Уравнения математической физики
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Ɋɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ y 2uxx x2u yy 0 , x > 0, y > 0. Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɣ ɬɢɩ ɜ ɩɟɪɜɨɦ ɤɜɚɞɪɚɧɬɟ.
Ⱦɚɥɟɟ, ɨɩɪɟɞɟɥɹɟɦ ɧɨɜɵɟ ɤɨɨɪɞɢɧɚɬɵ, ɱɬɨɛɵ ɩɪɢɜɟɫɬɢ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɟ.
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ɝɪɚɥɚ |
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ɇɨɜɵɟ ɤɨɨɪɞɢɧɚɬɵ ɜɜɨɞɢɦ ɩɨ ɮɨɪɦɭɥɚɦ [ y2 x2, K |
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ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɨɥɭɱɢɦ |
ɧɨɜɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ |
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B 16x2 y2 , C 0 , D 2(x2 y2) , E 2(y2 x2) , F 0 , G 0 .
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Au[[ Bu[K CuKK Du[ EuK Fu G ɢ ɩɨ- |
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ɩɨɥɭɱɢɦ ɤɚɧɨɧɢɱɟɫɤɢɣ ɜɢɞ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: u |
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ɇɚ ɫɚɦɨɦ ɞɟɥɟ ɞɥɹ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɫɭɳɟɫɬɜɭɟɬ ɞɜɟ ɤɚɧɨɧɢɱɟɫɤɢɟ ɮɨɪɦɵ. ȼɬɨɪɭɸ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɢɡ ɩɟɪɜɨɣ ɡɚɦɟɧɨɣ ɩɟɪɟɦɟɧɧɵɯ ɜɢɞɚ D D([,K) [ K , E E([,K) [ K. ɇɚɣɞɹ
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1.3 ɉɈɇəɌɂȿ Ɉ ɇȺɑȺɅɖɇɕɏ ɂ ȽɊȺɇɂɑɇɕɏ ɍɋɅɈȼɂəɏ
ɂɡ ɤɭɪɫɚ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ, ɤɚɤ ɩɪɚɜɢɥɨ, ɢɦɟɸɬ ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɪɟɲɟɧɢɣ. ɗɬɨ ɫɜɹɡɚɧɨ ɫ ɩɨɹɜɥɟɧɢɟɦ ɜ ɩɪɨɰɟɫɫɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɤɨɧɫɬɚɧɬ, ɩɪɢ
10
ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ ɤɨɬɨɪɵɯ ɪɟɲɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɢɫɯɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ. Ɋɟɲɟɧɢɟ ɡɚɞɚɱ ɦɚɬɮɢɡɢɤɢ ɫɜɹɡɚɧɨ ɫ ɧɚɯɨɠɞɟɧɢɟɦ ɡɚɜɢɫɢɦɨɫɬɟɣ ɨɬ ɤɨɨɪɞɢɧɚɬ ɢ ɜɪɟɦɟɧɢ ɨɩɪɟɞɟɥɟɧɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɤɨɬɨɪɵɟ, ɛɟɡɭɫɥɨɜɧɨ, ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɬɪɟɛɨɜɚɧɢɹɦ ɨɞɧɨɡɧɚɱɧɨɫɬɢ, ɤɨɧɟɱɧɨɫɬɢ ɢ ɧɟɩɪɟɪɵɜɧɨɫɬɢ. ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɥɸɛɚɹ ɡɚɞɚɱɚ ɦɚɬɮɢɡɢɤɢ ɩɪɟɞɩɨɥɚɝɚɟɬ ɩɨɢɫɤ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɪɟɲɟɧɢɹ (ɟɫɥɢ ɨɧɨ ɜɨɨɛɳɟ ɫɭɳɟɫɬɜɭɟɬ). ɉɨɷɬɨɦɭ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɨɪɦɭɥɢɪɨɜɤɚ ɮɢɡɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɞɨɥɠɧɚ ɩɨɦɢɦɨ ɨɫɧɨɜɧɵɯ ɭɪɚɜɧɟɧɢɣ (Ⱦɍɑɉ), ɨɩɢɫɵɜɚɸɳɢɯ ɢɫɤɨɦɵɟ ɮɭɧɤɰɢɢ ɜɧɭɬɪɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɨɛɥɚɫɬɢ, ɜɤɥɸɱɚɬɶ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ (ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɢɥɢ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ), ɨɩɢɫɵɜɚɸɳɢɟ ɢɫɤɨɦɵɟ ɮɭɧɤɰɢɢ ɧɚ ɝɪɚɧɢɰɚɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɨɛɥɚɫɬɢ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢ ɜɨ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɬɨɱɤɚɯ ɨɛɥɚɫɬɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ. ɗɬɢ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɧɚɡɵɜɚɸɬ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɝɪɚɧɢɱɧɵɦɢ ɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɡɚɞɚɱɢ.
