧɢɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (2.4) ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɜɫɟɦ ɦɭɥɶɬɢɢɧɞɟɤɫɚɦ s ( s1 , ! , sn ) , ɞɥɢɧɵ ɤɨɬɨɪɵɯ
s
s1 ! sn ɧɟ ɩɪɟɜɨɫɯɨɞɹɬ r , ɢ ɩɨ ɜɫɟɦ P 9
0 , 1 ,! , m 1 );
L ( x, Dx )
¦ a ( x )D s
s x
s dr
– ɥɢɧɟɣɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ ɩɨɪɹɞɤɚ r ; a s , P ( x , t ) , a s ( x ) ,
f ( x , t ) , g( x ) – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. Ɏɭɧɤɰɢɢ a s , P ( x , t ) (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ a s ( x ) ) ɧɚɡɵɜɚɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɨɩɟɪɚɬɨɪɚ A ( x , t , D x , Dt ) (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɩɟɪɚɬɨɪɚ L ( x , D x ) ). Ɉɛɥɚɫɬɶɸ ɡɚɞɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (2.2) (ɫɦ. §1) ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɰɢɥɢɧɞɪ ( x , t ) R n 1 : x : , 0 t T , ɝɞɟ T – ɧɟɤɨɬɨɪɚɹ ɩɨɥɨɠɢɬɟɥɶɧɚɹ ɩɨɫɬɨɹɧɧɚɹ ɢɥɢ f . Ɉɛɥɚɫɬɶɸ ɡɚɞɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (2.3) ɹɜɥɹɟɬɫɹ : .
QT
^
`
Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɱɚɫɬɧɵɦɢ ɫɥɭɱɚɹɦɢ ɭɪɚɜɧɟɧɢɹ (2.2) ɹɜɥɹɸɬɫɹ ɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɣ ɢ ɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɚ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɹ (2.3) ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɉɭɚɫɫɨɧɚ. Ɋɚɡɥɢɱɚɸɬ ɫɥɟɞɭɸɳɢɟ ɬɪɢ ɨɫɧɨɜɧɵɯ ɬɢɩɚ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɣ (2.2) ɢ (2.3). Ⱥ) Ɂɚɞɚɱɚ Ʉɨɲɢ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (2.2) ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ : R n . ɋɬɚɜɹɬɫɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
u( x , 0 )
u0 ( x ) ,
w u ( x ,0 ) wt
u1 ( x ) , ! ,
w m 1 u ( x , 0 ) w t m 1
um 1 ( x ) ,
(2.5) ɝɞɟ x R n ɢ u0 ( x ) , … , um 1 ( x ) – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɨɬɫɭɬɫɬɜɭɸɬ. Ȼ) ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (2.3) ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ : z R n . Ɂɚɞɚɸɬɫɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (2.5) ɜ ɬɨɱɤɚɯ x : ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɝɪɚɧɢɰɟ ɨɛɥɚɫɬɢ : . ȼ) Ʉɪɚɟɜɚɹ ɡɚɞɚɱɚ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (2.3) ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ : z R n . Ɂɚɞɚɸɬɫɹ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɝɪɚɧɢɰɟ ɨɛɥɚɫɬɢ : , ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɟɫɬɟɫɬɜɟɧɧɨ, ɨɬɫɭɬɫɬɜɭɸɬ. 3. Ʉɥɚɫɫɢɱɟɫɤɢɟ ɢ ɨɛɨɛɳɟɧɧɵɟ ɩɨɫɬɚɧɨɜɤɢ ɤɪɚɟɜɵɯ ɡɚɞɚɱ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɨɫɬɚɧɨɜɤɢ ɤɪɚɟɜɵɯ ɡɚɞɚɱ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟɫɹ ɬɟɦ, ɱɬɨ ɢɯ ɪɟɲɟɧɢɹ ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɝɥɚɞɤɢɦɢ (ɬɨ ɟɫɬɶ ɧɟɩɪɟɪɵɜɧɵɦɢ ɢ ɢɦɟɸɳɢɦɢ ɧɟɩɪɟɪɵɜɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɩɨɫɬɚɧɨɜɤɟ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ), ɢ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚɱɚɥɶɧɵɦ ɢ (ɢɥɢ) ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ) ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɨɛɥɚɫɬɢ ɡɚɞɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɦɧɨɠɟɫɬɜɚ, ɧɚ ɤɨɬɨɪɨɦ ɫɬɚɜɹɬɫɹ ɧɚɱɚɥɶɧɵɟ ɢ (ɢɥɢ) ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ). Ɍɚ10
ɤɢɟ ɪɟɲɟɧɢɹ ɧɚɡɵɜɚɸɬɫɹ ɤɥɚɫɫɢɱɟɫɤɢɦɢ (ɪɟɝɭɥɹɪɧɵɦɢ), ɚ ɩɨɫɬɚɧɨɜɤɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɪɚɟɜɵɯ ɡɚɞɚɱ – ɤɥɚɫɫɢɱɟɫɤɢɦɢ ɩɨɫɬɚɧɨɜɤɚɦɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɥɚɫɫɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɩɪɟɞɩɨɥɚɝɚɟɬ, ɧɚɩɪɢɦɟɪ, ɧɟɩɪɟɪɵɜɧɨɫɬɶ 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 ɪɚɡɥɢɱɧɵɦɢ ɫɩɨɫɨɛɚɦɢ. Ɉɞɢɧ ɢɡ ɧɢɯ ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ ɨɛɨɛɳɟɧɧɵɯ ɮɭɧɤɰɢɣ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɯ ɫɨɛɨɣ ɥɢɧɟɣɧɵɟ ɧɟɩɪɟɪɵɜɧɵɟ ɮɭɧɤɰɢɨɧɚɥɵ ɧɚɞ ɩɪɨɫɬɪɚɧɫɬɜɚɦɢ ɛɟɫɤɨɧɟɱɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɯ ɮɭɧɤɰɢɣ [4]. Ⱦɪɭɝɢɟ ɩɨɞɯɨɞɵ ɤ ɨɛɨɛɳɟɧɧɵɦ ɩɨɫɬɚɧɨɜɤɚɦ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɢ ɩɨɧɹɬɢɹɦ ɨɛɨɛɳɟɧɧɵɯ ɪɟɲɟɧɢɣ ɫɨɞɟɪɠɚɬɫɹ, ɧɚɩɪɢɦɟɪ, ɜ ɭɱɟɛɧɨɦ ɩɨɫɨɛɢɢ [5].
§3. ɉɨɫɬɚɧɨɜɤɚ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ Ɋɚɫɫɦɨɬɪɢɦ ɫɬɟɪɠɟɧɶ ɞɥɢɧɵ l , ɜ ɤɨɬɨɪɨɦ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ. Ȼɨɤɨɜɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɫɬɟɪɠɧɹ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɨɣ (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 ɜɪɟɦɟɧɢ t ɬɟɦɩɟɪɚɬɭɪɭ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɥɸɛɨɝɨ ɟɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɨɞɢɧɚɤɨɜɨɣ. ɉɭɫɬɶ u u ( x , t ) – 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 u ( x , t ) ɬɟɦɩɟɪɚɬɭɪɨɣ ɜ ɬɨɱɤɟ x ɫɬɟɪɠɧɹ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ). Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ [ɝɥ. III, §1], ɱɬɨ ɮɭɧɤɰɢɹ u ( x , t ) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɞɧɨɦɟɪɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
cU
wu wt
w § wu· ¨k ¸ F, 0 x l , t ! 0, w x ¨© w x ¸¹
(3.1)
ɝɞɟ c ! 0 , U ! 0 , k ! 0 – ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɭɞɟɥɶɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ, ɩɥɨɬɧɨɫɬɶ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɬɟɪɠɧɹ, F – ɩɥɨɬɧɨɫɬɶ ɜɧɭɬɪɟɧɧɢɯ 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 ɦɚɫ11
ɫɟ ɫɬɟɪɠɧɹ, ɩɪɢɯɨɞɹɳɟɣɫɹ ɧɚ ɟɞɢɧɢɰɭ ɟɝɨ ɨɛɴɟɦɚ; ɜɟɥɢɱɢɧɚ k ɹɜɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ ɜ ɡɚɤɨɧɟ Ɏɭɪɶɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ (ɫɦ. ɧɢɠɟ); ɧɚɤɨɧɟɰ, ɜɟɥɢɱɢɧɚ F ɱɢɫɥɟɧɧɨ ɪɚɜɧɚ ɤɨɥɢɱɟɫɬɜɭ ɬɟɩɥɚ, ɜɵɞɟɥɹɟɦɨɝɨ ɢɥɢ ɩɨɝɥɨɳɚɟɦɨɝɨ ɜ ɟɞɢɧɢɰɟ ɨɛɴɟɦɚ ɫɬɟɪɠɧɹ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ȼɨɨɛɳɟ ɝɨɜɨɪɹ, ɜɟɥɢɱɢɧɵ c , U , k , F ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɨɬ x , t ɢ ɬɟɦɩɟɪɚɬɭɪɵ u . Ɉɞɧɚɤɨ ɜɨ ɦɧɨɝɢɯ ɜɚɠɧɵɯ ɞɥɹ ɩɪɢɥɨɠɟɧɢɣ ɡɚɞɚɱɚɯ ɮɭɧɤɰɢɢ 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 ɡɚɜɢɫɹɬ ɬɨɥɶɤɨ ɨɬ x , ɚ ɮɭɧɤɰɢɹ F ɡɚɜɢɫɢɬ ɨɬ x , t . ȼ ɫɥɭɱɚɟ ɨɞɧɨɪɨɞɧɨɝɨ ɫɬɟɪɠɧɹ ɮɭɧɤɰɢɢ c , U , k ɹɜɥɹɸɬɫɹ ɩɨɫɬɨɹɧɧɵɦɢ, ɢ ɭɪɚɜɧɟɧɢɟ (3.1) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
wu wt
a2
w 2u f , 0 x l , t ! 0, w x2
k ( cU ) , f F ( cU ) . ɝɞɟ a 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 q( x , t )
k ( x )
w u( x , t ) . wx
(3.2)
Ɏɭɧɤɰɢɹ q( x , t ) ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ. Ɂɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ q ɜ ɬɨɱɤɟ ( x 0 , t 0 ) ɫɱɢɬɚɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɵɦ, ɟɫɥɢ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ x0 ɫɬɟɪɠɧɹ ɜ ɬɟɱɟɧɢɟ ɧɟɤɨɬɨɪɨɝɨ ɩɪɨɦɟɠɭɬɤɚ ɜɪɟɦɟɧɢ > t 0 , t 0 d t @ 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 x 0 ɡɚ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ > t 0 , t 0 d t @ ɬɟɩɥɨ ɩɪɨɬɟɤɚɟɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ox, ɬɨ ɬɟɦɩɟɪɚɬɭɪɚ u ( x , t 0 ) ɭɛɵɜɚɟɬ ɜ ɭɤɚɡɚɧɧɨɣ ɨɤɪɟɫɬɧɨɫɬɢ, ɬɚɤ ɤɚɤ ɬɟɩɥɨ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɢɡ ɦɟɫɬ ɫ ɛɨɥɟɟ ɜɵɫɨɤɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ ɤ ɦɟɫɬɚɦ ɫ ɛɨɥɟɟ ɧɢɡɤɨɣ. ɉɨ-
w u ( x0 , t 0 ) 0 , ɢ, ɜ ɫɢɥɭ (3.2), q ( x 0 , t 0 ) ! 0 , ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɨwx ɝɥɚɲɟɧɢɸ ɨ ɡɧɚɤɟ ɜɟɥɢɱɢɧɵ q . Ⱥɧɚɥɨɝɢɱɧɨ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɢ ɫɥɭɱɚɣ, ɤɨ-
ɷɬɨɦɭ
ɝɞɚ ɬɟɩɥɨ ɩɪɨɬɟɤɚɟɬ ɩɪɨɬɢɜ ɧɚɩɪɚɜɥɟɧɢɹ ɨɫɢ Ox. Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɩɨɥɧɨɫɬɶɸ ɨɩɢɫɚɬɶ ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ ɜ ɫɬɟɪɠɧɟ, ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɧɚɱɚɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɬɨɱɟɤ ɫɬɟɪɠɧɹ ɢ ɬɟɩ12
ɥɨɜɨɣ ɪɟɠɢɦ ɧɚ ɟɝɨ ɤɨɧɰɚɯ, ɬɨ ɟɫɬɶ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (3.1) ɧɭɠɧɨ ɡɚɞɚɬɶ ɧɚɱɚɥɶɧɵɟ ɢ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ. ɉɭɫɬɶ M ( x ) – ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɬɨɱɤɟ x ɫɬɟɪɠɧɹ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t 0 . Ɍɨɝɞɚ ɫɬɚɜɢɬɫɹ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ
u ( x ,0 ) M ( x ) , 0 d x d l .
(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 s ( t ) , ɚ ɧɚ ɩɪɚɜɨɦ ɤɨɧɰɟ ɡɚɞɚɧɚ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ w ( t ) , ɬɨ ɫɬɚɜɹɬɫɹ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ
u ( 0, t )
s( t ), k ( l )
wu( l , t ) wx
w( t ).
(3.4)
Ʉɨɧɟɰ x 0 ɢɥɢ x l ɫɬɟɪɠɧɹ ɧɚɡɵɜɚɟɬɫɹ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɵɦ, ɟɫɥɢ ɱɟɪɟɡ ɧɟɝɨ ɧɟ ɩɪɨɬɟɤɚɟɬ ɬɟɩɥɨ (ɬɨ ɟɫɬɶ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɧɚ ɷɬɨɦ ɤɨɧɰɟ ɪɚɜɧɚ ɧɭɥɸ). ȿɫɥɢ, ɧɚɩɪɢɦɟɪ, ɤɨɧɟɰ x 0 ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧ, ɬɨ ɩɨɥɭɱɚɟɦ ɤɪɚɟɜɨɟ ɭɫɥɨɜɢɟ
wu( 0, t ) wx
0.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɧɚ ɤɨɧɰɚɯ ɫɬɟɪɠɧɹ ɩɪɨɢɫɯɨɞɢɬ ɬɟɩɥɨɨɛɦɟɧ ɩɨ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ. ɗɬɨɬ ɡɚɤɨɧ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɩɟɪɟɞɚɜɚɟɦɨɟ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɫ ɟɞɢɧɢɰɵ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɜ ɨɤɪɭɠɚɸɳɭɸ ɫɪɟɞɭ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɢ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ, ɬɨ ɟɫɬɶ ɪɚɜɧɨ
~ ), H(U u ~ – ɬɟɦɩɟɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɣ ɝɞɟ U – ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ, u ɫɪɟɞɵ, H – ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ, ɧɚɡɵɜɚɟɦɵɣ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɨɛɦɟɧɚ. ɂɫɩɨɥɶɡɭɹ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɢ ɡɚɤɨɧ ɇɶɸɬɨɧɚ, ɩɨɥɭɱɚɟɦ ɭɫɥɨɜɢɹ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɤɨɧɰɚɯ ɫɬɟɪɠɧɹ:
w u ( 0 , t) H 1 > u ( 0 , t ) p1( t ) @, wx wu( l , t ) k(l ) H 2 > u ( l , t ) p2 ( t ) @ , wx k( 0 )
(3.5) (3.6)
ɝɞɟ H 1 ! 0 (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ H 2 ! 0 ) – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɥɟɜɨɦ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ ɩɪɚɜɨɦ) ɤɨɧɰɟ, p1 ( t ) (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ p2 ( t ) ) – ɬɟɦɩɟɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɨɤɨɥɨ ɥɟɜɨɝɨ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɤɨɥɨ ɩɪɚɜɨɝɨ) 13
ɤɨɧɰɚ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ H 1 ɢ H 2 ɩɨɫɬɨɹɧɧɵ. ɍɫɥɨɜɢɹ (3.5), (3.6) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
wu( 0 , t ) h1 u ( 0 , t ) J 1 ( t ) , wx
(3.7)
wu( l , t ) h2 u ( l , t ) J 2 ( t ) , wx
(3.8)
ɝɞɟ
h1
H1 ! 0 , h2 k(0 )
J 1( t )
H2 ! 0, k( l )
H 1 p1( t ) , J 2( t ) k( 0 )
H 2 p2 ( t ) k(l )
.
ȼɫɟ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɜɵɲɟ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (3.1) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
D1
wu( 0 , t ) E1 u( 0 , t ) J 1 ( t ) , t ! 0 , wx
(3.9)
D2
wu( l , t ) E 2 u( l , t ) J 2 ( t ), t ! 0 , wx
(3.10)
ɝɞɟ D 1 , E 1 , D 2 , E 2 – ɡɚɞɚɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɪɢɱɟɦ D 12 E 12 ! 0 ,
D 22 E 22 ! 0 ; J 1 ( t ) , J 2 ( t ) – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. ȿɫɥɢ D 1 0 , E 1 1 , ɬɨ ɭɫɥɨɜɢɟ (3.9) ɧɚɡɵɜɚɟɬɫɹ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɟɦ ɩɟɪɜɨɝɨ ɪɨɞɚ (ɧɚ ɥɟɜɨɦ ɤɨɧɰɟ); ɟɫɥɢ D 1 z 0 , E 1 0 , ɬɨ ɭɤɚɡɚɧɧɨɟ ɭɫɥɨɜɢɟ ɧɚɡɵɜɚɟɬɫɹ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɟɦ ɜɬɨɪɨɝɨ ɪɨɞɚ; ɟɫɥɢ ɠɟ D 1 z 0 , E 1 z 0 , ɬɨ ɨɧɨ ɧɚɡɵɜɚɟɬɫɹ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɟɦ ɬɪɟɬɶɟɝɨ ɪɨɞɚ. Ⱥɧɚɥɨɝɢɱɧɨ ɜɜɨɞɹɬɫɹ ɢ ɩɨɧɹɬɢɹ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ ɩɟɪɜɨɝɨ, ɜɬɨɪɨɝɨ ɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɧɚ ɩɪɚɜɨɦ ɤɨɧɰɟ. 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 D 1 , E 1 , D 2 , E 2 ɢ ɮɭɧɤɰɢɢ J 1 ( t ) , J 2 ( t ) ɡɚɞɚɸɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɚɦɢ ɜɢɞɚ (3.4) – (3.6). ɉɪɢ ɷɬɨɦ ɜɨɡɦɨɠɧɚ ɩɨɫɬɚɧɨɜɤɚ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ ɪɚɡɥɢɱɧɨɝɨ ɪɨɞɚ ɧɚ ɪɚɡɥɢɱɧɵɯ ɤɨɧɰɚɯ ɫɬɟɪɠɧɹ. 14
§4. ɉɨɫɬɚɧɨɜɤɚ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ Ʉɚɤ ɛɵɥɨ ɨɬɦɟɱɟɧɨ ɜ §1, ɤ ɭɪɚɜɧɟɧɢɸ ɤɨɥɟɛɚɧɢɣ ɩɪɢɜɨɞɢɬ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɤɨɥɟɛɚɬɟɥɶɧɵɯ ɩɪɨɰɟɫɫɨɜ ɪɚɡɥɢɱɧɨɣ ɮɢɡɢɱɟɫɤɨɣ ɩɪɢɪɨɞɵ. ɉɪɢɦɟɪɚɦɢ ɬɚɤɢɯ ɩɪɨɰɟɫɫɨɜ ɹɜɥɹɸɬɫɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɧɢɠɟ ɦɚɥɵɟ ɩɨɩɟɪɟɱɧɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ ɢ ɦɚɥɵɟ ɩɪɨɞɨɥɶɧɵɟ ɤɨɥɟɛɚɧɢɹ ɭɩɪɭɝɨɝɨ ɫɬɟɪɠɧɹ. ɉɨɞ ɫɬɪɭɧɨɣ ɩɨɧɢɦɚɸɬ ɬɨɧɤɭɸ ɧɢɬɶ, ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɫɜɨɛɨɞɧɨ ɢɡɝɢɛɚɬɶɫɹ, ɬɨ ɟɫɬɶ ɧɟ ɨɤɚɡɵɜɚɟɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢɡɦɟɧɟɧɢɸ ɟɺ ɮɨɪɦɵ, ɧɟ ɫɜɹɡɚɧɧɨɦɭ ɫ ɢɡɦɟɧɟɧɢɟɦ ɞɥɢɧɵ. Ɋɚɫɫɦɨɬɪɢɦ ɧɚɬɹɧɭɬɭɸ ɫɬɪɭɧɭ ɞɥɢɧɵ l, ɡɚɤɪɟɩɥɟɧɧɭɸ ɧɚ ɤɨɧɰɚɯ. ɉɭɫɬɶ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɫɬɪɭɧɚ ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɨɫɢ ɚɛɫɰɢɫɫ Ox ɢ ɟɟ 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 ɤɨɥɟɛɚɧɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɞɜɢɠɟɧɢɟ ɬɨɱɟɤ ɫɬɪɭɧɵ ɩɪɨɢɫɯɨɞɢɬ ɜ ɨɞɧɨɣ ɩɥɨɫɤɨɫɬɢ xOu ɢ ɜɫɟ ɬɨɱɤɢ ɞɜɢɠɭɬɫɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɨɫɢ Ox. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ u u ( x , t ) ɜɟɥɢɱɢɧɭ ɨɬɤɥɨɧɟɧɢɹ ɫɬɪɭɧɵ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜ ɬɨɱɤɟ ɫ ɚɛɫɰɢɫɫɨɣ x, 0 d x d l ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t , ɨɬɫɱɢɬɵɜɚɟɦɭɸ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ou. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɪɢ ɤɚɠɞɨɦ ɮɢɤɫɢɪɨɜɚɧɧɨɦ t t 0 ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ M ( x ) u ( x , t 0 ) ɞɚɟɬ ɮɨɪɦɭ ɫɬɪɭɧɵ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t 0 . ɉɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ ɡɧɚɱɟɧɢɢ x x0 ɮɭɧɤɰɢɹ s ( t ) u ( x 0 , t ) ɞɚɟɬ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɫɬɪɭɧɵ ɫ ɚɛɫɰɢɫɫɨɣ x0 ɜɞɨɥɶ ɩɪɹɦɨɣ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɨɫɢ Ou, ɩɪɨɢɡɜɨɞɧɚɹ sc( t ) ut ( x 0 , t ) – ɫɤɨɪɨɫɬɶ ɷɬɨɝɨ ɞɜɢɠɟɧɢɹ, ɚ ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ scc( t ) ut t ( x0 ,t ) – ɭɫɤɨɪɟɧɢɟ. Ɇɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɦɚɥɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ, ɬɨ ɟɫɬɶ ɬɚɤɢɟ ɤɨɥɟɛɚɧɢɹ, ɤɨɝɞɚ ɜɟɥɢɱɢɧɚ u ɩɪɢɧɢɦɚɟɬ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɟ ɡɧɚɱɟɧɢɹ ɢ ɤɜɚɞɪɚɬɨɦ ɜɟɥɢɱɢɧɵ u x ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɟɞɢɧɢɰɟɣ. Ʌɢɧɟɣɧɨɣ ɩɥɨɬɧɨɫɬɶɸ ɫɬɪɭɧɵ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɮɭɧɤɰɢɸ U ( x ) , ɨɩɪɟɞɟɥɟɧɧɭɸ ɧɚ ɨɬɪɟɡɤɟ >0 , l @ ɢ ɨɛɥɚɞɚɸɳɭɸ ɬɟɦ ɫɜɨɣɫɬɜɨɦ, ɱɬɨ ɦɚɫɫɚ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɚɥɨɝɨ ɭɱɚɫɬɤɚ > x , x d x @ ɫɬɟɪɠɧɹ ɪɚɜɧɚ U ( x ) d x (ɢɧɵɦɢ 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. 15
( x , u ( x , t )) – ɬɨɱɤɚ ɫɬɪɭɧɵ ɫ ɚɛɫɰɢɫɫɨɣ x ɜ ɦɨɦɟɧɬ ɜɪɟG ɦɟɧɢ t . ɑɟɪɟɡ T ( x , t ) ɨɛɨɡɧɚɱɢɦ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɭɱɚɫɬɨɤ P0t Pxt ɫɨ ɫɬɨɪɨɧɵ ɭɱɚɫɬɤɚ Pxt Pl t . Ɉɬɦɟɬɢɦ, ɱɬɨ ɫɨɝɥɚɫɧɨ ɬɪɟɬɶɟɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ, ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɭɱɚɫɬɨɤ Pxt Pl t ɫɨ ɫɬɨɪɨɧɵ G ɭɱɚɫɬɤɚ P0t Pxt ɪɚɜɧɚ T ( x , t ) . ɉɭɫɬɶ Pxt
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: G T ( x , t ) { T0 const ɞɥɹ ɜɫɟɯ 0 d x d l , t t 0 . ɉɥɨɬɧɨɫɬɶɸ ɜɧɟɲɧɢɯ ɫɢɥ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɮɭɧɤɰɢɸ F ( x , t ) , ɨɛɥɚɞɚɸɳɭɸ ɬɟɦ ɫɜɨɣɫɬɜɨɦ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ t t 0 ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ Ou ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɜɟɤɬɨɪɚ ɜɧɟɲɧɢɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɧɚ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɚɥɵɣ ɭɱɚɫɬɨɤ Pxt Pxt dx ɪɚɜɧɚ F ( x , t ) d x . ɂɫɩɨɥɶɡɭɹ ɜɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ, ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ [3, ɝɥ. 1, §1], ɱɬɨ ɮɭɧɤɰɢɹ u ( x , t ) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ
U
w 2u wt 2
T0
w 2u F, 0 x l , t ! 0. w x2
(4.1)
ɍɪɚɜɧɟɧɢɟ (4.1) ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɭɪɚɜɧɟɧɢɹ ɦɚɥɵɯ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɬɪɭɧɵ. ȿɫɥɢ ɜɧɟɲɧɢɟ ɫɢɥɵ ɨɬɫɭɬɫɬɜɭɸɬ ( F { 0 ), ɬɨ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ ɧɚɡɵɜɚɸɬɫɹ ɫɜɨɛɨɞɧɵɦɢ, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ – ɜɵɧɭɠɞɟɧɧɵɦɢ. ȿɫɥɢ ɫɬɪɭɧɚ ɨɞɧɨɪɨɞɧɚ, ɬɨ ɟɟ ɩɥɨɬɧɨɫɬɶ U ɩɨɫɬɨɹɧɧɚ, ɢ ɭɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɣ (4.1) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
w 2u wt 2
a2
w 2u f , 0 x l , t ! 0, w x2
(4.2)
ɝɞɟ a T0 U , f F U . Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ (4.1) ɹɜɥɹɟɬɫɹ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ (1.4) ɩɪɢ n 1 , p T0 , 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 >0 , l @ ɨɫɢ Ox. ɉɪɨɰɟɫɫ ɩɪɨɞɨɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɬɟɪɠɧɹ ɦɨɠɟɬ ɛɵɬɶ ɨɩɢɫɚɧ ɮɭɧɤɰɢɟɣ u ( x , t ) , ɩɪɟɞɫɬɚɜɥɹɸɳɟɣ ɫɨɛɨɣ ɫɦɟɳɟɧɢɟ ɤ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ t ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɫɬɟɪɠɧɹ, ɢɦɟɜɲɟɣ ɜ ɩɨɥɨɠɟɧɢɢ 16
ɪɚɜɧɨɜɟɫɢɹ ɚɛɫɰɢɫɫɭ x (ɫɦɟɳɟɧɢɹ ɨɬɫɱɢɬɵɜɚɸɬɫɹ ɜ ɩɨɥɨɠɢɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ox). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɤɨɨɪɞɢɧɚɬɚ ɭɤɚɡɚɧɧɨɣ ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɫɬɟɪɠɧɹ ɪɚɜɧɚ x u ( x , t ) . ɉɭɫɬɶ > x , x ǻx @ – ɧɟɤɨɬɨɪɵɣ ɭɱɚɫɬɨɤ ɫɬɟɪɠɧɹ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ. Ɉɬɧɨɫɢɬɟɥɶɧɵɦ ɭɞɥɢɧɟɧɢɟɦ ɷɬɨɝɨ ɭɱɚɫɬɤɚ ɤ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ t ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ ( l 2 l1 ) l1 , ɝɞɟ l 2 – ɞɥɢɧɚ ɭɱɚɫɬɤɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t , l1 ǻ x – ɞɥɢɧɚ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ. ɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɢ u ( x , t ) ɫɥɟɞɭɟɬ, ɱɬɨ
l2
x ǻ x u ( x ǻ x , t ) ( x u ( x , t )) ǻ x u( x ǻ x , t ) u ( x , t ) ,
ɩɨɷɬɨɦɭ
l 2 l1 l1
u( x ǻ x , t ) u ( x , t ) . ǻx
ȼ ɩɪɟɞɟɥɟ ɩɪɢ ǻ x o 0 (ɬɨ ɟɫɬɶ ɫɱɢɬɚɹ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɭɱɚɫɬɨɤ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɭɸ ɞɥɢɧɭ) ɩɨɥɭɱɢɦ, ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɭɞɥɢɧɟɧɢɟ ɫɬɟɪɠɧɹ ɜ ɬɨɱɤɟ x ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ u x ( x , t ) .
G
ɉɭɫɬɶ T ( x , t ) – ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɜ ɩɨɩɟɪɟɱɧɨɦ ɫɟɱɟɧɢɢ ɫɬɟɪɠɧɹ, ɨɬɜɟɱɚɸɳɟɦ ɬɨɱɤɟ x , ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t , ɢ T ( x , t ) – ɩɪɨɟɤɰɢɹ ɜɟɤɬɨɪɚ
G T ( x , t ) 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
T( x,t )
ES
w u( x , t ) , wx
(4.3)
ɝɞɟ E E ( x ) – ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ, ɧɚɡɵɜɚɟɦɵɣ ɦɨɞɭɥɟɦ ɘɧɝɚ ɜ ɬɨɱɤɟ x , S – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɹ. ɂɫɩɨɥɶɡɭɹ ɫɨɨɬɧɨɲɟɧɢɟ (4.3), ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ [2, ɝɥ. II, §1], ɱɬɨ ɮɭɧɤɰɢɹ u ( x , t ) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ
U
w2 u wt2
w § wu· ¨¨ E S ¸ F, 0 x l , t ! 0, wx © w x ¸¹
(4.4)
ɝɞɟ U U ( x ) – ɥɢɧɟɣɧɚɹ ɩɥɨɬɧɨɫɬɶ ɫɬɟɪɠɧɹ, F F ( x , t ) – 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 17
a
ES U , f
F U . 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 u ( x , 0 ) , ɢ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ u t ( x , 0 ) , ɬɨ ɞɥɹ ɨɩɢɫɚɧɢɹ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ ɧɭɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
u ( x ,0 ) M ( x ) ,
w u( x , 0 ) \ ( x ), 0 d x d l , wt
(4.5)
ɝɞɟ M ( x ) ɢ \ ( x ) – ɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ. ȿɫɥɢ ɤɨɧɰɵ x 0 ɢɥɢ x l ɫɬɪɭɧɵ ɡɚɤɪɟɩɥɟɧɵ, ɬɨ ɫɬɚɜɹɬɫɹ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ
u( 0 , t )
0 , u( l , t ) 0 , t ! 0 .
ȿɫɥɢ ɤɨɧɰɵ ɫɬɪɭɧɵ ɞɜɢɠɭɬɫɹ ɩɨ ɢɡɜɟɫɬɧɨɦɭ ɡɚɤɨɧɭ, ɬɨ ɫɬɚɜɹɬɫɹ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ
u( 0 , t )
P 1 ( t ), u( l , t ) P 2 ( t ) , t ! 0 ,
(4.6)
ɝɞɟ P 1 ( t ) ɢ P 2 ( t ) – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. ȼɨɡɦɨɠɧɵ ɢ ɞɪɭɝɢɟ ɬɢɩɵ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ. ɉɨɫɬɚɧɨɜɤɭ ɬɚɤɢɯ ɭɫɥɨɜɢɣ ɪɚɫɫɦɨɬɪɢɦ ɧɚ ɩɪɢɦɟɪɟ ɩɪɨɰɟɫɫɚ ɩɪɨɞɨɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɨɞɧɨɪɨɞɧɨɝɨ ɫɬɟɪɠɧɹ. ȿɫɥɢ ɨɞɢɧ ɢɡ ɤɨɧɰɨɜ ɫɬɟɪɠɧɹ, ɧɚɩɪɢɦɟɪ, x 0 , ɞɜɢɠɟɬɫɹ ɩɨ ɢɡɜɟɫɬɧɨɦɭ ɡɚɤɨɧɭ, ɨɩɪɟɞɟɥɹɟɦɨɦɭ ɮɭɧɤɰɢɟɣ P ( t ) , ɚ ɧɚ ɤɨɧɰɟ x l ɡɚɞɚɧɚ ɜɧɟɲɧɹɹ ɫɢɥɚ, ɩɪɨɟɤɰɢɹ ɤɨɬɨɪɨɣ ɧɚ ɨɫɶ Ox ɪɚɜɧɚ Z ( t ) , ɬɨ ɩɨɥɭɱɚɟɦ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ
u( 0 , t ) ES
P 1 ( t ),
w u( l , t ) wx
(4.7)
Z ( t ), t ! 0.
(4.8)
ɍɫɥɨɜɢɟ (4.8) ɫɥɟɞɭɟɬ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ (4.3) ɢ ɬɨɝɨ ɮɚɤɬɚ, ɱɬɨ ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɧɚ ɤɨɧɰɟ x l ɪɚɜɧɚ ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɷɬɨɦɭ ɤɨɧɰɭ ɜɧɟɲɧɟɣ ɫɢɥɟ. ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɤɨɧɟɰ x 0 ɡɚɤɪɟɩɥɟɧ, ɚ ɤɨɧɟɰ x l ɫɜɨɛɨɞɟɧ (ɬɨ ɟɫɬɶ ɧɚ ɧɟɝɨ ɧɟ ɞɟɣɫɬɜɭɸɬ ɜɧɟɲɧɢɟ ɫɢɥɵ), ɬɨ ɭɫɥɨɜɢɹ (4.7), (4.8) ɩɪɢɧɢɦɚɸɬ ɜɢɞ
u( 0 , t )
0,
18
w u( l , t ) wx
0.
Ɍɢɩɢɱɧɵɦ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɭɫɥɨɜɢɟ ɭɩɪɭɝɨɝɨ ɡɚɤɪɟɩɥɟɧɢɹ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɤɨɧɰɚ x l :
w u( l , t ) V u ( l , t ) . wx ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɤɨɧɟɰ x l ɦɨɠɟɬ ɩɟɪɟɦɟɳɚɬɶɫɹ, ɧɨ ɭɩɪɭɝɚɹ ɫɢɥɚ ES
ɡɚɤɪɟɩɥɟɧɢɹ ɜɵɡɵɜɚɟɬ ɧɚ ɷɬɨɦ ɤɨɧɰɟ ɧɚɬɹɠɟɧɢɟ, ɫɬɪɟɦɹɳɟɟɫɹ ɜɟɪɧɭɬɶ ɫɦɟɫɬɢɜɲɢɣɫɹ ɤɨɧɟɰ ɜ ɩɪɟɠɧɟɟ ɩɨɥɨɠɟɧɢɟ. ɗɬɚ ɫɢɥɚ, ɫɨɝɥɚɫɧɨ ɡɚɤɨɧɭ Ƚɭɤɚ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɫɦɟɳɟɧɢɸ u ( l , t ) ; ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ V ! 0 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
ES
w u( l , t ) wx
V > u ( l , t ) T ( t )@ .
(4.9)
ɍɫɥɨɜɢɟ ɭɩɪɭɝɨɝɨ ɡɚɤɪɟɩɥɟɧɢɹ ɧɚ ɥɟɜɨɦ ɤɨɧɰɟ ɢɦɟɟɬ ɜɢɞ
ES
w u( 0 , t ) wx
V~ > u ( 0 , t ) K ( t )@ ,
(4.10)
ɝɞɟ V~ ! 0 – ɤɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɡɚɤɪɟɩɥɟɧɢɹ ɧɚ ɷɬɨɦ ɤɨɧɰɟ; ɮɭɧɤɰɢɹ K ( t ) ɨɩɪɟɞɟɥɹɟɬɫɹ ɚɧɚɥɨɝɢɱɧɨ T ( t ) . ɍɫɥɨɜɢɹ (4.9), (4.10) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
wu( 0 , t ) h1 u ( 0 , t ) J 1 ( t ) , wx
wu( l , t ) h2 u ( l , t ) J 2 ( t ) , wx ɝɞɟ
h1
V~ ES
J 1( t )
! 0 , h2
V~K ( t ) ES
V ES , J 2( t )
! 0,
VT(t ) ES
(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 ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ:
D1
wu( 0 , t ) E1 u( 0 , t ) J 1 ( t ) , t ! 0 , wx 19
(4.11)
w u( l , t ) E 2 u( l , t ) J 2 ( t ) , t ! 0 , (4.12) wx ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬɵ D 1 , E 1 , D 2 , E 2 ɢ ɮɭɧɤɰɢɢ J 1 ( t ) , J 2 ( t ) ɡɚɞɚɸɬɫɹ ɜ ɫɨ-
D2
ɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɚɦɢ ɜɢɞɚ (4.6) – (4.10) (ɩɪɢ ɷɬɨɦ, ɨɱɟɜɢɞɧɨ, ɜɵɩɨɥɧɹɸɬɫɹ ɭɫɥɨɜɢɹ D 12 E 12 ! 0 , D 22 E 22 ! 0 ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ (4.2) ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɧɚɯɨɠɞɟɧɢɢ ɪɟɲɟɧɢɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ (4.5) ɢ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ (4.11), (4.12).