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
ɉɪɟɞɩɨɥɨɠɢɦ, ɧɟɨɛɯɨɞɢɦɨ ɪɟɲɢɬɶ ɨɩɪɟɞɟɥɟɧɧɭɸ ɡɚɞɚɱɭ, ɨɩɢɫɵɜɚɟɦɭɸ Ⱦɍɑɉ, ɞɥɹ ɧɟɤɨɬɨɪɨɣ ɨɛɥɚɫɬɢ. Ɍɨɝɞɚ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (Ƚɍ), ɬ. ɟ. ɫɜɹɡɚɬɶ ɢɫɤɨɦɵɟ ɩɟɪɟɦɟɧɧɵɟ ɧɚ ɝɪɚɧɢɰɟ ɨɛɥɚɫɬɢ ɧɟɤɨɬɨɪɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ.
ɉɨ ɜɢɞɭ ɭɪɚɜɧɟɧɢɣ, ɡɚɞɚɸɳɢɯ Ƚɍ, ɪɚɡɥɢɱɚɸɬ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ:
-ɩɟɪɜɨɝɨ ɪɨɞɚ (ɭɫɥɨɜɢɹ Ⱦɢɪɢɯɥɟ), ɡɚɞɚɸɳɢɟ ɡɧɚɱɟɧɢɹ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ ɱɟɪɟɡ ɡɚɞɚɧɧɭɸ ɧɚ ɝɪɚɧɢɰɟ ɨɛɥɚɫɬɢ ɢɡɜɟɫɬɧɭɸ ɮɭɧɤɰɢɸ;
-ɜɬɨɪɨɝɨ ɪɨɞɚ (ɭɫɥɨɜɢɹ ɇɟɣɦɚɧɚ), ɡɚɞɚɸɳɢɟ ɡɧɚɱɟɧɢɹ ɧɨɪɦɚɥɶɧɨɣ
ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ ɱɟɪɟɡ ɡɚɞɚɧɧɭɸ ɧɚ ɝɪɚɧɢɰɟ ɨɛɥɚɫɬɢ ɢɡɜɟɫɬɧɭɸ ɮɭɧɤɰɢɸ;
-ɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ (ɭɫɥɨɜɢɹ Ɋɨɛɟɧɚ), ɤɨɝɞɚ ɧɚ ɝɪɚɧɢɰɟ ɡɚɞɚɟɬɫɹ ɥɢɧɟɣɧɚɹ ɤɨɦɛɢɧɚɰɢɹ Ƚɍ 1-ɝɨ ɢ 2-ɝɨ ɜɢɞɚ.
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɜ ɡɚɞɚɱɚɯ, ɨɩɢɫɵɜɚɸɳɢɯ ɧɟɫɬɚɰɢɨɧɚɪɧɵɟ, ɬ.ɟ. ɢɡɦɟɧɹɸɳɢɟɫɹ ɜɨ ɜɪɟɦɟɧɢ ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ, ɩɨɦɢɦɨ ɝɪɚɧɢɱɧɵɯ ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɜɚɬɶ ɟɳɟ ɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɨɩɪɟɞɟɥɹɸɳɢɟ ɡɧɚɱɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɢɥɢ ɢɯ ɝɪɚɞɢɟɧɬɨɜ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɨɛɥɚɫɬɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
11
Ʌɟɤɰɢɹ ʋ2 2.1. ȼɕȼɈȾ ɍɊȺȼɇȿɇɂə ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ
Ɉɞɧɨɦɟɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ut Dux x f (x,t)
ɦɨɠɧɨ ɜɵɜɟɫɬɢ, ɨɩɢɪɚɹɫɶ ɧɚ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɣ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɚ. ɉɪɢ ɷɬɨɦ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɡɚɜɢɫɢɦɨɫɬɶ ɫɤɨɪɨɫɬɢ ɬɟɩɥɨɨɛɦɟɧɚ ɨɬ ɨɫɧɨɜɧɵɯ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ: ɬɟɩɥɨɩɪɨ-
ɜɨɞɧɨɫɬɢ, ɬɟɩɥɨɟɦɤɨɫɬɢ ɢ ɩɥɨɬɧɨɫɬɢ ɦɚɬɟɪɢɚɥɚ.
ȼ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɨɫɧɨɜɧɵɦ ɩɪɢɧɰɢɩɨɦ ɹɜɥɹɟɬɫɹ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ (ɬɟɩɥɨɜɨɣ ɷɧɟɪɝɢɢ). ȼɫɟ ɨɫɬɚɥɶɧɵɟ ɭɬɜɟɪɠɞɟɧɢɹ ɜɵɜɨɞɹɬɫɹ ɢɡ ɷɬɨɝɨ ɨɫɧɨɜɧɨɝɨ ɩɪɢɧɰɢɩɚ. ȼ ɱɚɫɬɧɨɫɬɢ, ɢɡ ɭɪɚɜɧɟɧɢɹ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɢ ɜɵɜɨɞɢɬɫɹ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɨɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɵ L, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɝɨ ɫɞɟɥɚɟɦ ɫɥɟɞɭɸɳɢɟ ɩɪɟɞɩɨɥɨɠɟɧɢɹ:
1. ɋɬɟɪɠɟɧɶ ɫɞɟɥɚɧ ɢɡ ɨɞɧɨɝɨ ɨɞɧɨɪɨɞɧɨɝɨ ɩɪɨɜɨɞɹɳɟɝɨ ɦɚɬɟɪɢɚɥɚ.
Рɢɫ. 2. Ɍɨɧɤɢɣ ɬɟɩɥɨɩɪɨɜɨɞɹɳɢɣ ɫɬɟɪɠɟɧɶ
2.Ȼɨɤɨɜɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɫɬɟɪɠɧɹ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɚ (ɬɟɩɥɨ ɦɨɠɟɬ ɪɚɫɩɪɨɫɬɪɚɧɹɬɫɹ ɬɨɥɶɤɨ ɜɞɨɥɶ ɨɫɢ ɯ).
3.ɋɬɟɪɠɟɧɶ ɬɨɧɤɢɣ, ɷɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɬɟɦɩɟɪɚɬɭɪɚ ɜɫɟɯ ɬɨɱɟɤ ɜ ɤɚɠɞɨɦ ɩɨɩɟɪɟɱɧɨɦ ɫɟɱɟɧɢɢ ɫɬɟɪɠɧɹ ɩɨɫɬɨɹɧɧɚ. ȿɫɥɢ ɪɚɫɫɦɨɬɪɟɬɶ ɱɚɫɬɶ ɫɬɟɪɠɧɹ ɧɚ ɨɬɪɟɡɤɟ [ x,x 'x] ɢ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɡɚɤɨɧɨɦ ɫɨ-
ɯɪɚɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɚ, ɬɨ ɦɨɠɧɨ ɧɚɩɢɫɚɬɶ:
Ɉɛɳɟɟ ɢɡɦɟɧɟɧɢɟ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɚ ɧɚ ɨɬɪɟɡɤɟ [ x,x 'x]
= ɉɨɥɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɩɪɨɲɟɞɲɟɝɨ ɱɟɪɟɡ ɝɪɚɧɢɰɵ + ɉɨɥɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɜɧɭɬɪɢ ɨɬɪɟɡɤɚ [ x,x 'x] .
Ɉɛɳɟɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ ɜɧɭɬɪɢ ɨɬɪɟɡɤɚ [ x,x 'x] ɜ ɥɸɛɨɣ ɦɨ-
x'x
ɦɟɧɬ ɜɪɟɦɟɧɢ t ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ ³ cUAu(s,t)ds , ɝɞɟ:
x
12
ɫ – ɭɞɟɥɶɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ ɦɚɬɟɪɢɚɥɚ (ɩɨɤɚɡɵɜɚɟɬ ɫɩɨɫɨɛɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɡɚɩɚɫɚɬɶ ɬɟɩɥɨ), U – ɩɥɨɬɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ, A – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɹ.
Ɍɨɝɞɚ ɡɚɤɨɧɭ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɦɨɠɧɨ ɩɪɢɞɚɬɶ ɫɥɟɞɭɸɳɭɸ ɦɚɬɟɦɚɬɢɱɟɫɤɭɸ ɮɨɪɦɭ:
d |
x'x |
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³ cUAu(s,t)ds cUA |
³ ut (s,t)ds |
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dt |
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k A[ux(x 'x,t) ux(x,t)] A ³ |
f (s,t)ds |
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ɝɞɟ k – ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ (ɩɨɤɚɡɵɜɚɟɬ ɫɩɨɫɨɛɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɩɪɨɜɨɞɢɬɶ ɬɟɩɥɨ), f(ɯ, t) – ɨɛɴɟɦɧɚɹ ɦɨɳɧɨɫɬɶ ɜɧɟɲɧɟɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ.
Ɂɚɞɚɱɚ ɬɟɩɟɪɶ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ ɡɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɟ (2.1) ɜ ɮɨɪɦɟ, ɧɟ ɫɨɞɟɪɠɚɳɟɣ ɢɧɬɟɝɪɚɥɨɜ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ ɩɪɨɛɥɟɦɵ ɫɥɟɞɭɟɬ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɬɟɨɪɟɦɨɣ ɨ ɫɪɟɞɧɟɦ ɡɧɚɱɟɧɢɢ ɢɡ ɤɭɪɫɚ ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɢɫɱɢɫɥɟɧɢɹ.