§5. ɇɟɨɞɧɨɪɨɞɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ. ɉɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ ɉɭɫɬɶ : – ɨɛɥɚɫɬɶ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ R n ɫ ɝɪɚɧɢɰɟɣ * , ɫɨɫɬɨɹɳɟɣ ɢɡ ɭɱɚɫɬɤɨɜ ī 1 ,! , ī Į . ɉɭɫɬɶ T – ɡɚɞɚɧɧɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ ɢɥɢ f ɢ
QT
^ ( x,t ) R
n 1
: x: , 0 t T `
n 1
– ɰɢɥɢɧɞɪ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ R , ɨɫɧɨɜɚɧɢɟɦ ɤɨɬɨɪɨɝɨ ɹɜɥɹɟɬɫɹ : . ȼ ɨɛɥɚɫɬɢ QT ɪɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ
w m u( x,t ) Au( x , t ) wt m
f ( x , t ) , ( x , t ) QT ,
(5.1)
ɝɞɟ m t 1 – ɡɚɞɚɧɧɨɟ ɰɟɥɨɟ, m 1
A ( x , D x , Dt )
¦¦ a
s, ȝ
( x ) D xs Dtȝ
s dr ȝ 0
ɥɢɧɟɣɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ, ɩɨɪɹɞɨɤ ɤɨɬɨɪɨɝɨ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ r m 1 (ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ ɫɦ. ɜ §1), a s , P ( x ) , f ( x , t ) – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. Ⱦɥɹ ɭɪɚɜɧɟɧɢɹ (5.1) ɫɬɚɜɹɬɫɹ ɨɞɧɨɪɨɞɧɵɟ ɧɚɱɚɥɶɧɵɟ ɢ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ:
u( x , 0 ) 0 ,
w u ( x ,0 ) wt
0, ! ,
w m 1 u ( x , 0 ) w t m 1
0, x : ,
l i , j ( x , Dx ) u ( x , t ) 0 , x * i , 0 x T , i
1, ! , D , j
(5.2) (5.3)
1, ! , ri ,
ɝɞɟ l i , j ( x , D x ) – ɥɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɨɩɟɪɚɬɨɪɵ, ri t 1 – ɡɚɞɚɧɧɵɟ ɰɟɥɵɟ (ɬɨ ɟɫɬɶ ɧɚ ɤɚɠɞɨɦ ɭɱɚɫɬɤɟ * i , i 1, ! , D ɝɪɚɧɢɰɵ * ɦɨɠɟɬ ɫɬɚɜɢɬɶɫɹ ɧɟɫɤɨɥɶɤɨ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ, ɫɜɨɢɯ ɞɥɹ ɤɚɠɞɨɝɨ ɭɱɚɫɬɤɚ). 20
ɑɚɫɬɧɵɦɢ ɫɥɭɱɚɹɦɢ ɡɚɞɚɱɢ (5.1) – (5.3) ɹɜɥɹɸɬɫɹ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɜ §§3, 4 ɫɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
w 2u( x , t ) w u( x , t ) a2 w x2 wt u ( x ,0 ) 0 , 0 d x d l ,
f ( x , t ), 0 x l , t ! 0 ,
D1
wu( 0, t ) E 1 u( 0 , t ) wx
0, t ! 0,
D2
w u( l , t ) E 2 u( l , t ) wx
0, t ! 0
(5.4)
ɢ ɫɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ 2 w 2 u( x , t ) 2 w u( x , t ) f ( x , t ), 0 x l , t ! 0 , a w x2 wt2 w u( x , 0 ) 0, 0 d x d l , u ( x ,0 ) 0 , wt wu( 0, t ) D1 E 1 u( 0 , t ) 0 , t ! 0 , wx
D2
w u( l , t ) E 2 u( l , t ) wx
(5.5)
0, t ! 0.
ɉɪɢ ɷɬɨɦ n 1 , A a 2 w 2 w x 2 , : ( 0 , l ) , ɝɪɚɧɢɰɚ * ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɭɱɚɫɬɤɨɜ: ɬɨɱɟɤ x 0 ɢ x l ; s 2 , r1 r2 1 . ȼ ɫɥɭɱɚɟ ɡɚɞɚɱɢ (5.4) m 1 , ɚ ɜ ɫɥɭɱɚɟ ɡɚɞɚɱɢ (5.5) m 2 . Ɍ ɟ ɨ ɪ ɟ ɦ ɚ (ɩɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ). ɉɭɫɬɶ ɩɪɢ ɤɚɠɞɨɦ ɮɢɤɫɢɪɨɜɚɧɧɨɦ ɡɧɚɱɟɧɢɢ ɩɚɪɚɦɟɬɪɚ W , 0 W T , ɫɭɳɟɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɤɥɚɫɫɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ v v ( x , t ; W ) ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ
wmv A v 0 , ( x , t ) QT , wt m w m 2 v( x , 0 ;W ) w v ( x , 0; W ) v ( x , 0; W ) 0 , 0, !, w t m 2 wt w m 1v ( x , 0 ;W ) f ( x ,W ) , x : , w t m 1 l i , j ( x , Dx ) v ( x , t ; W ) 0 , x * i , 0 x T , 21
(5.6)
0, (5.7) (5.8)
i
1, ! , s , j
1, ! , ri .
Ɍɨɝɞɚ ɡɚɞɚɱɚ (5.1) – (5.3) ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɤɥɚɫɫɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ t
u( x, t )
³ v ( x , t W ; W ) dW . (5.9) 0
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɉɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɩɨɦɧɢɦ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ ɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɢ ɫɨɛɫɬɜɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ, ɡɚɜɢɫɹɳɟɝɨ ɨɬ ɩɚɪɚɦɟɬɪɚ (ɫɦ., ɧɚɩɪɢɦɟɪ, [1, §53]): ɉɭɫɬɶ ɮɭɧɤɰɢɹ g ( t ,W ) ɢ ɟɟ ɱɚɫɬɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ g t ( t ,W ) ɧɟɩɪɟɪɵɜɧɵ ɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ > a1 , b1 @ u > a 2 , b2 @, ɮɭɧɤɰɢɢ [ ( t ) ɢ K ( t ) ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵ ɧɚ ɨɬɪɟɡɤɟ > a1 , b1 @ ɢ a 2 d [ ( t ) d K ( t ) d b2 ɞɥɹ ɜɫɟɯ t > a1 , b1 @ . Ɍɨɝɞɚ ɮɭɧɤɰɢɹ K( t )
³ g ( t , W ) dW
I( t )
[(t )
ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɧɚ ɨɬɪɟɡɤɟ > a1 , b1 @ ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɚɹ ɪɚɜɧɚ K( t )
I c( t )
³g
t
( t , W ) dW g ( t ,K ( t )) K c( t ) g ( t ,[ ( t ))[ c( t ) .
[(t )
ɂɫɩɨɥɶɡɭɹ ɭɤɚɡɚɧɧɨɟ ɩɪɚɜɢɥɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɫɨɛɫɬɜɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ, ɡɚɜɢɫɹɳɟɝɨ ɨɬ ɩɚɪɚɦɟɬɪɚ, ɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (5.7), ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɢɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɩɨ ɩɟɪɟɦɟɧɧɨɣ t ɮɭɧɤɰɢɢ u ( x , t ) , ɨɩɪɟɞɟɥɹɟɦɨɣ ɮɨɪɦɭɥɨɣ (5.9): t
wPu( x,t ) wt P wmu ( x,t ) wt m
³ 0
t
³ 0
wPv ( x, t W ;W ) dW , P wtP
1, ! , m - 1 ,
w m 1v ( x , 0 ; t ) w mv ( x , t W ;W ) dW m w t m 1 wt t
³ 0
w mv ( x , t W ;W ) dW f ( x , t ) . wtm
ɂɡ (5.6), (5.9), (5.10) ɫɥɟɞɭɟɬ, ɱɬɨ m
w u( x,t ) A u( x , t ) wt m 22
(5.10)
t
³ 0
ª w m v ( x ,T ;W ) A v ( x ,T ;W « wT m ¬
º )» ¼T
dW f ( x , t )
f ( x,t ),
t W
ɬɨ ɟɫɬɶ ɮɭɧɤɰɢɹ u ( x , t ) 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). ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɷɬɚ ɡɚɞɚɱɚ ɢɦɟɟɬ ɟɳɟ ɨɞɧɨ ɪɟɲɟɧɢɟ u 1 ( x , t ) , ɢ ɪɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ
v~( x , t )
u ( x , t ) u1 ( x , t ) . ~ Ɏɭɧɤɰɢɹ v ( x , t ) ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ w m v~ ( x , t ) A v~ ( x , t ) 0 , ( x , t ) QT , wt m w m 1 v~( x , 0 ) w v~ ( x , 0 ) 0, ! , 0, v~ ( x , 0 ) 0 , w t m 1 wt x: , l ( x , D ) v~ ( x , t ) 0 , x * , 0 x T , i,j
x
i
i
1, ! , s , j
(5.11) (5.12)
(5.13)
1, ! , ri ,
ɤɨɬɨɪɭɸ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɡɚɞɚɱɢ (5.6) – (5.8) ɩɪɢ f ( x , t ) { 0 . Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (5.11) – (5.13) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ v~( x , t ) { 0 , ɢ ɷɬɨ ɪɟɲɟɧɢɟ ɟɞɢɧɫɬɜɟɧɧɨ ɜ ɫɢɥɭ ɟɞɢɧɫɬɜɟɧɧɨɫɬɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (5.6) – (5.8). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, u( x , t ) { u1 ( x , t ) , ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɩɨɤɚɡɚɬɶ. Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ. Ɂ ɚ ɦ ɟ ɱ ɚ ɧ ɢ ɟ. ɂɡ ɩɪɢɜɟɞɟɧɧɨɝɨ ɜɵɲɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɟɫɥɢ ɩɪɢ ɤɚɠɞɨɦ ɡɧɚɱɟɧɢɢ ɩɚɪɚɦɟɬɪɚ W , 0 W T , ɫɭɳɟɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɤɥɚɫɫɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ v v ( x , t ; W ) ɡɚɞɚɱɢ Ʉɨɲɢ
wm w Aw 0 , x Rn , 0 t T , wt m w m 2 w ( x , 0 ; IJ ) w w ( x , 0; IJ ) w ( x , 0; IJ ) 0 , 0, !, w t m 2 wt w m 1 w ( x , 0 ; IJ ) f ( x, IJ ) , x R n , w t m 1 ɬɨ ɡɚɞɚɱɚ Ʉɨɲɢ
23
(5.6)
0, (5.7)
w m u( x,t ) Au( x , t ) wt m u( x , 0 )
0,
w u ( x ,0 ) wt
f ( x , t ) , x Rn , 0 t T ,
0, ! ,
w m 1 u ( x , 0 ) w t m 1
0 , x Rn ,
ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɤɥɚɫɫɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ t
u( x, t )
³ w ( x, t IJ ; IJ )dIJ . 0
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ ɩɨɡɜɨɥɹɟɬ ɫɜɨɞɢɬɶ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɤ ɪɟɲɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɡɚɞɚɱɢ ɞɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ, ɤɨɬɨɪɚɹ ɨɛɵɱɧɨ ɩɪɨɳɟ, ɱɟɦ ɢɫɯɨɞɧɚɹ ɡɚɞɚɱɚ.
§6. Ɂɚɞɚɱɚ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ Ɂɚɞɚɱɚ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɜɨɡɧɢɤɚɟɬ ɩɪɢ ɪɟɲɟɧɢɢ ɦɧɨɝɢɯ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɜ ɬɨɦ ɱɢɫɥɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ. ȼ ɧɚɫɬɨɹɳɟɦ ɩɚɪɚɝɪɚɮɟ ɦɵ ɨɝɪɚɧɢɱɢɦɫɹ ɪɚɫɫɦɨɬɪɟɧɢɟɦ ɜɚɠɧɨɝɨ ɱɚɫɬɧɨɝɨ ɫɥɭɱɚɹ ɷɬɨɣ ɡɚɞɚɱɢ, ɩɪɟɞɫɬɚɜɥɹɸɳɟɝɨ ɧɚɢɛɨɥɶɲɢɣ ɢɧɬɟɪɟɫ ɞɥɹ ɩɪɢɥɨɠɟɧɢɣ. ɋɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɡɚɞɚɱɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɜɫɟ ɡɧɚɱɟɧɢɹ ɱɢɫɥɨɜɨɝɨ ɩɚɪɚɦɟɬɪɚ O , ɩɪɢ ɤɨɬɨɪɵɯ ɫɭɳɟɫɬɜɭɸɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ (ɬɨ ɟɫɬɶ ɨɬɥɢɱɧɵɟ ɨɬ ɬɨɠɞɟɫɬɜɟɧɧɨɝɨ ɧɭɥɹ) ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
X cc( x ) O X ( x ) 0 , 0 x l ,
(6.1)
ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1 X c( 0 ) E 1 X ( 0 ) 0 ,
(6.2)
D 2 X c( l ) E 2 X ( l ) 0 ,
(6.3)
ɢ ɧɚɣɬɢ ɷɬɢ ɪɟɲɟɧɢɹ. Ɂɞɟɫɶ l, D 1 , E 1 , D 2 , E 2 – ɡɚɞɚɧɧɵɟ ɜɟɳɟɫɬɜɟɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɪɢɱɟɦ
D 12 E 12 ! 0 , D 22 E 22 ! 0 .
Ɍɟ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɚ O , ɩɪɢ ɤɨɬɨɪɵɯ ɡɚɞɚɱɚ (6.1) – (6.3) ɢɦɟɟɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ, ɧɚɡɵɜɚɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ, ɚ ɨɬɜɟɱɚɸɳɢɟ ɢɦ ɪɟɲɟɧɢɹ – ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ. 24
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɢɥɭ ɨɞɧɨɪɨɞɧɨɫɬɢ ɭɪɚɜɧɟɧɢɹ (6.1) ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ (6.2), (6.3), ɤɚɠɞɚɹ ɢɡ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ, ɨɬɥɢɱɧɨɝɨ ɨɬ ɧɭɥɹ (ɬɨ ɟɫɬɶ ɟɫɥɢ X ( x ) – ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ, ɬɨ ɮɭɧɤɰɢɹ c X ( x ) , ɝɞɟ c const z 0 , 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 1 ( x ) ɢ X 2 ( x ) ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ (6.1) – (6.3). Ɍɨɝɞɚ ɮɭɧɤɰɢɹ
X( x )
c1 X 1 ( x ) c2 X 2 ( x ) ,
ɝɞɟ c1 ɢ c 2 – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɹɜɥɹɟɬɫɹ ɨɛɳɢɦ ɪɟɲɟɧɢɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ 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 Y ( x ) ɭɪɚɜɧɟɧɢɹ (6.1), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
Y(0 )
E1 ,
Y c( 0 ) D 1 ,
ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɝɪɚɧɢɱɧɨɦɭ ɭɫɥɨɜɢɸ (6.2), ɬɚɤ ɤɚɤ
D 1 Y c( 0 ) E 1Y ( 0 ) D 12 E 12 ! 0. ɜɨ 1.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɢɥɢ ɩɪɨɬɢɜɨɪɟɱɢɟ, ɱɬɨ ɢ ɞɨɤɚɡɵɜɚɟɬ ɫɜɨɣɫɬ-
2. ɉɭɫɬɶ P ɢ Q – ɪɚɡɥɢɱɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɢ X ( x ) ɢ Y ( x ) – ɨɬɜɟɱɚɸɳɢɟ ɢɦ ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ. Ɍɨɝɞɚ l
³ X ( x )Y ( x ) d x 0
25
0.
(6.4)
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɉɨ ɭɫɥɨɜɢɸ, ɮɭɧɤɰɢɢ X ( x ) ɢ Y ( x ) ɹɜɥɹɸɬɫɹ ɧɟɬɪɢɜɢɚɥɶɧɵɦɢ ɪɟɲɟɧɢɹɦɢ ɤɪɚɟɜɵɯ ɡɚɞɚɱ
X cc( x ) P X ( x ) 0 , 0 x l ,
(6.5)
D 1 X c( 0 ) E 1 X ( 0 ) 0 ,
(6.6)
D 2 X c( l ) E 2 X ( l ) 0 ,
(6.7)
ɢ
Y cc( x ) Q Y ( x ) 0 , 0 x l ,
(6.5' )
D 1 Y c( 0 ) E 1 Y ( 0 ) 0 ,
(6.6' )
D 2 Y c( l ) E 2 Y ( l ) 0 .
(6.7' )
ɍɦɧɨɠɚɹ ɭɪɚɜɧɟɧɢɟ (6.5) ɧɚ ɮɭɧɤɰɢɸ Y ( x ) , ɭɪɚɜɧɟɧɢɟ (6.5' ) ɧɚ ɮɭɧɤɰɢɸ X ( x ) ɢ ɜɵɱɢɬɚɹ ɢɡ ɨɞɧɨɝɨ ɩɨɥɭɱɢɜɲɟɝɨɫɹ ɪɚɜɟɧɫɬɜɚ ɞɪɭɝɨɟ, ɢɦɟɟɦ:
Y( x )X cc( x ) X ( x )Y cc( x ) ( ȝ Ȟ ) X ( x )Y ( x ) 0. (6.8) Ɋɚɜɟɧɫɬɜɨ (6.8) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
d >Y ( x ) X c( x ) X ( x )Y c( x )@ ( ȝ Ȟ ) X ( x )Y ( x ) 0. (6.9) dx ɂɧɬɟɝɪɢɪɭɹ ɪɚɜɟɧɫɬɜɨ (6.9) ɩɨ ɨɬɪɟɡɤɭ [ 0, l ] , ɧɚɯɨɞɢɦ, ɱɬɨ l
(Q P ) ³ X ( x )Y ( x ) d x
Y ( l ) X c( x ) X ( l )Y c( l )
0
Y ( 0 ) X c( 0 ) X ( 0 )Y c( 0 ) .
(6.10)
ɍɩɪɚɠɧɟɧɢɟ 1. ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɹ (6.6), (6.7), (6.6' ) , (6.7' ) , ɚ ɬɚɤɠɟ ɭɫɥɨɜɢɹ D 12 E 12 ! 0 , D 22 E 22 ! 0 , ɩɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɚɜɚɹ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ (6.10) ɪɚɜɧɚ ɧɭɥɸ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɢɥɢ, ɱɬɨ l
(Q P ) ³ X ( x )Y ( x ) d x 0
0.