Ɍɟɨɪɟɦɚ ɨ ɫɪɟɞɧɟɦ ɡɧɚɱɟɧɢɢ
ȿɫɥɢ ɮɭɧɤɰɢɹ f(x) ɧɟɩɪɟɪɵɜɧɚ ɧɚ ɨɬɪɟɡɤɟ [a,b] , ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɩɨ
b
ɤɪɚɣɧɟɣɦɟɪɟɨɞɧɚɬɨɱɤɚ [ [a,b] ɬɚɤɚɹ, ɱɬɨ ³ f (x)dx f ([)(b a) .
a
ɉɪɢɦɟɧɹɹ ɷɬɨɬ ɪɟɡɭɥɶɬɚɬ ɤ ɭɪɚɜɧɟɧɢɸ (2.1), ɩɪɢɯɨɞɢɦ ɤ ɫɥɟ-
ɞɭɸɳɟɦɭ ɭɪɚɜɧɟɧɢɸ: |
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cUAut ([,t)'x k A[ux(x 'x,t) ux(x,t)] A f ([,t)'x , |
x [ x 'x , |
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ªux |
(x 'x,t) ux |
(x,t)º |
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ɢɥɢ ut ([,t) |
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f ([,t) . |
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cU |
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ɍɫɬɪɟɦɢɦ 'x ɤ ɧɭɥɸ, ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ |
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ut (x,t) D2ux x(x,t) F(x,t) , |
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ɝɞɟ D2 |
k(cU) 1 – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ, |
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F(x,t) |
(cU) 1 f (x,t) |
– ɩɥɨɬɧɨɫɬɶ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ. |
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ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɛɨɤɨɜɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɫɬɟɪɠɧɹ ɧɟ ɹɜɥɹɟɬɫɹ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɨɣ, ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɜɟɥɢɱɢɧɚ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɱɟɪɟɡ
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ɛɨɤɨɜɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɫɬɟɪɠɧɹ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ (ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ E – ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ) ɪɚɡɧɨɫɬɢ ɦɟɠɞɭ ɬɟɦɩɟɪɚɬɭ-
ɪɨɣ ɫɬɟɪɠɧɹ ɢ(ɯ, t) ɢ ɬɟɦɩɟɪɚɬɭɪɨɣ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ, ɤɨɬɨɪɚɹ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɢ ɪɚɜɧɨɣ ɧɭɥɸ (ɩɪɢ ɥɸɛɨɣ ɞɪɭɝɨɣ ɩɨɫɬɨɹɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɪɟɡɭɥɶɬɚɬ ɛɭɞɟɬ ɚɧɚɥɨɝɢɱɟɧ). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɚ ɩɪɢɜɨɞɢɬ ɤ ɭɪɚɜɧɟɧɢɸ
ut (x,t) D2ux x(x,t) Eu F(x,t) .
ȼɩɥɨɫɤɨɦ ɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɦ ɫɥɭɱɚɹɯ ɩɪɢ ɜɵɜɨɞɟ ɭɪɚɜɧɟɧɢɹ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɥɟɞɭɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɪɚɧɟɟ ɢɡɭɱɟɧɧɵɣ ɭɱɟɛɧɵɣ ɦɚɬɟɪɢɚɥ ɩɨ ɬɟɦɟ ɤɪɚɬɧɵɟ ɢɧɬɟɝɪɚɥɵ ɢ ɬɟɨɪɢɹ ɩɨɥɹ.
ȼɨɫɧɨɜɭ ɜɵɜɨɞɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɨɥɨɠɟɧ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ, ɤɨɬɨɪɵɣ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɫɥɭɱɚɟ ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɤɨɥɢɱɟɫɬɜɨ
ɬɟɩɥɨɬɵ dQ, ɜɜɟɞɟɧɧɨɟ ɜ ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ ɢɡɜɧɟ ɡɚ ɜɪɟɦɹ dW ɜɫɥɟɞɫɬɜɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɚ ɬɚɤɠɟ ɨɬ ɜɧɭɬɪɟɧɧɢɯ ɢɫɬɨɱɧɢɤɨɜ, ɪɚɜɧɨ ɢɡɦɟɧɟɧɢɸ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɢɥɢ ɷɧɬɚɥɶɩɢɢ ɜɟɳɟɫɬɜɚ (ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɪɚɫɫɦɨɬɪɟɧɢɹ ɢɡɨɯɨɪɧɨɝɨ ɢɥɢ ɢɡɨɛɚɪɧɨɝɨ ɩɪɨɰɟɫɫɚ), ɫɨɞɟɪɠɚɳɟɝɨɫɹ ɜ ɷɥɟɦɟɧɬɚɪɧɨɦ ɨɛɴɟɦɟ:
dQ1 dQ2 dQ , |
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ɝɞɟ dQ1 – ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɜɜɟɞɟɧɧɨɟ ɜ ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ ɩɭɬɟɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɡɚ ɜɪɟɦɹ dt ; dQ2 – ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɡɚ ɜɪɟɦɹ dt ɜɵɞɟɥɢɥɨɫɶ ɜ ɷɥɟɦɟɧɬɚɪɧɨɦ ɨɛɴɟɦɟ dV ɡɚ ɫɱɟɬ ɜɧɭɬɪɟɧɧɢɯ ɢɫɬɨɱɧɢɤɨɜ; dQ – ɢɡɦɟɧɟɧɢɟ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɢɥɢ ɷɧɬɚɥɶɩɢɢ ɜɟɳɟɫɬɜɚ, ɫɨɞɟɪɠɚɳɟɝɨɫɹ ɜɷɥɟɦɟɧɬɚɪɧɨɦ ɨɛɴɟɦɟ dV , ɡɚɜɪɟɦɹ dt .
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɫɨɫɬɚɜɥɹɸɳɢɯ ɭɪɚɜɧɟɧɢɹ (2.2) ɜɵɞɟɥɢɦ ɜ ɬɟɥɟ ɷɥɟɦɟɧɬɚɪɧɵɣ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞ ɫɨ ɫɬɨɪɨɧɚɦɢ dx, dy, dz (ɪɢɫ. 3). ɉɚɪɚɥɥɟɥɟɩɢɩɟɞ ɞɨɥɠɟɧ ɛɵɬɶ ɪɚɫɩɨɥɨɠɟɧ ɬɚɤ, ɱɬɨɛɵ ɟɝɨ ɝɪɚɧɢ ɛɵɥɢ ɩɚɪɚɥɥɟɥɶɧɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɤɨɨɪɞɢɧɚɬɧɵɦ ɩɥɨɫɤɨɫɬɹɦ. Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɩɨɞɜɨɞɢɬɫɹ ɤ ɝɪɚɧɹɦ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɨɛɴɟɦɚ ɡɚ ɜɪɟɦɹ dt ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɟɣ Ox, Ɉɭ, Oz, ɨɛɨɡɧɚɱɢɦ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ dQx, dQy, dQz. Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɛɭɞɟɬ ɨɬɜɨɞɢɬɶɫɹ ɱɟɪɟɡ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɝɪɚɧɢ ɜ ɬɟɯ ɠɟ ɧɚɩɪɚɜɥɟɧɢɹɯ, ɨɛɨɡɧɚɱɢɦ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ dQx+dx, dQy+dy, dQz+dz. Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɨɞɜɟɞɟɧɧɨɟ ɤ ɝɪɚɧɢ dydz ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ɉɯ ɡɚ ɜɪɟɦɹ dt ɫɨɫɬɚɜɥɹɟɬ
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dQx qxdydzdt , ɝɞɟ qx – ɩɪɨɟɤɰɢɹ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɧɨɪɦɚɥɢ ɤ ɭɤɚɡɚɧɧɨɣ ɝɪɚɧɢ.
Рɢɫ. 3. Ʉ ɜɵɜɨɞɭ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɨɬɜɟɞɟɧɧɨɟ ɱɟɪɟɡ ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɝɪɚɧɶ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ɉɯ, ɡɚɩɢɲɟɬɫɹ ɤɚɤ dQx dx qx dxdydzdt . Ɋɚɡɧɢɰɚ ɦɟɠɞɭ ɤɨɥɢɱɟɫɬɜɨɦ ɬɟɩɥɨɬɵ, ɩɨɞ-
ɜɟɞɟɧɧɨɝɨ ɤ ɷɥɟɦɟɧɬɚɪɧɨɦɭ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɭ, ɢ ɤɨɥɢɱɟɫɬɜɨɦ ɬɟɩɥɨ-
ɬɵ ɨɬɜɟɞɟɧɧɨɝɨ ɨɬ ɧɟɝɨ ɡɚ ɜɪɟɦɹ dt ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ɉɯ, |
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ɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ: |
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dQ |
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dQ |
x dx |
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x dx |
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Ɍɚɤ ɤɚɤ ɮɭɧɤɰɢɹ qx+dx ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɢɧɬɟɪɜɚɥɟ dx, ɬɨ ɤ ɧɟɣ ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɮɨɪɦɭɥɭ Ʌɚɝɪɚɧɠɚ.
Ⱥɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚɡɨɦ ɦɨɠɧɨ ɧɚɣɬɢ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɨɞɜɨɞɢɦɨɟ ɤ ɷɥɟɦɟɧɬɚɪɧɨɦɭ ɨɛɴɟɦɭ ɢ ɜ ɧɚɩɪɚɜɥɟɧɢɹɯ ɞɜɭɯ ɞɪɭɝɢɯ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ Ɉɭ ɢ Ɉz. Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ dQ, ɩɨɞɜɟɞɟɧɧɨɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɤ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦɭ ɨɛɴɟɦɭ, ɛɭɞɟɬ ɪɚɜɧɨ:
dQ1 |
§wq |
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wq |
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wq |
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· |
G |
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¨ wx |
wy |
wz |
¸dxdydzdt |
div(q)dxdydzdt |
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x |
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y |
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© |
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¹ |
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Ɍɟɩɟɪɶ ɨɩɪɟɞɟɥɢɦ ɜɬɨɪɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ ɭɪɚɜɧɟɧɢɹ (2.3). Ɉɛɨɡɧɚɱɢɦ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɜɵɞɟɥɹɟɦɨɟ ɜɧɭɬɪɟɧɧɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɜ ɟɞɢɧɢɰɟ ɨɛɴɟɦɚ ɫɪɟɞɵ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɢ ɧɚɡɵɜɚɟɦɨɟ ɦɨɳɧɨ-
ɫɬɶɸ |
ɜɧɭɬɪɟɧɧɢɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɨɬɵ, ɱɟɪɟɡ qV , ɬɨɝɞɚ |
dQ2 |
qV dxdydzdt . |
15
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɭɪɚɜɧɟɧɢɸ
cUwu |
div(qG) q |
wt |
V |
ȼ ɬɜɟɪɞɵɯ ɬɟɥɚɯ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ Ɏɭ- |
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G |
JJJJG G |
k(x, y,z) – ɤɨɷɮɮɢɰɢɟɧɬ (ɜɨɨɛɳɟ ɝɨɜɨ- |
ɪɶɟ q |
k(x, y,z)grad u , ɝɞɟ |
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ɪɹ, ɩɟɪɟɦɟɧɧɵɣ) ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɢɡɨɬɪɨɩɧɨɝɨ ɬɟɥɚ. ɉɨɞɫɬɚɜɥɹɹ ɜ (2.4), ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɢɡɨɬɪɨɩɧɨɝɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ
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cUwu |
JJJJG |
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div(k grad u) F(x, y,z,t) , |
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ɝɞɟ F(x, y,z,t) |
– ɬ.ɧ. ɮɭɧɤɰɢɹ ɢɫɬɨɱɧɢɤɚ. ȼ ɫɥɭɱɚɟ ɨɞɧɨɪɨɞɧɨɝɨ ɢɡɨ- |
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ɬɪɨɩɧɨɝɨ ɬɟɥɚ (k = const) ɩɪɢɯɨɞɢɦ ɤ ɛɨɥɟɟ ɩɪɨɫɬɨɦɭ ɭɪɚɜɧɟɧɢɸ |
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wu |
a2'u f (x, y,z,t) , |
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ɝɞɟ 'u 2u |
JJJJG |
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div(grad u) – ɥɚɩɥɚɫɢɚɧ, f (x, y,z,t) – ɦɨɞɢɮɢɰɢɪɨ- |
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ɜɚɧɧɚɹ ɮɭɧɤɰɢɹ ɢɫɬɨɱɧɢɤɚ, ɚ a k /(cU) .
ȼ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ (2.5), ɤɨɬɨɪɨɟ ɦɵ ɜɵɜɟɥɢ ɞɥɹ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɫɬɨɢɬ ɥɚɩɥɚɫɢɚɧ (ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ). ȼ ɰɢɥɢɧɞ-
ɪɢɱɟɫɤɨɣ |
ɫɢɫɬɟɦɟ |
ɤɨɨɪɞɢɧɚɬ |
ɥɚɩɥɚɫɢɚɧ |
ɢɦɟɟɬ |
ɜɢɞ: |
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'u |
w2u |
1 wu |
1 |
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w2u |
w2u , ɝɞɟ r – ɪɚɞɢɭɫ-ɜɟɤɬɨɪ; ij – ɩɨɥɹɪɧɵɣ |
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r 2 wM2 |
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wr 2 |
r wr |
wz 2 |
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ɭɝɨɥ; z – ɚɩɥɢɤɚɬɚ. ȼɵɪɚɠɟɧɢɟ ɥɚɩɥɚɫɢɚɧɚ ɜ ɫɮɟɪɢɱɟɫɤɢɯ ɤɨɨɪɞɢɧɚɬɚɯ ɢɦɟɟɬ ɜɢɞ
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1 w2(r u) |
1 |
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w |
wu |
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w2u |
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'u |
r wr 2 |
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sinTwT |
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– ɫɮɟɪɢɱɟ- |
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r 2 sinT |
wT |
r 2 sin2 T |
wM2 |
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ɫɤɢɣ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ; |
ij – ɩɨɥɹɪɧɵɣ (ɚɡɢɦɭɬɚɥɶɧɵɣ) ɭɝɨɥ, ș – ɧɨɪ- |
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ɦɚɥɶɧɵɣ (ɡɟɧɢɬɧɵɣ) ɭɝɨɥ. |
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ȼ ɥɸɛɨɦ ɫɥɭɱɚɟ ɜɧɟ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɵɛɨɪɚ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ (2.5), ɩɪɢ ɷɬɨɦ ɥɚɩɥɚ-
ɫɢɚɧ ɛɭɞɟɬ ɢɦɟɬɶ ɨɞɧɭ ɢɡ ɭɤɚɡɚɧɧɵɯ ɜɵɲɟ ɮɨɪɦ.