Ɍɚɤ ɤɚɤ P z Q , ɬɨ ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ ɪɚɜɟɧɫɬɜɨ (6.4). ɋɜɨɣɫɬɜɨ 2 ɞɨɤɚɡɚɧɨ. 3. ȼɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜɟɳɟɫɬɜɟɧɧɵ. ɍɩɪɚɠɧɟɧɢɟ 2. Ⱦɨɤɚɡɚɬɶ ɫɜɨɣɫɬɜɨ 3. 26
ɍɤɚɡɚɧɢɟ. ɉɪɟɞɩɨɥɨɠɢɦ ɩɪɨɬɢɜɧɨɟ: ɩɭɫɬɶ ɫɭɳɟɫɬɜɭɟɬ ɤɨɦɩɥɟɤɫɧɨɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ O [ iK , ɝɞɟ [ , K R , K z 0 , i – ɦɧɢɦɚɹ ɟɞɢɧɢɰɚ, ɢ ɟɦɭ ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ X ( x ) . ɉɨɤɚɡɚɬɶ, ɱɬɨ ɱɢɫɥɨ
O
[ iK ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦ, ɢ ɟɦɭ ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ X ( x ) (ɮɭɧɤɰɢɹ, ɤɨɦɩɥɟɤɫɧɨ ɫɨɩɪɹɠɟɧɧɚɹ ɤ X ( x ) ). Ⱦɥɹ ɡɚɜɟɪɲɟɧɢɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɫɥɟɞɭɟɬ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɫɜɨɣɫɬɜɨɦ 2 (ɩɪɢ ɷɬɨɦ ɭɱɢɬɵɜɚɟɬɫɹ, ɱɬɨ O z O ). ɋ ɥ ɟ ɞ ɫ ɬ ɜ ɢ ɟ. Ʌɸɛɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ, ɨɬɥɢɱɧɨɝɨ ɨɬ ɧɭɥɹ, ɫɨɜɩɚɞɚɟɬ ɫ ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɨɣ ɮɭɧɤɰɢɟɣ. Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɉɪɟɞɩɨɥɨɠɢɦ,
ɱɬɨ
ɫɭɳɟɫɬɜɭɟɬ
ɫɨɛɫɬɜɟɧɧɚɹ
ɮɭɧɤɰɢɹ
X ( x ) Y ( x ) i Z ( x ) , ɝɞɟ Y ( x ) ɢ Z ( x ) – ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɵɟ ɮɭɧɤɰɢɢ, ɨɬɥɢɱɧɵɟ ɨɬ ɬɨɠɞɟɫɬɜɟɧɧɨɝɨ ɧɭɥɹ, ɢ ɩɭɫɬɶ ɮɭɧɤɰɢɹ X ( x ) ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ O . ɉɨɞɫɬɚɜɥɹɹ ɜ ɪɚɜɟɧɫɬɜɚ (6.1) – (6.3) ɜɦɟɫɬɨ X ( x ) ɫɭɦɦɭ Y ( x ) i Z ( x ) ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ ɱɢɫɥɨ O ɜɟɳɟɫɬɜɟɧɧɨ, ɥɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ Y ( x ) ɢ Z ( x ) ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ, ɨɬɜɟɱɚɸɳɢɦɢ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ O . ɇɨ ɬɨɝɞɚ, ɜ ɫɢɥɭ ɫɜɨɣɫɬɜɚ 1, Z ( x ) cY ( x ) , ɝɞɟ c z 0 – ɜɟɳɟɫɬɜɟɧɧɚɹ ɩɨɫɬɨɹɧɧɚɹ (ɱɢɫɥɨ c ɜɟɳɟɫɬɜɟɧɧɨ, ɬɚɤ ɤɚɤ ɮɭɧɤɰɢɢ Y ( x ) ɢ Z ( x ) ɩɪɢɧɢɦɚɸɬ ɜɟɳɟɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, X ( x ) Y ( x ) i Z ( x ) ( 1 i c)Y ( x ) , ɬɨ ɟɫɬɶ ɮɭɧɤɰɢɹ X ( x ) ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ 1 i c ɫɨɜɩɚɞɚɟɬ ɫ ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɨɣ ɮɭɧɤɰɢɟɣ Y ( x ) . 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 O n , n 1, 2, ! , ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɢɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ^X n ( x )` . ȼɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɨɠɧɨ ɡɚɧɭɦɟɪɨɜɚɬɶ ɜ ɩɨɪɹɞɤɟ ɜɨɡɪɚɫɬɚɧɢɹ, ɬɨ ɟɫɬɶ
O 1 O 2 ! O n O n 1 ! , ɩɪɢɱɟɦ O n o f ɩɪɢ n o f . ɂɫɩɨɥɶɡɭɹ ɫɜɨɣɫɬɜɨ 4 ɢ ɫɥɟɞɫɬɜɢɟ ɢɡ ɫɜɨɣɫɬɜɚ 3, ɫɜɨɣɫɬɜɨ 2 ɦɨɠɧɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
27
2' . ɉɭɫɬɶ O n ɢ O m , n z m – ɪɚɡɥɢɱɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɢ X n ( x ) ɢ X m ( x ) – ɨɬɜɟɱɚɸɳɢɟ ɢɦ ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ. Ɍɨɝɞɚ ɮɭɧɤɰɢɢ X n ( x ) ɢ X m ( x ) ɨɪɬɨɝɨɧɚɥɶɧɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ L 2 ( 0 , l ) (ɫɦ. §1, ɩ. 3), ɬɨ ɟɫɬɶ l
³X
n
( x ) X m ( x )d x
0.
0
Ɇɨɠɧɨ ɞɚɬɶ ɢ ɛɨɥɟɟ ɤɨɪɨɬɤɭɸ ɮɨɪɦɭɥɢɪɨɜɤɭ ɷɬɨɝɨ ɫɜɨɣɫɬɜɚ:
2" . ɋɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ, ɨɬɜɟɱɚɸɳɢɟ ɪɚɡɥɢɱɧɵɦ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ, ɨɪɬɨɝɨɧɚɥɶɧɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ L2 ( 0 , l ) . ɋɜɨɣɫɬɜɨ 2 ɧɚɡɵɜɚɸɬ ɨɛɵɱɧɨ ɫɜɨɣɫɬɜɨɦ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ. ɉɪɢɜɟɞɟɦ ɬɟɩɟɪɶ ɛɟɡ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɨɞɧɨ ɢɡ ɨɫɧɨɜɧɵɯ ɫɜɨɣɫɬɜ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ. 5. (Ɍɟɨɪɟɦɚ ȼ. Ⱥ. ɋɬɟɤɥɨɜɚ). ȿɫɥɢ ɮɭɧɤɰɢɹ f ( x ) ɞɜɚɠɞɵ ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɧɚ ɨɬɪɟɡɤɟ [ 0, l ] ɢ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɞɧɨɪɨɞɧɵɦ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1 f c( 0 ) E 1 f ( 0 ) 0 , D 2 f c( l ) E 2 f ( l ) 0 , ɬɨ ɨɧɚ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɚ ɜ ɚɛɫɨɥɸɬɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɫɯɨɞɹɳɢɣɫɹ ɧɚ ɨɬɪɟɡɤɟ [ 0, l ] ɪɹɞ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ: f
f(x)
¦f
n
X n ( x ).
(6.11)
n 1
Ɋɚɡɥɨɠɟɧɢɟ (6.11) ɧɚɡɵɜɚɟɬɫɹ ɪɹɞɨɦ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f ( x ) ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ^X n ( x )` , ɚ ɱɢɫɥɚ f n , n 1, 2, ! – ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ Ɏɭɪɶɟ ɷɬɨɣ ɮɭɧɤɰɢɢ. Ʉɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f ( x ) ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. ɍɦɧɨɠɢɦ ɪɚɜɟɧɫɬɜɨ (6.11) ɧɚ ɮɭɧɤɰɢɸ X m ( x ) , ɝɞɟ ɧɨɦɟɪ m ( m t 1 ) ɮɢɤɫɢɪɨɜɚɧ, ɢ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ ɩɨ ɨɬɪɟɡɤɭ [ 0, l ] . Ɍɨɝɞɚ, ɢɫɩɨɥɶɡɭɹ ɫɜɨɣɫɬɜɨ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ, ɧɚɯɨɞɢɦ, ɱɬɨ l
³ f ( x )X 0
l m
( x )d x
f m ³ X m2 ( x ) d x 0
28
fm X m
2
,m
1, 2, ! .
Ɂɚɦɟɧɹɹ ɢɧɞɟɤɫ m ɧɚ n , ɩɨɥɭɱɢɦ, ɱɬɨ
fn
1
Xn
l
2
³ f ( x )X
n
( x ) d x , n 1, 2, ! .
(6.12)
0
Ʌ ɟ ɦ ɦ ɚ. ɉɭɫɬɶ X ( x ) – ɩɪɨɢɡɜɨɥɶɧɨɟ ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.1). Ɍɨɝɞɚ ɢɦɟɟɬ ɦɟɫɬɨ ɪɚɜɟɧɫɬɜɨ
O X
l
2
2
X c( l ) X ( l ) X c( 0 ) X ( 0 ) ³ X c( x ) d x . (6.13) 0
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 [ 0, l ] , ɢɦɟɟɦ: l
³ 0
l
X cc( x ) X ( x ) d x O ³ X 2 ( x ) d x
0.
0
ɉɪɨɢɡɜɨɞɹ ɜ ɩɟɪɜɨɦ ɫɥɚɝɚɟɦɨɦ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ, ɩɨɥɭɱɢɦ ɪɚɜɟɧɫɬɜɨ (6.13). Ʌɟɦɦɚ ɞɨɤɚɡɚɧɚ. ɋ ɥ ɟ ɞ ɫ ɬ ɜ ɢ ɟ. ȿɫɥɢ ɥɸɛɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ X ( x ) ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɸ
X c( l ) X ( l ) X c( 0 ) X ( 0 ) t 0 ,
(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 ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɱɢɫɥɚ D 1 , E 1 , D 2 , E 2 ). Ⱥ ɢɦɟɧɧɨ, ɦɵ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɥɟɞɭɸɳɢɟ ɬɢɩɵ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ:
I. X ( 0 )
0, X( l ) 0;
II. X c( 0 ) h 1 X ( 0 ) III. X ( 0 )
0, X( l ) 0;
0 , X c( l ) h 2 X ( l ) 0 ;
IV. X c( 0 ) h 1 X ( 0 )
0 , X c( l ) h 2 X ( l )
0,
ɝɞɟ h 1 t 0 ɢ h 2 t 0 – ɡɚɞɚɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ. Ʉ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɢɦɟɧɧɨ ɬɚɤɨɝɨ ɜɢɞɚ ɩɪɢɜɨɞɢɬ ɪɟɲɟɧɢɟ ɦɧɨɝɢɯ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɢ, ɜ ɱɚɫɬɧɨɫɬɢ, ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ, ɨɩɢɫɵɜɚɸɳɢɯ ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ ɜ 29
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 ɫɥɟɞɫɬɜɢɹ 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
X c( l ) X ( l ) X c( 0 ) X ( 0 ) 0
(6.15)
ɢ l
2
³ X c( x ) d x
0.
(6.16)
0
ɂɡ (6.16) ɜɵɬɟɤɚɟɬ, ɱɬɨ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ X ( x ) , ɨɬɜɟɱɚɸɳɚɹ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ O 0 , ɹɜɥɹɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ: X ( x ) { const z 0 ; ɩɪɢ ɷɬɨɦ ɭɫɥɨɜɢɟ (6.15) ɜɵɩɨɥɧɹɟɬɫɹ ɚɜɬɨɦɚɬɢɱɟɫɤɢ. Ɂɚɦɟɬɢɦ, ɧɚɤɨɧɟɰ, ɱɬɨ ɫɪɟɞɢ ɭɫɥɨɜɢɣ I – IV ɢɦɟɟɬɫɹ ɬɨɥɶɤɨ ɨɞɢɧ ɬɢɩ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ, ɤɨɬɨɪɵɦ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɮɭɧɤɰɢɹ X ( x ) { const z 0 , ɚ ɢɦɟɧɧɨ
X c( 0 ) 0 , X c( l )
0
(6.17)
(ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɭɫɥɨɜɢɣ IV ɩɪɢ h 1 h2 0 ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɩɪɚɜɟɞɥɢɜɨ ɫɥɟɞɭɸɳɟɟ ɫɜɨɣɫɬɜɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ 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, ɤɨɝɞɚ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɢɦɟɸɬ ɜɢɞ (6.17). ɇɭɥɟɜɨɦɭ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ X ( x ) { const z 0 . ɋ ɥ ɟ ɞ ɫ ɬ ɜ ɢ ɟ. 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 ɪɚɡɥɢɱɚɬɶ ɫɥɭɱɚɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ II – IV, ɤɨɝɞɚ ɩɨɫɬɨɹɧɧɵɟ h 1 ɢ (ɢɥɢ) h 2 ɪɚɜɧɵ ɧɭɥɸ, ɢ ɤɨɝɞɚ ɨɧɢ ɩɨɥɨɠɢɬɟɥɶɧɵ. Ⱥ ɢɦɟɧɧɨ, ɪɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɢɟ 9 ɬɢɩɨɜ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ: I. X ( 0 )
0, X( l ) 0;
II . ɚ) X c( 0 )
0 , X ( l ) 0 ( h1 30
0 );
ɛ) X c( 0 ) h 1 X ( 0 ) III. ɚ) X ( 0 ) ɛ) X ( 0 ) IV. ɚ) X c( 0 ) ɛ) X c( 0 )
0 , X c( l )
0 , X ( l ) 0 ( h 1 ! 0 ); 0 ; ( h2
0 );
0 , X c( l ) h 2 X ( l ) 0 ( h 2 ! 0 )
0 , X c( l ) 0 ( h 1 0 , X c( l ) h 2 X ( l )
h2
0 );
0 ( h1
0 , h 2 ! 0 );
ɜ) X c( 0 ) h 1 X ( 0 )
0 , X c( l )
0 ( h1 ! 0 , h 2
ɝ) X c( 0 ) h 1 X ( 0 )
0 , X c( l ) h 2 X ( l ) 0 ( h 1 ! 0 , h 2 ! 0 ).
0 );
ɉɪɢɦɟɪ 1. ɇɚɣɬɢ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɢ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ 1 ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ I:
X(0 )
0,
(6.18)
X( l ) 0.
(6.19)
Ɋɟɲɟɧɢɟ ȼ ɫɢɥɭ ɫɜɨɣɫɬɜɚ 6 ɜɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ O ɩɨɥɨɠɢɬɟɥɶɧɵ. Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.1) ɩɪɢ O ! 0 ɢɦɟɟɬ ɜɢɞ
X( x )
A cos O x B sin O x ,
(6.20)
ɝɞɟ A ɢ B – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ. ɇɚɣɞɟɦ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɜɢɞɚ (6.20), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɭɫɥɨɜɢɹɦ (6.18), (6.19). ɂɡ (6.18) ɫɥɟɞɭɟɬ, ɱɬɨ A 0 , ɢ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
X( x )
B sin O x ;
ɩɪɢ ɷɬɨɦ B z 0 . ɂɫɩɨɥɶɡɭɹ ɬɟɩɟɪɶ ɭɫɥɨɜɢɟ (6.19), ɢɦɟɟɦ
sin O l
0.
(6.21)
Ɂɧɚɱɟɧɢɹ O ! 0 , ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɭɪɚɜɧɟɧɢɸ (6.21), ɬɨ ɟɫɬɶ 1
Ɂɞɟɫɶ ɢ ɜ ɞɚɥɶɧɟɣɲɟɦ ɩɨɞ ɧɨɪɦɨɣ ɮɭɧɤɰɢɢ ɢɡ L 2 ( 0 , l ) ɩɨɧɢɦɚɟɬɫɹ ɟɟ ɧɨɪɦɚ ɜ ɞɚɧɧɨɦ
ɩɪɨɫɬɪɚɧɫɬɜɟ (ɡɚɦɟɬɢɦ, ɱɬɨ ɞɥɹ ɮɭɧɤɰɢɣ, ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɬɟɦ ɢɥɢ ɢɧɵɦ ɩɨɞɦɧɨɠɟɫɬɜɚɦ ɦɧɨɠɟɫɬɜɚ L 2 ( 0 , l ) , ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɢ ɞɪɭɝɢɟ ɧɨɪɦɵ; ɧɚɩɪɢɦɟɪ, ɟɫɥɢ ɮɭɧɤɰɢɹ
f ( x ) ɧɟɩɪɟɪɵɜɧɚ ɧɚ ɨɬɪɟɡɤɟ [0, l ] , ɧɚɪɹɞɭ ɫ ɟɟ ɧɨɪɦɨɣ ɜ L 2 ( 0 , l ) ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɧɨɪɦɭ f
C
max f ( x ) ). 0 d x dl
31
2
O
On
§ nS · ¨ ¸ , n 1, 2, ! , © l ¹
ɹɜɥɹɸɬɫɹ ɢɫɤɨɦɵɦɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ. ɋɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ O n ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ
Bn sin O n x
Xn( x )
Bn sin
nS x , n 1, 2, ! , l
ɝɞɟ Bn z 0 – ɩɪɨɢɡɜɨɥɶɧɚɹ ɩɨɫɬɨɹɧɧɚɹ. Ɍɚɤ ɤɚɤ ɥɸɛɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ, ɨɬɥɢɱɧɨɝɨ ɨɬ ɧɭɥɹ, ɬɨ ɛɟɡ ɨɝɪɚɧɢɱɟɧɢɹ ɨɛɳɧɨɫɬɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ Bn 1 , n 1, 2, ! . ɂɬɚɤ, ɢɫɤɨɦɵɦɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɢ
X n ( x ) sin
nS x , n 1, 2, ! . l
(6.22)
2
Ⱦɥɹ ɮɭɧɤɰɢɣ (6.22) ɧɚɣɞɟɦ ɡɧɚɱɟɧɢɹ X n , ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɮɨɪɦɭɥɟ (6.12):
Xn
2
l 2 ³ sin 0
1 § 2 nS x · ¸d x ¨ 1 cos ³ 20© l ¹ l
nS x dx l
l , 2
n 1, 2, ! . ɍɩɪɚɠɧɟɧɢɟ 3. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɪɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɢ f ( x ) , ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɣ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟɦɵ ɋɬɟɤɥɨɜɚ, ɜ ɪɹɞ Ɏɭɪɶɟ (6.11) ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ (6.22) ɫɨɜɩɚɞɚɟɬ ɧɚ ɨɬɪɟɡɤɟ [ 0, l ] ɫ ɪɚɡɥɨɠɟɧɢɟɦ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɣ ɪɹɞ Ɏɭɪɶɟ ɧɟɱɟɬɧɨɝɨ ɩɪɨɞɨɥɠɟɧɢɹ ɷɬɨɣ ɮɭɧɤɰɢɢ ɧɚ ɨɬɪɟɡɨɤ [ l , l ] (ɫɦ., ɧɚɩɪɢɦɟɪ, [1, §55]). ɉɪɢɦɟɪ 2. ɇɚɣɬɢ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɢ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ II ɛ):
X c( 0 ) h 1 X ( 0 )
0 , h1 ! 0 ,
X( l ) 0.