ȿɫɥɢ ɜɧɭɬɪɟɧɧɢɟ ɢɫɬɨɱɧɢɤɢ ɢɥɢ ɫɬɨɤɢ ɬɟɩɥɨɬɵ ɨɬɫɭɬɫɬɜɭɸɬ, ɬɨ ɭɪɚɜɧɟɧɢɟ (2.5) ɩɪɢɧɢɦɚɟɬ ɜɢɞ ɭɪɚɜɧɟɧɢɹ Ɏɭɪɶɟ
wu |
a2'u , |
(2 |
wt |
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16
ȼ ɫɥɭɱɚɟ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɨɬɫɭɬɫɬɜɢɢ ɜɧɭɬɪɟɧɧɢɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɨɬɵ ɜɵɪɚɠɟɧɢɟ (2.6) ɩɪɢɧɢɦɚɟɬ ɜɢɞ ɭɪɚɜɧɟɧɢɹ Ʌɚɩɥɚɫɚ 'u 0 .
Ⱦɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɪɚɡɥɢɱɚɸɬ 3 ɬɢɩɚ Ƚɍ, ɤɨɬɨɪɵɟ ɡɚɞɚɸɬ ɬɟɩɥɨɜɨɣ ɪɟɠɢɦ ɧɚ ɝɪɚɧɢɰɟ. Ɉɫɧɨɜɧɵɟ ɜɢɞɵ ɬɟɩɥɨɜɵɯ ɪɟɠɢɦɨɜ
-ɧɚ ɝɪɚɧɢɰɟ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɨɩɪɟɞɟɥɟɧɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ. ɗɬɨ Ƚɍ ɩɟɪɜɨɝɨ ɪɨɞɚ (ɡɚɞɚɱɚ Ⱦɢɪɢɯɥɟ);
-ɱɟɪɟɡ ɝɪɚɧɢɰɭ ɩɨɞɚɟɬɫɹ ɨɩɪɟɞɟɥɟɧɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ (ɡɚɞɚɟɬɫɹ ɡɧɚɱɟɧɢɟ ɜɧɟɲɧɟɣ ɧɨɪɦɚɥɶɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ). ɗɬɨ Ƚɍ ɜɬɨɪɨɝɨ ɪɨɞɚ
(ɡɚɞɚɱɚ ɇɟɣɦɚɧɚ);
-ɩɪɨɢɫɯɨɞɢɬ ɬɟɩɥɨɨɛɦɟɧ ɫ ɜɧɟɲɧɟɣ ɫɪɟɞɨɣ, ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɨɣ ɢɡɜɟɫɬɧɚ (ɩɨɬɨɤ, ɜɬɟɤɚɸɳɢɣ ɜ ɨɛɥɚɫɬɶ (ɢɥɢ ɜɵɬɟɤɚɸɳɢɣ ɢɡ ɨɛɥɚɫɬɢ)
ɱɟɪɟɡ ɝɪɚɧɢɰɭ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɪɚɡɧɨɫɬɢ ɦɟɠɞɭ ɬɟɦɩɟɪɚɬɭɪɨɣ ɜɧɭɬɪɢ ɬɟɥɚ ɢ ɧɟɤɨɬɨɪɨɣ ɡɚɞɚɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ). ɗɬɨ Ƚɍ ɬɪɟɬɶɟɝɨ ɪɨɞɚ (ɡɚɞɚɱɚ Ɋɨɛɟɧɚ).
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɡɚɞɚɸɬ ɧɚɱɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜɧɭɬɪɢ ɬɟɥɚ.
2.2. ɆȿɌɈȾ ɊȺɁȾȿɅȿɇɂə ɉȿɊȿɆȿɇɇɕɏ
ɐɟɥɶ ɞɚɧɧɨɝɨ ɢɡɥɨɠɟɧɢɹ ɨɡɧɚɤɨɦɢɬɶ ɫ ɦɨɳɧɵɦ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɢ ɩɨɤɚɡɚɬɶ, ɤɚɤ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɷɬɢɦ ɦɟɬɨɞɨɦ ɞɥɹ ɪɟɲɟɧɢɹ ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɨɣ ɞɢɮɮɭɡɢɨɧɧɨɣ ɡɚɞɚɱɢ.