(6.23) (6.24)
Ɋɟɲɟɧɢɟ Ɍɚɤ ɠɟ, ɤɚɤ ɢ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ, ɜɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ O ɩɨɥɨɠɢɬɟɥɶɧɵ, ɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.1) ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ (6.20). h1 O A , ɢ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɟ (6.23), ɧɚɯɨɞɢɦ, ɱɬɨ B
32
h § · A ¨¨ cos O x 1 sin O x ¸¸ , O © ¹
X( x )
ɝɞɟ A – ɩɪɨɢɡɜɨɥɶɧɚɹ ɩɨɫɬɨɹɧɧɚɹ, ɨɬɥɢɱɧɚɹ ɨɬ ɧɭɥɹ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ A 1 . ɂɫɩɨɥɶɡɭɹ ɬɟɩɟɪɶ ɭɫɥɨɜɢɟ (6.24), ɩɨɥɭɱɢɦ, ɱɬɨ ɢɫɤɨɦɵɦɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɹɜɥɹɸɬɫɹ ɜɟɥɢɱɢɧɵ O n W n2 , n 1, 2, ! , ɝɞɟ W n – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɪɧɢ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
tg W l
W
(6.25)
h1
(ɥɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ (6.25) ɢɦɟɟɬ ɫɱɟɬɧɨɟ ɦɧɨɠɟɫɬɜɨ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ, ɹɜɥɹɸɳɢɯɫɹ ɚɛɫɰɢɫɫɚɦɢ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɝɪɚɮɢɤɨɜ ɮɭɧɤɰɢɣ y tg W l ɢ y W h1 ).
W n2 ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ
ɋɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ O n
X n ( x ) cos W n x 2
h1
Wn
sin W n x , n 1, 2, ! .
(6.26)
1, 2, ! . ɂɫɩɨɥɶɡɭɹ ɢɡɜɟɫɬɧɭɸ ɮɨɪɦɭɥɭ
ɇɚɣɞɟɦ ɜɟɥɢɱɢɧɵ X n , n
a 2 b 2 sin ( Į ȝ ) ,
a cos D b sin D
ɝɞɟ a ɢ b – ɩɪɨɢɡɜɨɥɶɧɵɟ ɜɟɳɟɫɬɜɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɬɚɤɢɟ, ɱɬɨ a 2 b 2 ! 0 , ɢ tg P a b , ɡɚɩɢɲɟɦ ɮɭɧɤɰɢɸ X n ( x ) ɜ ɜɢɞɟ
Xn( x ) ɝɞɟ tg P n
W n2 h12 Wn
sin ( IJ n x ȝn ) , n 1, 2, ! ,
(6.27)
W n h1 . Ɍɨɝɞɚ Xn
2
W n2 h12 W n2
l
³ sin
2
( IJ n x ȝn ) d x .
(6.28)
0
Ⱦɚɥɟɟ, l
³
sin 2 ( IJ n x ȝn ) d x
0
1 2
l
³ >1 cos 2 ( IJ
n
x ȝn ) @ d x
0
l 1 1 sin 2 ( IJ n l ȝn ) sin 2 ȝn . 4W n 2 4W n ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɵ 33
(6.29)
2 tgD , tg ( Į ȕ ) 1 tg 2D
sin 2D ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ tg W n l
tg D tg E 1 tgD tg E
W n h 1 (ɜ ɫɢɥɭ (6.25)) ɢ tg P n
sin 2 ( IJ n l ȝn )
0 , sin 2 ȝn
W n h1 , ɢɦɟɟɦ:
2W n h 1 , n 1, 2, ! . (6.30) W n2 h 21
ɂɡ (6.28) – (6.30) ɫɥɟɞɭɟɬ, ɱɬɨ 2
Xn
l W n2 h12 h1
2 W n2
,n
1, 2, ! .
ɉɪɢɦɟɪ 3. ɇɚɣɬɢ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɢ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ IV ɚ):
X c( 0 ) 0 , X c( l )
0
(6.31)
Ɋɟɲɟɧɢɟ ȼ ɫɢɥɭ ɫɜɨɣɫɬɜɚ 6, ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɧɚɢɦɟɧɶɲɟɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɪɚɜɧɨ ɧɭɥɸ, ɢ ɟɦɭ ɨɬɜɟɱɚɟɬ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ X ( x ) { const z 0 . ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɭɤɚɡɚɧɧɨɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɱɟɪɟɡ O 0 ɢ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ X 0 ( x ) { 1 . ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ (6.20) ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (6.31), ɧɚɯɨɞɢɦ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɨɬɜɟɱɚɸɳɢɟ ɢɦ ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ: 2
nS x § nS · , ¸ , X n ( x ) cos ¨ l l ¹ ©
On
(6.32)
ɝɞɟ n t 1 . ȼɫɸ ɫɨɜɨɤɭɩɧɨɫɬɶ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɢ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɟ ɦɨɠɧɨ ɬɚɤɠɟ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ (6.32), ɩɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ n 0, 1, 2 ! . Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ
X0
2
l , Xn
2
l 2 ɞɥɹ n 1, 2 ! .
ɉɪɢɦɟɪ 4. ɇɚɣɬɢ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɢ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ IV ɝ):
X c( 0 ) h 1 X ( 0 )
0 , X c( l ) h 2 X ( l )
34
0,
ɝɞɟ h 1 ! 0 , h 2 ! 0 . Ɋɟɲɟɧɢɟ ɂɫɩɨɥɶɡɭɹ ɬɟ ɠɟ ɪɚɫɫɭɠɞɟɧɢɹ, ɱɬɨ ɢ ɜ ɩɪɢɦɟɪɟ 2, ɩɨɥɭɱɢɦ, ɱɬɨ ɢɫɤɨɦɵɦɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɹɜɥɹɸɬɫɹ ɜɟɥɢɱɢɧɵ O n W n2 , n 1, 2, ! , ɝɞɟ W n – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ
ctg W l p
q pW ,
(6.33)
W
1 ,q h1 h2
h1 h2 h1 h2
.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɮɭɧɤɰɢɹ y pW q W ɦɨɧɨɬɨɧɧɨ ɜɨɡɪɚɫɬɚɟɬ ɜ ɫɜɨɟɣ ɨɛɥɚɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ (ɩɪɢ W z 0 ) ɢ ɟɟ ɝɪɚɮɢɤ ɢɦɟɟɬ ɧɚɤɥɨɧɧɭɸ ɚɫɢɦɩɬɨɬɭ y pW ɢ ɜɟɪɬɢɤɚɥɶɧɭɸ ɚɫɢɦɩɬɨɬɭ W 0 . Ʌɟɝɤɨ ɜɢɞɟɬɶ ɬɚɤɠɟ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ (6.33) ɢɦɟɟɬ ɫɱɟɬɧɨɟ ɦɧɨɠɟɫɬɜɨ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɤɨɪɧɟɣ, ɹɜɥɹɸɳɢɯɫɹ ɚɛɫɰɢɫɫɚɦɢ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɝɪɚɮɢɤɨɜ ɮɭɧɤɰɢɣ y ctg W l ɢ y pW q W . ɋɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ O n
W n2 ɨɬɜɟɱɚɸɬ ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ
X n ( x ) cos W n x
h1
Wn
sin W n x , n 1, 2, ! .
ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɵ (6.27) – (6.29), ɢɦɟɟɦ:
Xn
2
W n2 h12 I , n 1, 2, ! , W n2
(6.34)
ɝɞɟ
I
l 1 1 sin 2 ( IJ n l ȝn ) sin 2 P n , 2 4W n 4W n
ctg P n
h1 W n .
(6.35) (6.36)
ɂɡ (6.33) ɫɥɟɞɭɟɬ, ɱɬɨ
ctg W n l
W n2 h 1 h 2 , n 1, 2, ! . W n ( h1 h2 )
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɬɟɩɟɪɶ ɮɨɪɦɭɥɚɦɢ 35
(6.37)
sin 2D
2 ctgD , ctg ( Į ȕ ) 1 ctg 2D
ctg D c tg E 1 , ctg D ctg E
ɫ ɭɱɟɬɨɦ (6.36), (6.37) ɧɚɯɨɞɢɦ, ɱɬɨ
ctg ( W n l P n ) sin 2 ( W n l P n )
2W n h2 2 n
W h
2 2
h2
Wn
,
, sin 2 P n
2W n h1
W n2 h 21
.
Ɉɬɫɸɞɚ ɢ ɢɡ (6.34), (6.35) ɫɥɟɞɭɟɬ, ɱɬɨ
Xn
2
l ( IJ n2 h12 ) ( IJ n2 h22 ) ( IJ n2 h1 h2 ) ( h1 h 2 )
2W n2 ( IJ n2 h22 )
,n
1, 2, ! .
ɍɩɪɚɠɧɟɧɢɟ 4. ɂɫɩɨɥɶɡɭɹ ɩɪɢɦɟɪɵ 1 – 4, ɧɚɣɬɢ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɢ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ II ɚ), III ɚ), III ɛ), IV ɛ), IV ɜ). ȼ ɩɪɢɥɨɠɟɧɢɢ ɩɪɢɜɟɞɟɧɵ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɢ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɞɥɹ ɜɫɟɯ ɬɢɩɨɜ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ I – IV.
§7. Ɋɟɲɟɧɢɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ 1. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. Ɋɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɡɚɞɚɱɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ u u ( x,t ) , ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
a 2 u xx , 0 x l , t ! 0 ,
(7.1)
u ( x, 0 ) M ( x ) , 0 d x d l
(7.2)
D 1 u x ( 0 ,t ) E 1 u ( 0 , t ) 0 , t ! 0 ,
(7.3)
D 2 u x ( l, t ) E 2 u ( l, t ) 0 , t ! 0 .
(7.4)
ut ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
36
Ɂɞɟɫɶ a ! 0 , l ! 0 , D 1 , E 1 , D 2 , E 2 – ɡɚɞɚɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɪɢɱɟɦ D 12 E 12 ! 0 , D 22 E 22 ! 0 ; M ( x ) – ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (7.1) – (7.4) ɪɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɭɸ ɜɫɩɨɦɨɝɚɬɟɥɶɧɭɸ ɡɚɞɚɱɭ: ɇɚɣɬɢ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ
a 2U xx , 0 x l , t ! 0 ,
Ut
(7.5)
ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1U x ( 0 ,t ) E 1 U ( 0 , t ) 0 , t ! 0 ,
(7.6)
D 2U x ( l, t ) E 2 U ( l, t ) 0 , t ! 0
(7.7)
ɢ ɩɪɟɞɫɬɚɜɢɦɵɟ ɜ ɜɢɞɟ
U( x, t )
X( x )T( t )
(7.8)
ɝɞɟ X ( x ) – ɮɭɧɤɰɢɹ, ɧɟ ɡɚɜɢɫɹɳɚɹ ɨɬ ɩɟɪɟɦɟɧɧɨɣ t , T ( t ) – ɮɭɧɤɰɢɹ, ɧɟ ɡɚɜɢɫɹɳɚɹ ɨɬ ɩɟɪɟɦɟɧɧɨɣ x . ɉɨɞɫɬɚɜɥɹɹ (7.8) ɜ (7.5) ɢ ɪɚɡɞɟɥɢɜ ɩɨɥɭɱɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ ɧɚ a 2 X ( x )T ( t ) , ɢɦɟɟɦ:
T c( t ) a 2T ( t )
X cc( x ) . X( x )
(7.9)
Ʌɟɜɚɹ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ (7.9) ɧɟ ɡɚɜɢɫɢɬ ɨɬ x , ɚ ɩɪɚɜɚɹ ɧɟ ɡɚɜɢɫɢɬ ɨɬ t . ɉɨɷɬɨɦɭ ɪɚɜɟɧɫɬɜɨ (7.9) ɜɨɡɦɨɠɧɨ ɥɢɲɶ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɟ ɟɝɨ ɱɚɫɬɢ ɪɚɜɧɵ ɩɨɫɬɨɹɧɧɨɣ:
T c( t ) a 2T ( t )
X cc( x ) X( x )
O const
(7.10)
(ɦɵ ɧɟ ɩɪɟɞɩɨɥɚɝɚɟɦ ɡɚɪɚɧɟɟ, ɤɚɤɨɣ ɡɧɚɤ ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɚɹ O , ɚ ɡɧɚɤ « – » ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (7.10) ɢɫɩɨɥɶɡɨɜɚɧ ɞɥɹ ɭɞɨɛɫɬɜɚ ɩɨɫɥɟɞɭɸɳɟɝɨ ɢɡɥɨɠɟɧɢɹ ɦɚɬɟɪɢɚɥɚ). ɂɡ (7.10) ɩɨɥɭɱɚɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɮɭɧɤɰɢɣ T ( t ) ɢ X ( x ) :
T c( t ) a 2 O T ( t ) 0 , X cc( x ) O X ( x ) 0 .
(7.11) (7.12)
ɉɨɞɫɬɚɜɥɹɹ (7.8) ɜ (7.6) ɢ ɜ (7.7), ɩɨɥɭɱɢɦ, ɱɬɨ ɮɭɧɤɰɢɹ X ( x ) ɞɨɥɠɧɚ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1 X c( 0 ) E 1 X ( 0 ) 0 , 37
(7.13)
D 2 X c( l ) E 2 X ( l ) 0 .
(7.14)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɡɚɞɚɱɟ: Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɜɫɟ ɡɧɚɱɟɧɢɹ ɱɢɫɥɨɜɨɝɨ ɩɚɪɚɦɟɬɪɚ O , ɩɪɢ ɤɨɬɨɪɵɯ ɫɭɳɟɫɬɜɭɸɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (7.12), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ (7.13), (7.14), ɢ ɧɚɣɬɢ ɷɬɢ ɪɟɲɟɧɢɹ. ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɜɵɲɟ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɨ ɧɚɯɨɠɞɟɧɢɢ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɢ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (7.13), (7.14) ɨɬɧɨɫɹɬɫɹ ɤ ɨɞɧɨɦɭ ɢɡ ɬɢɩɨɜ I – IV (ɫɦ. §6). Ɍɨɝɞɚ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɫɭɳɟɫɬɜɭɟɬ ɫɱɟɬɧɨɟ ɦɧɨɠɟɫɬɜɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ O n t 0 , n 1, 2,! , ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ^X n ( x )` (ɫɦ. ɉɪɢɥɨɠɟɧɢɟ). ɉɭɫɬɶ Tn ( t ) – ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (7.11) ɩɪɢ O O n :
Tnc( t ) a 2 O n Tn ( t )
0 , n 1, 2,! .
(7.15)
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (7.15) ɢɦɟɟɬ ɜɢɞ
Tn ( t )
An exp ( a 2 Ȝ n t ) ,
(7.16)
ɝɞɟ An – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ (ɜ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɫɱɢɬɚɟɦ ɢɯ ɨɬɥɢɱɧɵɦɢ ɨɬ ɧɭɥɹ, ɬɚɤ ɤɚɤ ɧɚɫ ɢɧɬɟɪɟɫɭɸɬ ɬɨɥɶɤɨ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɟɲɟɧɢɹɦɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɞɚɱɢ (7.5) – (7.8) ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɢ
U n( x, t )
An exp ( a 2 Ȝ n t ) X n ( x ) ,
X n ( x ) Tn ( t )
n 1, 2,! . Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ ɜ ɜɢɞɟ ɪɹɞɚ, ɱɥɟɧɚɦɢ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɢ U n ( x, t ) : f
u ( x, t )
f
¦ U ( x, t ) ¦ A n
n
n 1
exp ( a 2 Ȝ n t ) X n ( x ) .
(7.17)
n 1
ȿɫɥɢ ɪɹɞ (7.17) ɫɯɨɞɢɬɫɹ ɢ ɟɝɨ ɦɨɠɧɨ ɩɨɱɥɟɧɧɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ ɞɜɚɠɞɵ ɩɨ x ɢ ɨɞɢɧ ɪɚɡ ɩɨ t , ɬɨ ɥɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɟɝɨ ɫɭɦɦɚ u ( x, t ) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ (7.1) ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ (7.3), (7.4). Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɢɡ ɩɨɫɬɪɨɟɧɢɹ ɮɭɧɤɰɢɣ U n ( x, t ) ɫɥɟɞɭɟɬ, ɱɬɨ
u t ( x, t ) a 2 u xx ( x, t )
f
¦ >U n 1
38
nt
( x, t ) U nxx ( x, t ) @ 0 ,
ɬɨ ɟɫɬɶ ɮɭɧɤɰɢɹ u ( x, t ) ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (7.1). Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ (7.3), (7.4). ȼɵɛɟɪɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ An ɬɚɤ, ɱɬɨɛɵ ɮɭɧɤɰɢɹ u ( x, t ) ɭɞɨɜɥɟɬɜɨɪɹɥɚ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ (7.2). Ȼɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɹ M ( x ) ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɚ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ: f
¦M
M(x)
n
X n( x ) ,
(7.18)
n 1
ɝɞɟ
M
l
1 n
Xn
2
1, 2,! .
³ M ( x )X ( x ) d x , n n
0
ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɟ (7.2) ɢ ɪɚɜɟɧɫɬɜɚ (7.17), (7.18), ɢɦɟɟɦ: f
u ( x, 0 )
¦
f
An X n ( x ) M ( x )
n 1
¦M
n
X n( x ) .
n 1
Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɮɭɧɤɰɢɹ u ( x, t ) ɛɭɞɟɬ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ (7.2), ɟɫɥɢ
M n , n 1, 2,! .
An
ɂɬɚɤ, ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.1) – (7.4) ɢɦɟɟɬ ɜɢɞ f
u ( x, t )
¦M
n
exp ( a 2 Ȝ n t ) X n ( x ) .
(7.19)
n 1
Ɋɚɫɫɦɨɬɪɟɧɧɵɣ ɦɟɬɨɞ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɪɟɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɮɭɧɤɰɢɣ, ɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ (ɫɜɨɟɣ ɞɥɹ ɤɚɠɞɨɣ ɮɭɧɤɰɢɢ), ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɦɟɬɨɞɚ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ. 2. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. ɍɤɚɡɚɧɧɚɹ ɡɚɞɚɱɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ u u ( x,t ) , ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɧɟɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ut
a 2 u xx f ( x, t ) , 0 x l , t ! 0 ,
(7.20)
ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ
u ( x, 0 ) 0 , 0 d x d l 39
(7.21)
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1 u x ( 0 ,t ) E 1 u ( 0 , t ) 0 , t ! 0 ,
(7.22)
D 2 u x ( l, t ) E 2 u ( l, t ) 0 , t ! 0 .
(7.23)
Ɂɞɟɫɶ f ( x, t ) – ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ, ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ a 2 , D 1 , E 1 , D 2 , E 2 ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɬɟɦ ɠɟ ɭɫɥɨɜɢɹɦ, ɱɬɨ ɢ ɜ ɡɚɞɚɱɟ (7.1) – (7.4). ȼ ɫɢɥɭ ɩɪɢɧɰɢɩɚ Ⱦɸɚɦɟɥɹ (ɫɦ. §5) ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.20) – (7.23) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ t
u ( x, t )
³ v ( x, t IJ; IJ ) dW ,
(7.24)
0
ɝɞɟ v
v ( x, t; IJ ) – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ vt
a 2 v xx , 0 x l , t ! 0 ,
v ( x, 0 ; IJ )
(7.25)
f ( x, IJ ) , 0 d x d l ,
(7.26)
D1 v x( 0, t ; W ) E1 v ( 0, t ; W ) 0 , t ! 0 ,
(7.27)
D 2 v x ( l, t; W ) E 2 v ( l, t; W ) 0 , t ! 0 .
(7.28)
ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɩɪɢ ɤɚɠɞɨɦ ɮɢɤɫɢɪɨɜɚɧɧɨɦ
W ! 0 ɮɭɧɤɰɢɹ f ( x , W ) ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɚ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ: f
f ( x,W )
¦ f ( IJ ) X ( x ), n
n
n 1
ɝɞɟ
fn ( IJ )
l
1
Xn
2
1, 2,! .
³ f ( x,W ) X ( x ) d x , n n
0
Ɍɨɝɞɚ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɡɚɞɚɱɚ (7.25) – (7.28) ɢɦɟɟɬ ɪɟɲɟɧɢɟ f
v ( x, t ;W )
¦ f ( IJ ) exp ( a Ȝ t ) X ( x ) . 2
n
n
n
(7.29)
n 1
ɉɨɞɫɬɚɜɥɹɹ (7.29) ɜ (7.24) ɢ ɢɡɦɟɧɹɹ ɦɟɫɬɚɦɢ ɫɭɦɦɢɪɨɜɚɧɢɟ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ, ɧɚɯɨɞɢɦ, ɱɬɨ f
u ( x, t )
¦T ( t )X ( x ), n
n 1
ɝɞɟ 40
n
t
³ exp >a O ( IJ t )@ f ( IJ ) dW , n 2
Tn ( t )
n
n
1, 2,! .
0
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ:
a 2 u xx f ( x, t ) , 0 x l , t ! 0 ,
ut
u ( x, 0 ) M ( x ) , 0 d x d l ,
(7.30) (7.31)
D 1 u x ( 0 ,t ) E 1 u ( 0 , t ) 0 , t ! 0 ,
(7.32)
D 2 u x ( l, t ) E 2 u ( l, t ) 0 , t ! 0 .
(7.33)
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.30) – (7.33) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
u ( x, t )
u1( x, t ) u2( x, t ) ,
ɝɞɟ u1 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
u1 t
a 2 u1 xx , 0 x l , t ! 0 ,
u 1 ( x, 0 ) M ( x ) , 0 d x d l ,
D 1 u1 x ( 0 ,t ) E 1 u1 ( 0 , t ) 0 , t ! 0 , D 2 u1 x ( l, t ) E 2 u 1 ( l, t ) 0 , t ! 0 , ɚ u2 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
a 2 u2 xx f ( x, t ) , 0 x l , t ! 0 ,
u2 t
u 2 ( x, 0 )
0, 0d x d l ,
D 1 u2 x ( 0 ,t ) E 1 u2 ( 0 , t ) 0 , t ! 0 , D 2 u2 x ( l, t ) E 2 u 2 ( l, t ) 0 , t ! 0 . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɞɚɱɚ (7.30) – (7.33) ɫɜɨɞɢɬɫɹ ɤ ɪɚɫɫɦɨɬɪɟɧɧɵɦ ɜɵɲɟ ɡɚɞɚɱɚɦ. 3. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. ɍɤɚɡɚɧɧɚɹ ɡɚɞɚɱɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ u u ( x,t ) , ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɧɟɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ut
a 2 u xx f ( x, t ) , 0 x l , t ! 0 ,
ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ 41
(7.34)
u ( x, 0 ) M ( x ) , 0 d x d l ,
(7.35)
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1ux ( 0 , t ) E 1 u( 0 , t ) J 1 ( t ) , t ! 0 ,
(7.36)
D 2 u x ( l, t ) E 2 u ( l, t ) J 2 ( t ) , t ! 0 .
(7.37)
Ɂɞɟɫɶ f ( x, t ) , M ( x ) , J 1 ( t ) J 2 ( t ) – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ, ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ a 2 , D 1 , E 1 , D 2 , E 2 ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɬɟɦ ɠɟ ɭɫɥɨɜɢɹɦ, ɱɬɨ ɢ ɜ ɡɚɞɚɱɟ (7.1) – (7.4). ɋɜɟɞɟɦ ɡɚɞɚɱɭ (7.34) – (7.37) ɤ ɡɚɞɚɱɟ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.34) – (7.37) ɜ ɜɢɞɟ
u ( x, t ) v ( x, t ) w ( x, t ) , ɝɞɟ v ( x, t ) – ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɚ w ( x, t ) – ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤɰɢɹ, ɞɜɚɠɞɵ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɩɨ x ɢ ɨɞɢɧ ɪɚɡ ɩɨ t ɢ ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D1w x( 0, t ) E1 w ( 0, t ) J 1 ( t ) , t ! 0 ,
(7.38)
D 2 w x ( l, t ) E 2 w ( l, t ) J 2 ( t ) , t ! 0 .
(7.39)
ȿɫɥɢ E 12 E 22 ! 0 , ɬɨ ɮɭɧɤɰɢɸ w ( x, t ) ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
w( x, t )
c( t ) x d( t ) .
(7.40)
ɉɨɞɫɬɚɜɥɹɹ (7.40) ɜ (7.38), (7.39), ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɞɜɭɯ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɜɭɯ ɧɟɢɡɜɟɫɬɧɵɯ ɮɭɧɤɰɢɣ c ( t ) ɢ d ( t ) , ɪɟɲɢɜ ɤɨɬɨɪɭɸ, ɧɚɣɞɟɦ ɮɭɧɤɰɢɸ w( x, t ) . Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ D 1 D 2 1 , E 1 E 2 0 . Ɍɨɝɞɚ, ɟɫɥɢ J 1( t ) J 2 ( t ) J ( t ) , ɬɨ ɮɭɧɤɰɢɸ w ( x, t ) ɦɨɠɧɨ ɡɚɞɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: w ( x, t ) J ( t ) x . ȿɫɥɢ ɠɟ J 1( t ) z J 2 ( t ) , ɬɨ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɢɫɤɚɬɶ w ( x, t ) ɜ ɜɢɞɟ
w( x, t ) c ( t ) x 2 d ( t ) x . ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɹ w ( x, t ) ɢɡɜɟɫɬɧɚ. ɉɨɞɫɬɚɜɥɹɹ ɮɭɧɤɰɢɸ u v w ɜ (7.34) – (7.37) ɢ ɭɱɢɬɵɜɚɹ ɪɚɜɟɧɫɬɜɚ (7.38), (7.39), ɩɨɥɭɱɢɦ ɡɚɞɚɱɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɨɜɨɣ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ v ( x, t ) :
vt
~ a 2 v xx f ( x, t ) , 0 x l , t ! 0 , 42
(7.41)
v ( x, 0 ) M~ ( x ) , 0 d x d l ,
ɝɞɟ
(7.42)
D 1v x ( 0 , t ) E 1 v ( 0 , t ) 0 , t ! 0 ,
(7.43)
D 2 v x ( l, t ) E 2 v ( l, t ) 0 , t ! 0 ,
(7.44)
~ f ( x, t )
f ( x, t ) a 2 w xx ( x, t ) w t ( x, t ) ,
M~ ( x ) M ( x ) w ( x, 0 ) .
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜɢɞɚ (7.41) – (7.44) ɪɚɫɫɦɚɬɪɢɜɚɥɨɫɶ ɜ §7, ɩ. 2. ɉɪɢɦɟɪ 1. Ɋɟɲɢɬɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ
ut
3S x , 0 x 1, t ! 0 , 2 Sx 2, u ( x, 0 ) 3 cos 2 u x ( 0 ,t ) sin t , u(1, t ) sin t 2 .
a 2 u xx x cos t 2 cos
(7.45) (7.46) (7.47)
Ɋɟɲɟɧɢɟ Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.45) – (7.47) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
u ( x, t ) v ( x, t ) w ( x, t ) , ɝɞɟ v ( x, t ) – ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɚ ɮɭɧɤɰɢɹ w ( x, t ) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
w x ( 0 , t ) sin t , w ( 1, t ) sin t 2 .
(7.48)
Ɏɭɧɤɰɢɸ w ( x, t ) ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
w ( x, t ) c ( t ) x d ( t ) . ɂɡ (7.48) ɫɥɟɞɭɟɬ, ɱɬɨ c ( t )
sin t , d ( t )
2 , ɢ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
w ( x, t ) x sin t 2 . Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ v ( x, t ) ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ
~ a 2 v xx f ( x, t ) , 0 x 1 , t ! 0 , v ( x, 0 ) M~ ( x ) ,
vt
v x ( 0, t )
0 , v ( 1, t ) 0 ,
ɝɞɟ 43
(7.49) (7.50) (7.51)
~ f ( x, t )
f ( x, t ) a 2 w xx ( x, t ) w t ( x, t )
M~ ( x ) M ( x ) w ( x, 0 ) 3 cos
2 cos
Sx 2
3S x , 2
.
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.49) – (7.51) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
v1 ( x, t ) v 2 ( x, t ) ,
v ( x, t ) ɝɞɟ v1 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
a 2 v1 xx ,
v1 t v 1 ( x, 0 )
3 cos
(7.52)
Sx
,
2 v 1 x ( 0 , t ) 0 , v1 ( 1 , t ) 0 ,
(7.53) (7.54)
ɚ v 2 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
v2 t
3S x , 2
a 2 v 2 xx 2 cos v 2 ( x, 0 )
v 2 x ( 0, t )
(7.55)
0,
0 , v 2 ( 1, t )
(7.56)
0.
(7.57)
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ (7.52) – (7.54). Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚɅɢɭɜɢɥɥɹ ɹɜɥɹɸɬɫɹ
On
( 2n 1 )2S 2 ( 2n 1 )S x , X n ( x ) cos , n 1, 2,! . 4 2
Ɍɚɤ ɤɚɤ
M~ ( x ) 3 cos
Sx 2
3X1( x ),
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
M~1
3 , M~n
0 ɞɥɹ n
2, 3, ! .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §7, ɩ. 1 ɫɥɟɞɭɟɬ, ɱɬɨ
v1 ( x,t )
§ a 2S 2 t · Sx . 3 exp ¨ ¸ cos 4 ¹ 2 ©
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɡɚɞɚɱɭ (7.55) – (7.57). 44
Ɍɚɤ ɤɚɤ
3S x 2X2( x ), 2 cos 2 ~ ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f ( x,t ) ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ~ f ( x,t )
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
~ f2
~ 2 , fn
0 ɞɥɹ n z 2 .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §7, ɩ. 2 ɫɥɟɞɭɟɬ, ɱɬɨ
v 2 ( x,t ) T2( t ) X 2( x ) T2 ( t ) cos
3S x , 2
ɝɞɟ t
T2 ( t )
³
~ exp >a 2 O 2 ( IJ t ) @ f 2 dW
0
8 9a 2S 2
ª § 9a 2S 2 t · º exp 1 ¨ ¸» . « 4 ¹¼ © ¬
ɂɬɚɤ,
v 2 ( x,t )
§ 9a 2S 2 t · º 3S x 8 ª , 1 exp¨ ¸ » cos 2 2 « 4 ¹¼ 2 9a S ¬ ©
ɢ
u ( x ,t )
v1( x, t ) v 2 ( x, t ) w ( x, t )
§ a 2S 2 t · 8 Sx 3 exp ¨ ¸ cos 4 ¹ 2 9a 2S 2 ©
ª 3S x § 9a 2S 2 t · º exp 1 ¨ ¸ » cos « 4 ¹¼ 2 © ¬
x sin t 2 . ɉɪɢɦɟɪ 2. Ɋɟɲɢɬɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ
ut
a 2 u xx cos S x 2 , 0 x 1 , t ! 0 ,
(7.58)
u ( x, 0 ) 4 cos 2S x 1 ,
(7.59)
u x ( 0 ,t )
(7.60)
0 , u x ( 1,t )
0.
Ɋɟɲɟɧɢɟ Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.58) – (7.60) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ (ɫɦ. §7, ɩ. 2)
u ( x, t )
u1( x, t ) u2 ( x, t )
ɝɞɟ u1 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
u1 t
a 2 u1 xx , 0 x 1 , t ! 0 , 45
(7.61)
u 1 ( x, 0 ) M ( x ) ,
u1 x ( 0 ,t )
0 , u1 x ( l, t )
(7.62)
0,
(7.63)
M ( x ) 4 cos 2S x 1 , ɚ u2 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
a 2 u2 xx f ( x, t ) , 0 x 1 , t ! 0 ,
u2 t
u 2 ( x, 0 )
0,
(7.64) (7.65)
u2 x ( 0 ,t ) 0 , u2 x ( l, t ) 0 ,
(7.66)
f ( x, t ) cos S x 2 . ɋɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɹɜɥɹɸɬɫɹ
On
( nʌ )2 , X n ( x ) cos nS x , n
0, 1, 2, ! .
Ɍɚɤ ɤɚɤ
M ( x ) 4 cos 2S x 1 1 X 0 ( x ) 4 X 2 ( x ) , ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
M 0 1 , M 2 4 , M n 0 ɞɥɹ n z 0, 2 . Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §7, ɩ. 1 ɫɥɟɞɭɟɬ, ɱɬɨ f
¦M
u 1 ( x, t )
n
exp ( a 2 Ȝ n t ) X n ( x )
n 0
1 4 exp ( 4 a 2 ʌ 2 t ) cos 2S x . Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ (7.64) – (7.66). Ɍɚɤ ɤɚɤ
f ( x, t )
cos S x 2
2X 0( x ) X1 ( x ) ,
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f ( x, t ) ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
f0
2 , f1
1 , fn
0 ɞɥɹ n t 2 .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §7, ɩ. 2 ɫɥɟɞɭɟɬ, ɱɬɨ f
u2( x, t )
¦T ( t )X ( x ) n
n
n 0
46
T0 ( t ) T1 ( t ) X 1( x ) ,
ɝɞɟ t
t
³ exp>a O ( IJ t )@ f 2
T 0( t )
0
0
³ f dW
dW
0
0
t
T 1( t )
1
³ exp>a O ( IJ t )@ f dW 2
1
2t ,
0
1
2
a S
0
>1 exp ( a
2
2
ʌ 2 t )@ .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
2t
u2 ( x, t )
1 2
a S2
>1 exp ( a ʌ t )@ cos S x 2
2
ɢ
u( x, t ) 1 2t 4 exp ( 4 a 2 ʌ 2 t ) cos 2S x +
1 2
a S2
>1 exp ( a ʌ t )@ cos S x . 2
2
§8. Ɋɟɲɟɧɢɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ 1. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. Ɋɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɡɚɞɚɱɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ u u ( x,t ) , ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ
a 2 u xx , 0 x l , t ! 0 ,
(8.1)
u ( x, 0 ) M ( x ) , 0 d x d l ,
(8.2)
ut ( x , 0) \ ( x ) , 0 d x d l
(8.3)
D 1 u x ( 0 ,t ) E 1 u ( 0 , t ) 0 , t ! 0 ,
(8.4)
utt ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 2 u x ( l, t ) E 2 u ( l, t ) 0 , t ! 0 . Ɂɞɟɫɶ a ! 0 , l ! 0 , D 1 , E 1 , D 2 , E 2 – ɡɚɞɚɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɪɢɱɟɦ
D 12 E 12 ! 0 , D 22 E 22 ! 0 ; M ( x ) , \ ( x ) – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. 47
(8.5)
Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (8.1) – (8.5) ɪɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɭɸ ɜɫɩɨɦɨɝɚɬɟɥɶɧɭɸ ɡɚɞɚɱɭ: ɇɚɣɬɢ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ
Utt
a 2U xx , 0 x l , t ! 0 ,
(8.6)
ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1U x ( 0 ,t ) E 1 U ( 0 , t ) 0 , t ! 0 ,
(8.7)
D 2U x ( l, t ) E 2 U ( l, t ) 0 , t ! 0
(8.8)
ɢ ɩɪɟɞɫɬɚɜɢɦɵɟ ɜ ɜɢɞɟ
U( x, t )
X( x )T( t )
(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 ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɮɭɧɤɰɢɣ T ( t ) ɢ X ( x ) :
T cc( t ) a 2 O T ( t )
0,
(8.10)
X cc( x ) O X ( x ) 0 .
(8.11)
ɉɨɞɫɬɚɜɥɹɹ (8.9) ɜ (8.7) ɢ ɜ (8.8), ɩɨɥɭɱɢɦ, ɱɬɨ ɮɭɧɤɰɢɹ X ( x ) ɞɨɥɠɧɚ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1 X c( 0 ) E 1 X ( 0 ) 0 , D 2 X c( l ) E 2 X ( l ) 0 .
(8.12) (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, ɫɭɳɟɫɬɜɭɟɬ ɫɱɟɬɧɨɟ ɦɧɨɠɟɫɬɜɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ O n t 0 , n 1, 2,! , ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ^X n ( x )` (ɫɦ. ɉɪɢɥɨɠɟɧɢɟ). ɉɭɫɬɶ Tn ( t ) – ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (7.11) ɩɪɢ O O n :
Tncc ( t ) a 2 O n Tn ( t ) 48
0 , n 1, 2,! .
(8.14)
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ E 12 E 22 ! 0 . Ɍɨɝɞɚ ɜɫɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ O n ! 0 , n 1, 2,! , ɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (8.14) ɢɦɟɟɬ ɜɢɞ
Tn ( t )
An cos a O n t Bn sin a O n t ,
(8.15)
ɝɞɟ An , Bn – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ (ɜ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɨɩɭɫɤɚɬɶ ɫɤɨɛɤɢ ɩɪɢ ɡɚɩɢɫɢ ɚɪɝɭɦɟɧɬɚ ɮɭɧɤɰɢɣ cos a O n t ɢ sin a O n t ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɟɲɟɧɢɹɦɢ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɨɣ ɜɵɲɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɞɚɱɢ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɢ
U n ( x,t ) Tn ( t ) X n ( x ) , n 1, 2,! . Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.1) – (8.5) ɜ ɜɢɞɟ ɪɹɞɚ f
¦
u ( x, t )
f
U n ( x, t )
n 1
¦T ( t )X n
n
( x ).
n 1
ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɢ M ( x ) ɢ \ ( x ) ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɵ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ: f
M(x)
¦
f
M n X n( x ) , \ ( x )
n 1
¦\
n
X n( x ) ,
n 1
ɝɞɟ
Mn
l
1 Xn
2
³
M ( x ) X n( x ) d x , \ n
0
l
1 Xn
2
³\ ( x )X ( x ) d x , n
0
n 1, 2,! . Ɍɨɝɞɚ, ɩɨɬɪɟɛɨɜɚɜ, ɱɬɨɛɵ ɮɭɧɤɰɢɹ u ( x, t ) ɭɞɨɜɥɟɬɜɨɪɹɥɚ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ (8.2), (8.3), ɩɨɥɭɱɢɦ ɪɚɜɟɧɫɬɜɚ f
u ( x, 0 )
¦
f
Tn ( 0 ) X n ( x ) M ( x )
n 1
¦M
f
u t ( x, 0 )
¦
n
X n( x ) ,
n 1 f
¦\
Tnc ( 0 ) X n ( x ) \ ( x )
n 1
n
X n( x ) .
n 1
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɭɸ ɡɚɞɚɱɭ Ʉɨɲɢ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɣ Tn ( t ) :
Tncc ( t ) a 2 O n Tn ( t ) 49
0,
(8.16)
Tn ( 0 ) M n , Tnc( 0 ) \ n , n 1, 2,! .
(8.17)
ɂɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɟ (8.15) ɞɥɹ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (8.14), ɧɚɯɨɞɢɦ, ɱɬɨ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (8.16), (8.17) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
Tn ( t ) M n cos a O n t
\n sin a O n t . a On
(8.18)
ɂɬɚɤ, ɜ ɫɥɭɱɚɟ E 12 E 22 ! 0 ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.1) – (8.5) ɢɦɟɟɬ ɜɢɞ f
§
¦ ¨¨©M
u ( x, t )
n
cos a On t
n 1
· \n sin a On t ¸¸ X n ( x ) . (8.19) a On ¹
ɉɪɟɞɩɨɥɨɠɢɦ ɬɟɩɟɪɶ, ɱɬɨ D 1 D 2 1 , E 1 E 2 0 . ɋɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɹɜɥɹɸɬɫɹ 2
§ nS · ¨ ¸ , X n ( x) © l ¹
On
cos
nS x ,n l
0, 1, 2 ! ;
ɩɪɢ ɷɬɨɦ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɪɚɜɧɵ
X0
2
l , Xn
2
l 2 , n 1, 2,! .
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.1) – (8.5) ɜ ɜɢɞɟ f
u ( x, t )
¦T ( t )X n
n
( x ).
n 0
Ɇɵ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɢ M ( x ) ɢ \ ( x ) ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɵ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ^X n ( x )`: f
M(x)
¦
f
M n X n( x ) , \ ( x )
n 0
¦\
n
X n( x ) ,
n 0
ɝɞɟ
Mn
\0
X0
2
l
³
M ( x ) X 0( x ) d x
0
l
1
Xn
l
1
M0
2
1 M ( x )d x , l 0
³
l
³ M ( x )X ( x ) d x n
0
nS x 2 , n 1, 2,! , M ( x ) cos l 0 l
³
l
l
³
³
nS x 1 2 \ ( x )d x , \ n , n 1, 2,! . \ ( x ) cos l 0 l l 0 50
Ɍɚɤ ɠɟ, ɤɚɤ ɢ ɪɚɧɶɲɟ, ɦɵ ɩɨɥɭɱɚɟɦ ɡɚɞɚɱɭ Ʉɨɲɢ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɣ Tn ( t ) :
Tncc ( t ) a 2 O n Tn ( t )
0,
(8.20)
0, 1, 2, ! .
Tn ( 0 ) M n , Tnc( 0 ) \ n , n ȼ ɫɥɭɱɚɟ n
(8.21)
0 ɭɪɚɜɧɟɧɢɟ (8.20) ɩɪɢɧɢɦɚɟɬ ɜɢɞ T0cc( t ) 0 .
(8.22)
Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (8.22) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
T0 ( t )
A0 B0 t ,
ɝɞɟ A0 , B0 – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ. ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (8.21), ɧɚɯɨɞɢɦ, ɱɬɨ
M 0 , B0 \ 0 .
A0 Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɫɥɭɱɚɟ n
0 ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.20), (8.21) ɢɦɟɟɬ ɜɢɞ
T0 ( t ) M 0 \ 0 t . ȼ ɫɥɭɱɚɟ n t 1 ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ O n ɩɨɥɨɠɢɬɟɥɶɧɵ, ɢ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.20), (8.21) ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ (8.18):
Tn ( t ) M n cos a O n t
M n cos
\n a On
nS a t l \ n nS a t sin , n 1, 2,! . l nS a l
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ D 1 (8.5) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
D2
1, E1
E2
f
u ( x, t )
¦
0 ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (8.1) – f
Tn ( t ) X n ( x ) T0 ( t )
n 0
f
M0 \ 0 t
sin a O n t
n
n
(x)
n 1
§
¦ ¨¨©M n 1
¦T ( t )X
n
cos
nS a t l \ n nS a t · nS x ¸¸ cos sin . l nS a l ¹ l (8.23)
2. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. ɍɤɚɡɚɧɧɚɹ ɡɚɞɚɱɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ u u ( x,t ) , ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɧɟɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ 51
ut t
a 2 u xx f ( x, t ) , 0 x l , t ! 0 ,
(8.24)
u ( x, 0 ) 0 , 0 d x d l ,
(8.25)
ut ( x , 0)
(8.26)
ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
0, 0 d x d l
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1 u x ( 0 ,t ) E 1 u ( 0 , t ) 0 , t ! 0 ,
(8.27)
D 2 u x ( l, t ) E 2 u ( l, t ) 0 , t ! 0 .
(8.28) 2
Ɂɞɟɫɶ f ( x, t ) – ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ, ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ a , D 1 , E 1 , D 2 , E 2 ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɬɟɦ ɠɟ ɭɫɥɨɜɢɹɦ, ɱɬɨ ɢ ɜ ɡɚɞɚɱɟ (8.1) – (8.5).
ȼ ɫɢɥɭ ɩɪɢɧɰɢɩɚ Ⱦɸɚɦɟɥɹ (ɫɦ. §5) ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.24) – (8.28) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ t
³ v ( x, t IJ; IJ ) dW ,
u ( x, t ) ɝɞɟ v
v ( x, t; IJ ) – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
a 2 v xx , 0 x l , t ! 0 ,
vt
(8.29)
0
(8.30)
v ( x, 0 ; IJ ) 0 , 0 d x d l , v t ( x, 0 ; IJ )
(8.31)
f ( x, IJ ) , 0 d x d l ,
(8.32)
D1 v x( 0, t ; W ) E1 v ( 0, t ; W ) 0 , t ! 0 ,
(8.33)
D 2 v x ( l, t;W ) E 2 v ( l, t; W ) 0 , t ! 0 .
(8.34)
ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɩɪɢ ɤɚɠɞɨɦ ɮɢɤɫɢɪɨɜɚɧɧɨɦ
W ! 0 ɮɭɧɤɰɢɹ f ( x , W ) ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɚ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ: f
f ( x,W )
¦ f ( IJ ) X ( x ), n
n
n 1
ɝɞɟ
fn ( IJ )
l
1
Xn
2
³ f ( x,W ) X ( x ) d x , n n
1, 2,! .
0
ɉɭɫɬɶ E 12 E 22 ! 0 . Ɍɨɝɞɚ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɡɚɞɚɱɚ (8.30) – (8.34) ɢɦɟɟɬ ɪɟɲɟɧɢɟ 52
f
fn ( IJ )
¦a
v ( x, t ;W )
On
n 1
sin a Ȝn t X n ( x ) .
(8.35)
ɉɨɞɫɬɚɜɥɹɹ (8.35) ɜ (8.29) ɢ ɢɡɦɟɧɹɹ ɦɟɫɬɚɦɢ ɫɭɦɦɢɪɨɜɚɧɢɟ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ, ɧɚɯɨɞɢɦ, ɱɬɨ f
¦T ( t )X ( x ),
u ( x, t )
n
n
n 1
ɝɞɟ
Tn ( t )
t
1
³ sin >a
a On
@
On ( IJ t ) f n ( IJ ) dW , n 1, 2,! .
0
ɉɪɟɞɩɨɥɨɠɢɦ ɬɟɩɟɪɶ, ɱɬɨ D 1 D 2 1 , E 1 E 2 0 . ȼ ɫɢɥɭ ɩɪɢɧɰɢɩɚ Ⱦɸɚɦɟɥɹ ɪɟɲɟɧɢɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ (8.29), ɝɞɟ v v ( x, t; IJ ) – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
a 2 v xx , 0 x l , t ! 0 ,
vt
v ( x, 0 ; IJ )
(8.36)
0, 0 d x d l ,
(8.37)
f ( x, IJ ) , 0 d x d l ,
v t ( x, 0 ; IJ )
(8.38)
v x( 0, t ; W ) 0 , t ! 0 , v x ( l, t; W ) 0 , t ! 0 .
(8.39) (8.40)
Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.36) – (8.40) ɢɦɟɟɬ ɜɢɞ f
v ( x, t; IJ )
f0 ( IJ ) t
n 1 f
l Sa
¦
f0 ( IJ )
1 l
f0 ( IJ ) t
fn ( IJ )
¦a
n 1
On
sin a Ȝn t cos On x
f n (W ) nS a t nS x sin cos , n l l
(8.41)
ɝɞɟ
fn ( IJ )
2 l
l
³ f ( x,IJ ) d x , 0
l
³ f ( x,IJ ) cos 0
nS x d x , n 1, 2,! . l
ɉɨɞɫɬɚɜɥɹɹ (8.41) ɜ (8.29) ɢ ɢɡɦɟɧɹɹ ɦɟɫɬɚɦɢ ɫɭɦɦɢɪɨɜɚɧɢɟ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ, ɧɚɯɨɞɢɦ, ɱɬɨ ɜ ɫɥɭɱɚɟ D 1 D 2 1 , E 1 E 2 0 ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (8.24) – (8.28) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ 53
f
¦ T (t ) cos
u ( x, t ) T0 ( t )
n
n 1
nS x , l
(8.42)
ɝɞɟ t
T0 ( t )
³ f ( IJ )( t IJ ) d W , 0
0
Tn ( t )
l
t
f ( IJ ) sin nS a ³ n
0
nS a ( t IJ ) dW , n 1, 2,! . l
Ɋɚɫɫɦɨɬɪɢɦ ɬɚɤɠɟ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ:
ut
a 2 u xx f ( x, t ) , 0 x l , t ! 0 ,
(8.43)
u ( x, 0 ) M ( x ) , 0 d x d l ,
(8.44)
ut ( x, 0 ) \ ( x ) , 0 d x d l ,
(8.45)
D 1 u x ( 0 ,t ) E 1 u ( 0 , t ) 0 , t ! 0 ,
(8.46)
D 2 u x ( l, t ) E 2 u ( l, t ) 0 , t ! 0 .
(8.47)
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.30) – (7.33) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
u ( x, t )
u1( x, t ) u2 ( x, t ) ,
ɝɞɟ u1 ɢ u2 – ɪɟɲɟɧɢɹ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɪɚɧɟɟ ɡɚɞɚɱ:
u1 t
a 2 u1 xx , 0 x l , t ! 0 ,
u 1 ( x, 0 ) M ( x ) , 0 d x d l ,
u1 t ( x, 0 ) \ ( x ) , 0 d x d l ,
D 1 u1 x ( 0 ,t ) E 1 u1 ( 0 , t ) 0 , t ! 0 , D 2 u1 x ( l, t ) E 2 u 1 ( l, t ) 0 , t ! 0 , ɢ
u2 t
a 2 u2 xx f ( x, t ) , 0 x l , t ! 0 ,
u 2 ( x, 0 )
0, 0d x d l ,
u2 t ( x, 0 ) 0 , 0 d x d l ,
D 1 u2 x ( 0 ,t ) E 1 u2 ( 0 , t ) 0 , t ! 0 , D 2 u2 x ( l, t ) E 2 u 2 ( l, t ) 0 , t ! 0 . 54
3. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. ɍɤɚɡɚɧɧɚɹ ɡɚɞɚɱɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ u u ( x,t ) , ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɧɟɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ
ut t
a 2 u xx f ( x, t ) , 0 x l , t ! 0 ,
(8.48)
u ( x, 0 ) M ( x ) , 0 d x d l ,
(8.49)
ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
ut ( x, 0 ) \ ( x ) , 0 d x d l
(8.50)
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D 1ux ( 0 , t ) E 1 u( 0 , t ) J 1 ( t ) , t ! 0 ,
(8.51)
D 2 u x ( l, t ) E 2 u ( l, t ) J 2 ( t ) , t ! 0 .
(8.52)
Ɂɞɟɫɶ f ( x, t ) , M ( x ) , \ ( x ) , J 1 ( t ) J 2 ( t ) – ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ, ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ a 2 , D 1 , E 1 , D 2 , E 2 ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɬɟɦ ɠɟ ɭɫɥɨɜɢɹɦ, ɱɬɨ ɢ ɜ ɡɚɞɚɱɟ (8.1) – (8.5). Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.48) – (8.52) ɜ ɜɢɞɟ
u ( x, t ) v ( x, t ) w ( x, t ) , ɝɞɟ v ( x, t ) – ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɚ w ( x, t ) – ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤɰɢɹ, ɞɜɚɠɞɵ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɩɨ x ɢ ɩɨ t ɢ ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
D1w x( 0, t ) E1 w ( 0, t ) J 1 ( t ) , t ! 0 ,
(8.53)
D 2 w x ( l, t ) E 2 w ( l, t ) J 2 ( t ) , t ! 0 .
(8.54)
ɇɚɯɨɠɞɟɧɢɟ ɮɭɧɤɰɢɢ w ( x, t ) ɪɚɫɫɦɨɬɪɟɧɨ ɜ §7, ɩ. 3. ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɢɡɜɟɫɬɧɚ. ɉɨɞɫɬɚɜɥɹɹ ɮɭɧɤɰɢɸ u v w ɜ (8.48) – (8.52) ɢ ɭɱɢɬɵɜɚɹ ɪɚɜɟɧɫɬɜɚ (8.53), (8.54), ɩɨɥɭɱɢɦ ɡɚɞɚɱɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɨɜɨɣ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ v ( x, t ) :
~ a 2 v xx f ( x, t ) , 0 x l , t ! 0 , v ( x, 0 ) M~ ( x ) , 0 d x d l ,
(8.56)
v t ( x, 0 ) \~ ( x ) , 0 d x d l ,
(8.57)
D 1v x ( 0 , t ) E 1 v ( 0 , t ) 0 , t ! 0 ,
(8.58)
vt t
55
(8.55)
D 2 v x ( l, t ) E 2 v ( l, t ) 0 , t ! 0 , ɝɞɟ
(8.59)
~ f ( x, t )
f ( x, t ) a 2 w xx ( x, t ) w t ( x, t ) , M~ ( x ) M ( x ) w ( x, 0 ) , \~ ( x ) \ ( x ) w t ( x, 0 ) . Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜɢɞɚ (8.55) – (8.59) ɪɚɫɫɦɚɬɪɢɜɚɥɨɫɶ ɜ §8, ɩ. 2. ɉɪɢɦɟɪ 1. Ɋɟɲɢɬɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ
ut t
a 2 u xx 2 x cos t 3 sin
Sx 2
, 0 x 1, t ! 0 ,
3S x 1, 2 5S x , ut ( x, 0 ) sin 2
u ( x, 0 )
2 sin
u( 0 , t ) cos t , u x (1, t )
(8.60) (8.61) (8.62)
t2 .
(8.63)
Ɋɟɲɟɧɢɟ Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.60) – (8.63) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
u ( x, t ) v ( x, t ) w ( x, t ) , ɝɞɟ v ( x, t ) – ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɚ ɮɭɧɤɰɢɹ w ( x, t ) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
w ( 0 , t ) cos t , w x ( 1 , t )
t2 .
(8.64)
Ɏɭɧɤɰɢɸ w ( x, t ) ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
w ( x, t ) c ( t ) x d ( t ) . ɂɡ (8.64) ɫɥɟɞɭɟɬ, ɱɬɨ d ( t )
cos t , c ( t )
t 2 , ɢ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
w ( x, t ) x t 2 cos t . Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ v ( x, t ) ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ
~ a 2 v xx f ( x, t ) , 0 x 1 , t ! 0 , v ( x, 0 ) M~ ( x ) ,
(8.66)
v t ( x, 0 ) \~ ( x ) ,
(8.67)
v ( 0 , t ) 0 , v x ( 1, t ) 0 ,
(8.68)
vt
56
(8.65)
ɝɞɟ
~ f ( x, t )
f ( x, t ) a 2 w xx ( x, t ) w t t ( x, t )
3 sin
Sx 2
,
3S x , 2 5S x \~ ( x ) \ ( x ) w t ( x, 0 ) sin . 2
M~ ( x ) M ( x ) w ( x, 0 ) 2 sin
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.65) – (8.68) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
v1 ( x, t ) v 2 ( x, t ) ,
v ( x, t ) ɝɞɟ v1 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
v1 t
a 2 v1 xx ,
(8.69)
3S x , 2 5S x , v 1 t ( x, 0 ) sin 2 v 1 ( 0 , t ) 0 , v1 x ( 1 , t ) 0 ,
v 1 ( x, 0 )
2 sin
(8.70) (8.71) (8.72)
ɚ v 2 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
v2 t
a 2 v 2 xx 3 sin
v 2 ( x, 0 )
0,
v 2 t ( x, 0 )
0,
v 2 ( 0, t )
Sx 2
,
(8.73) (8.74) (8.75)
0 , v 2 x ( 1, t )
0.
(8.76)
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ (8.69) – (8.72). Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚɅɢɭɜɢɥɥɹ ɹɜɥɹɸɬɫɹ
On
( 2n 1 )2S 2 ( 2n 1 )S x , X n ( x ) sin , n 1, 2,! . 4 2
Ɍɚɤ ɤɚɤ
M~ ( x ) 2 sin
3S x 2
ɢ 57
2 X 2 ( x ),
\~ ( x ) sin
5S x 2
X3( x )
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɞɚɧɧɵɯ ɮɭɧɤɰɢɣ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
M~2
\~
2 , M~n 1 , \~
3
0 ɞɥɹ n z 2 , 0 ɞɥɹ n z 3 .
n
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §8, ɩ. 1 ɫɥɟɞɭɟɬ, ɱɬɨ
v1 ( x,t ) M 2 cos a O 2 t X 2 ( x ) 2 cos
\3 a O3
sin a O 3 t X 3 ( x )
3S a t 3S x 2 5S a t 5S x . sin sin sin 2 2 5S a 2 2
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɡɚɞɚɱɭ (8.73) – (8.76). Ɍɚɤ ɤɚɤ
~ f ( x,t )
Sx 3 sin 3X1( x ), 2 ~ ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f ( x,t ) ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
~ f1
~ 3 , fn
0 ɞɥɹ n z 1 .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §8, ɩ. 2 ɫɥɟɞɭɟɬ, ɱɬɨ
v 2 ( x,t ) T1( t ) X 1( x ) T1 ( t ) sin
Sx 2
,
ɝɞɟ
T1 ( t ) 3 2
a O1
1
a O1
t
³ sin >a
@
O1 ( IJ t ) f 1 dW
0
1 cos a
O1t
12 § Sat · 1 cos ¸. 2 2 ¨ 2 ¹ S a ©
ɂɬɚɤ,
v 2 ( x,t )
Sat · S x 12 § , 1 cos ¸ sin 2 2 ¨ S a © 2 ¹ 2
ɢ 58
v1( x, t ) v 2 ( x, t ) w ( x, t )
u ( x ,t ) 2 cos
3S a t 3S x 2 5S a t 5S x sin sin sin 2 2 5S a 2 2
Sat · S x 12 § x t 2 cos t . ¸ sin ¨ 1 cos 2 ¹ 2
S 2a 2 ©
ɉɪɢɦɟɪ 2. Ɋɟɲɢɬɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ
ut
a 2 u xx 2 cos 3S x 4 , 0 x 1 , t ! 0 ,
(8.77)
u ( x, 0 ) cos 2S x 2 ,
(8.78)
ut ( x, 0 )
3,
(8.79)
ux ( 0 , t )
0 , ux ( 1, t )
0.
(8.80)
Ɋɟɲɟɧɢɟ Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.77) – (8.80) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ (ɫɦ. §8, ɩ. 2)
u ( x, t )
u1( x, t ) u2 ( x, t )
ɝɞɟ u1 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
u1 t
a 2 u1 xx , 0 x 1 , t ! 0 ,
(8.81)
u 1 ( x, 0 ) M ( x ) ,
(8.82)
u1 t ( x, 0 ) \ ( x ) ,
(8.83)
u1 x ( 0 ,t ) 0 , u1 x ( l, t ) 0 ,
(8.84)
M ( x ) cos 2S x 2 , \ ( x ) 3 , ɚ u2 – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
u2 t
a 2 u2 xx f ( x, t ) , 0 x 1 , t ! 0 ,
(8.85)
u 2 ( x, 0 )
0,
(8.86)
u 2 t ( x, 0 )
0,
(8.87)
u2 x ( 0 ,t ) 0 , u2 x ( l, t ) 0 ,
(8.88)
f ( x, t )
2 cos 3S x 4 .
ɋɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɹɜɥɹɸɬɫɹ 59
( nʌ )2 , X n ( x ) cos nS x , n
On
0, 1, 2, ! .
Ɍɚɤ ɤɚɤ
M ( x ) cos 2S x 2 2 X 0 ( x ) X 2 ( x ) , ɢ
\ ( x ) 3 3X 0( x ), ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɞɚɧɧɵɯ ɮɭɧɤɰɢɣ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
M 0 2 , M 2 1 , M n 0 ɞɥɹ n z 0, 2 , \ 0 3 , \ n 0 ɞɥɹ n t 1 . Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §8, ɩ. 1 ɫɥɟɞɭɟɬ, ɱɬɨ
u1 ( x, t ) f
M0 \ 0 t
§
¦ ¨¨©M
n
cos nS a t
n 1
\n nS a
· sin nS a t ¸¸ cos nSx ¹
2 3t cos 2S a t cos 2S x . Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ (8.85) – (8.88). Ɍɚɤ ɤɚɤ
2 cos 3S x 4
f ( x, t )
4X0( x ) 2X3 ( x ),
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f ( x, t ) ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
f0
4, f3
0 ɞɥɹ n z 0, 3 .
2 , fn
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §8, ɩ. 2 ɫɥɟɞɭɟɬ, ɱɬɨ f
¦T ( t )X ( x )
u2( x, t )
n
n
T0 ( t ) T3 ( t ) X 3( x ) ,
n 0
ɝɞɟ t
T 0( t )
³f
0
( t IJ ) dW
2t 2 ,
0
T 3( t )
1 3S a
t
³f
3
sin>3S a (t IJ)@ dW
0
60
2 1 cos 3S a t . 9S 2 a 2
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
2t 2
u2 ( x, t )
2 9S 2 a 2
1 cos 3S a t cos 3S x
ɢ
2 3t 2t 2 cos 2S a t cos 2S x
u ( x, t )
2 9S 2 a 2
1 cos 3S a t cos 3S x .
Ɂɚɞɚɱɢ ɇɚɣɬɢ ɪɟɲɟɧɢɟ u 1. ut
a 2 u xx x sin t 3 cos
u ( x, 0 ) ux ( 0 , t ) 2. ut
u ( x, t ) ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ: 2
2 , 0 x 1, t ! 0 ,
3S x x, 2 cos t , u( 1 , t ) cos t 2 t .
2 cos
a 2 u xx x sin t cos t 2 sin
u ( x, 0 ) sin
Sx
x, 2 u ( 0 , t ) sin t , u x ( 1 , t )
3. ut
Sx
3Sx 2 , 0 x 1, t ! 0 , 2
cos t .
a 2 u xx x cos t 2 sin t 2 sin 5S x , 0 x 1 , t ! 0 ,
u ( x, 0 )
3 sin 2S x 2 ,
u ( 0, t )
2 cos t , u ( 1, t ) sin t 2 cos t .
4. ut
a 2 u xx 2a 2 cos t x 2 sin t 2 cos S x 3 , 0 x 1 , t ! 0 ,
u ( x, 0 )
x2 2x 4,
u x ( 0, t )
2 , u x ( 1, t )
5. ut
2 cos t 2 .
a 2 u xx 2 x cos t 3 sin
Sx 2
2 , 0 x 1, t ! 0 , 61
2 sin
u ( x, 0 )
Sx
,
2 u ( 0 , t ) sin t , u x ( 1, t )
a 2 u xx x 2 sin t 2 cos
6. ut
ux ( 0 , t )
5S x 2, 2 t , u( 1, t ) t 2 cos t .
2
, 0 x 1, t ! 0 ,
a 2 u xx 2a 2 sin t x 2 cos t 3 cos 2S x , 0 x 1 , t ! 0 ,
u ( x, 0 )
x 2,
u x ( 0 , t ) 1 , u x ( 1, t ) 8. ut
Sx
3 cos
u ( x, 0 )
7. ut
2t .
2 sin t 1 .
a 2 u xx 2 x cos t 3 sin S x 1 , 0 x 1 , t ! 0 ,
u ( x, 0 )
3 sin 3S x 2 ,
u ( 0, t )
t , u ( 1, t )
a 2 u xx x cos t 3 sin
9. ut
3S x , 2 2t , u x ( 1 , t ) sin t .
u ( 0, t )
2
2 , 0 x 1, t ! 0 ,
a 2 u xx 2 x cos t 2t 3 cos
u ( x, 0 ) ux ( 0 , t ) 11. ut t
Sx
2 sin
u ( x, 0 )
10. ut
2 sin t t .
3S x , 2 2 sin t , u ( 1 , t )
2 cos
2 sin t t 2 .
a 2 u xx 2 x 2 sin t 3 cos
Sx
3S x , 2 5S x 2, u t ( x, 0 ) 2 cos 2 u x ( 0 , t ) t 2 , u ( 1 , t ) t 2 2 sin t .
u ( x, 0 )
5Sx , 0 x 1, t ! 0 , 2
2
cos
62
, 0 x 1, t ! 0 ,
a 2 u xx ( x 2 2 a 2 ) sin t 2 x cos t 3 , 0 x 1 , t ! 0 , u ( x, 0 ) 2 cos 2S x 2 x 1 , u t ( x, 0 ) x 2 4 , u x ( 0 , t ) 2 cos t , u x ( 1 , t ) 2 sin t 2 cos t .
12. ut t
13. ut t
a 2 u xx 2 x 2 cos t 3 sin
3S x 2, 2 5S x , u t ( x, 0 ) 4 sin 2 u ( 0 , t ) 2 cos t , u x ( 1, t )
u ( x, 0 )
Sx
, 0 x 1, t ! 0 ,
2
2 sin
t2.
a 2 u xx x sin t 3 sin 2S x , 0 x 1 , t ! 0 , u ( x, 0 ) sin S x 2 sin 3S x 2 , u t ( x, 0 ) sin 5S x x , u ( 0 , t ) 2 , u ( 1 , t ) sin t 2 .
14. ut t
15. ut t
a 2 u xx x cos t 3 cos
Sx 2
2 , 0 x 1, t ! 0 ,
3S x 5S x 2 cos x, 2 2 Sx , u t ( x, 0 ) 4 cos 2 u x ( 0 , t ) cos t , u ( 1 , t ) cos t t 2 .
u ( x, 0 )
16. ut t
cos
a 2 u xx 2 x cos t sin t 3 sin
Sx
3S x 2x , 2 5S x 1, u t ( x, 0 ) sin 2 u ( 0 , t ) sin t , u x ( 1, t ) 2 cos t .
u ( x, 0 )
2
, 0 x 1, t ! 0 ,
2 sin
a 2 u xx 2 a 2 t 2 x sin t 4 , 0 x 1 , t ! 0 , u ( x, 0 ) 2 cos 3S x 3 , u t ( x, 0 ) x 2 2 x 2 , u x ( 0 , t ) 2 sin t , u x ( 1 , t ) 2t 2 sin t .
17. ut t
63
3S x , 0 x 1, t ! 0 , 2 5S x Sx 2 sin , u ( x, 0 ) 3 sin 2 2 3S x u t ( x, 0 ) sin 2x 1, 2 u ( 0 , t ) t , u x ( 1 , t ) 2 sin t .
a 2 u xx 2 x sin t sin
18. ut t
a 2 u xx 2 x cos 2t sin 2S x , 0 x 1 , t ! 0 , u ( x, 0 ) 2 sin S x 4 sin 3S x , u t ( x, 0 ) 3 sin 5S x 2 , u ( 0 , t ) 2t , u ( 1 , t ) sin 2 t 2t .
19. ut t
a 2 u xx x sin t 3 sin
20. ut t
Sx 2
4 , 0 x 1, t ! 0 ,
3S x 5S x sin , 2 2 Sx u t ( x, 0 ) 4 sin x, 2 u ( 0 , t ) 2t 2 , u x ( 1, t ) sin t .
u ( x, 0 )
2 sin
ɉɪɢɥɨɠɟɧɢɟ ɋɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɫɨɛɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɢ ɤɜɚɞɪɚɬɵ ɧɨɪɦ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɜ ɡɚɞɚɱɟ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ I – IV 1 2
nS x l 2 § nS · , Xn , ¨ ¸ , X n ( x ) sin l 2 © l ¹ n 1, 2, ! . 2 2 ( 2n 1 ) S ( 2n 1 )S x II. ɚ) O n , X n ( x ) cos , 2 4l 2l l 2 , n 1, 2, ! . Xn 2 I. O n
1
ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ ɢ ɨɩɪɟɞɟɥɟɧɢɹ ɫɦ. ɜ §6.
64
ɛ) O n W n2 , n ɭɪɚɜɧɟɧɢɹ
1, 2, ! , ɝɞɟ W n – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɪɧɢ tg W l
W
;
h1 h X n ( x ) cos W n x 1 sin W n x ,
Wn
Xn
2
l W n2 h12 h1 2 W n2
,
n 1, 2, ! .
( 2n 1 )2S 2 ( 2n 1 )S x , X n ( x ) sin , 2 4l 2l l 2 , n 1, 2, ! . Xn 2 2 W n , n 1, 2, ! , ɝɞɟ W n – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɪɧɢ
III. ɚ) O n
ɛ) O n ɭɪɚɜɧɟɧɢɹ
tg W l
W
;
h2 X n ( x ) sin W n x , Xn
2
l W n2 h22 h2
2
IV. ɚ) O n
§ nS · ¨ ¸ , Xn( x ) © l ¹
X0 ɛ) O n W n2 , n ɭɪɚɜɧɟɧɢɹ
2
l , Xn
,n
2 W n2 h22
cos 2
nS x ,n l
1, 2, ! .
0, 1, 2 ! ,
l ɞɥɹ n 1, 2, ! . 2
1, 2, ! , ɝɞɟ W n – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɪɧɢ
tg W l
h2
W
;
X n ( x ) cos W n x ,
65
ɜ) O n W n2 , n ɭɪɚɜɧɟɧɢɹ
1, 2, ! , ɝɞɟ W n – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɪɧɢ
tg W l
h1
W
X n ( x ) cos W n x
Xn ɝ)
On
2
;
h1
Wn
l W n2 h12 h1 2 W n2
sin W n x ,
,n
1, 2, ! .
W n2 , n 1, 2, ! , ɝɞɟ W n – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɤɨɪɧɢ
ɭɪɚɜɧɟɧɢɹ
ctg W l
W 2 h 1h 2 ; W ( h1 h2 )
X n ( x ) cos W n x Xn
2
h1
Wn
sin W n x ,
l ( IJ n2 h12 ) ( IJ n2 h22 ) ( IJ n2 h1 h2 ) ( h1 h 2 ) 2W n2 ( IJ n2 h22 )
,
n 1, 2, ! .
Ʌɢɬɟɪɚɬɭɪɚ 1. Ʉɭɞɪɹɜɰɟɜ Ʌ.Ⱦ. Ʉɭɪɫ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ : ɜ 2-ɯ ɬ. / Ʌ.Ⱦ. Ʉɭɞɪɹɜɰɟɜ. – Ɇ. : ȼɵɫɲ. ɲɤ., 1981. – Ɍ. 2. – 584 ɫ. 2. Ɍɢɯɨɧɨɜ Ⱥ.ɇ. ɍɪɚɜɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ / Ⱥ.ɇ. Ɍɢɯɨɧɨɜ, Ⱥ.Ⱥ. ɋɚɦɚɪɫɤɢɣ. – Ɇ. : ɇɚɭɤɚ, 1977. – 736 ɫ. 3. Ʉɨɲɥɹɤɨɜ ɇ.ɋ. ɍɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ / ɇ.ɋ. Ʉɨɲɥɹɤɨɜ, ɗ.Ȼ. Ƚɥɢɧɟɪ, Ɇ.Ɇ. ɋɦɢɪɧɨɜ. – Ɇ. : ȼɵɫɲ. ɲɤ., 1970. – 712 ɫ. 4. ȼɥɚɞɢɦɢɪɨɜ ȼ.ɋ. ɍɪɚɜɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ / ȼ.ɋ. ȼɥɚɞɢɦɢɪɨɜ. – Ɇ. : ɇɚɭɤɚ, 1976. – 528 ɫ. 5. Ɇɢɯɥɢɧ ɋ.Ƚ. Ʌɢɧɟɣɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ / ɋ.Ƚ. Ɇɢɯɥɢɧ. – Ɇ. : ȼɵɫɲ. ɲɤ., 1977. – 432 ɫ. 6. ɋɜɟɲɧɢɤɨɜ Ⱥ.Ƚ. Ʌɟɤɰɢɢ ɩɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɟ / Ⱥ.Ƚ. ɋɜɟɲɧɢɤɨɜ, Ⱥ.ɇ. Ȼɨɝɨɥɸɛɨɜ, ȼ.ȼ. Ʉɪɚɜɰɨɜ. – Ɇ. : ɂɡɞ-ɜɨ ɆȽɍ, 1993. – 352 ɫ. 7. ɑɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ : ɭɱɟɛ.-ɦɟɬɨɞ. ɩɨɫɨɛɢɟ / ɫɨɫɬ. Ⱥ.Ⱥ. Ʉɭɥɢɤɨɜ. – ȼɨɪɨɧɟɠ : ɅɈɉ ȼȽɍ, 2006. – 63 ɫ. 66
ɋɈȾȿɊɀȺɇɂȿ ɉɪɟɞɢɫɥɨɜɢɟ . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3 §1. ɉɨɧɹɬɢɟ ɨɛ ɭɪɚɜɧɟɧɢɹɯ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɢ ɨɛ ɭɪɚɜɧɟɧɢɹɯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ . . . . . . . . . . . . . . . . . . . . . . . 3 §2. Ʉɪɚɟɜɵɟ ɡɚɞɚɱɢ ɞɥɹ ɭɪɚɜɧɟɧɢɣ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 §3. ɉɨɫɬɚɧɨɜɤɚ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 §4. ɉɨɫɬɚɧɨɜɤɚ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 §5. ɇɟɨɞɧɨɪɨɞɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ. ɉɪɢɧɰɢɩ Ⱦɸɚɦɟɥɹ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 §6. Ɂɚɞɚɱɚ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 §7. Ɋɟɲɟɧɢɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ . . . . . . . . . 36 §8. Ɋɟɲɟɧɢɟ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ . . . . . . . . . . . . . . . . . . 47 Ɂɚɞɚɱɢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ɉɪɢɥɨɠɟɧɢɟ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Ʌɢɬɟɪɚɬɭɪɚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
67
ɍɱɟɛɧɨɟ ɢɡɞɚɧɢɟ
ɋɆȿɒȺɇɇɕȿ ɁȺȾȺɑɂ ȾɅə ɍɊȺȼɇȿɇɂə ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ ɂ ɍɊȺȼɇȿɇɂə ɄɈɅȿȻȺɇɂɃ ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɜɭɡɨɜ ɋɨɫɬɚɜɢɬɟɥɶ Ʉɭɥɢɤɨɜ Ⱥɥɟɤɫɚɧɞɪ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ ɂɡɞɚɧɨ ɜ ɚɜɬɨɪɫɤɨɣ ɪɟɞɚɤɰɢɢ
ɉɨɞɩɢɫɚɧɨ ɜ ɩɟɱɚɬɶ 11.09.07. Ɏɨɪɦɚɬ 60×84/16. ɍɫɥ. ɩɟɱ. ɥ. 3,95. Ɍɢɪɚɠ 50 ɷɤɡ. Ɂɚɤɚɡ 1028. ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. 394000, ɝ. ȼɨɪɨɧɟɠ, ɩɥ. ɢɦ. Ʌɟɧɢɧɚ, 10. Ɍɟɥ. 208-298, 598-026 (ɮɚɤɫ) http://www.ppc.vsu.ru; e-mail:
[email protected]
.
Ɉɬɩɟɱɚɬɚɧɨ ɜ ɬɢɩɨɝɪɚɮɢɢ ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɨɝɨ ɰɟɧɬɪɚ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. 394000, ɝ. ȼɨɪɨɧɟɠ, ɭɥ. ɉɭɲɤɢɧɫɤɚɹ, 3. Ɍɟɥ. 204-133.
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