Ɉɫɧɨɜɧɚɹ ɢɞɟɹ ɦɟɬɨɞɚ ɫɨɫɬɨɢɬ ɜ ɪɚɡɥɨɠɟɧɢɢ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ ɜ ɪɹɞ Ɏɭɪɶɟ, ɧɚɯɨɠɞɟɧɢɢ ɨɬɤɥɢɤɚ ɫɢɫɬɟɦɵ ɧɚ ɤɚɠɞɵɣ ɱɥɟɧ ɷɬɨɝɨ ɪɹɞɚ ɢ ɩɨɫɥɟɞɭɸɳɟɝɨ ɫɭɦɦɢɪɨɜɚɧɢɹ ɜɫɟɯ ɨɬɤɥɢɤɨɜ. Ɍɚɤ ɦɨɠɧɨ ɧɚɣɬɢ ɨɬɤɥɢɤ ɧɚ ɩɪɨɢɡɜɨɥɶɧɨɟɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ.
Ɇɟɬɨɞ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ – ɨɞɢɧ ɢɡ ɧɚɢɛɨɥɟɟ ɩɨɱɬɟɧɧɵɯ ɩɨ ɜɨɡɪɚɫɬɭ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɫɦɟɲɚɧɧɵɯ ɡɚɞɚɱ ɢ ɩɪɢɦɟɧɹɟɬɫɹ, ɤɨɝɞɚ:
1.ɍɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɢ ɨɞɧɨɪɨɞɧɵɦ (ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ).
2.Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɡɚɞɚɧɵ ɜ ɜɢɞɟ:
Dux(0,t) Eu(0,t) |
0 |
Jux(1,t) Gu(1,t) |
0 , |
ɝɞɟ D , E, J ɢ G – ɤɨɧɫɬɚɧɬɵ (ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɡɚɞɚɧɧɵɟ ɜ ɬɚɤɨɦ
17
ɜɢɞɟ, ɧɚɡɵɜɚɸɬɫɹ ɥɢɧɟɣɧɵɦɢɨɞɧɨɪɨɞɧɵɦɢɝɪɚɧɢɱɧɵɦɢɭɫɥɨɜɢɹɦɢ). Ɇɟɬɨɞ ɛɵɥ ɫɨɡɞɚɧ ɜɨ ɜɪɟɦɟɧɚ Ɏɭɪɶɟ (ɨɛɵɱɧɨ ɨɧ ɧɚɡɵɜɚɟɬɫɹ ɦɟɬɨɞɨɦ Ɏɭɪɶɟ) ɢ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɟɟ ɩɨɩɭɥɹɪɧɵɦ
(ɜ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɨɧ ɩɪɢɦɟɧɢɦ).
ȼɦɟɫɬɨ ɢɡɭɱɟɧɢɹ ɦɟɬɨɞɚ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɚɡɛɟɪɟɦ ɫɧɚɱɚɥɚ ɱɚɫɬɧɭɸ ɡɚɞɚɱɭ (ɩɨɡɠɟ ɨɛɫɭɞɢɦ ɢ ɨɛɳɢɣ ɫɥɭɱɚɣ). Ɋɚɫɫɦɨɬɪɢɦ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɬɢɩɚ: ɧɚɣɬɢ ɪɟɲɟɧɢɟ Ⱦɍɑɉ
ut D2uxx , 0< x <1, 0< t < f ,
ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ (Ƚɍ) |
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^uu(1,(0,tt)) |
00 , 0< t < f , |
ɢ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ (ɇɍ)
u(x,0) M(x) , 0 d x d1.
Рɢɫ. 4. ɋɯɟɦɚɬɢɱɟɫɤɨɟ ɢɡɨɛɪɚɠɟɧɢɟ ɞɢɮɮɭɡɢɨɧɧɨɣ ɡɚɞɚɱɢ
Ɏɢɡɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɬɚɤɨɜɚ. ɂɦɟɟɬɫɹ ɫɬɟɪɠɟɧɶ ɤɨɧɟɱɧɨɣ ɞɥɢɧɵ, ɤɨɧɰɵ ɤɨɬɨɪɨɝɨ ɩɨɞɞɟɪɠɢɜɚɸɬɫɹ ɩɪɢ ɩɨɫɬɨɹɧɧɨɣ, ɪɚɜɧɨɣ ɧɭɥɸ ɬɟɦɩɟɪɚɬɭɪɟ (ɧɚ ɫɚɦɨɦ ɞɟɥɟ ɤɨɧɰɵ ɦɨɝɭɬ ɩɨɞɞɟɪɠɢɜɚɬɶɫɹ ɩɪɢ ɝɨɪɚɡɞɨ ɛɨɥɟɟ ɜɵɫɨɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ, ɡɧɚɱɟɧɢɟ ɤɨɬɨɪɨɣ ɩɪɢɧɢɦɚɟɬɫɹ ɡɚ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ). Ⱦɨɩɨɥɧɢɬɟɥɶɧɵɟ ɞɚɧɧɵɟ ɨ ɡɚɞɚɱɟ ɩɪɟɞɫɬɚɜɥɟɧɵ ɜ ɜɢɞɟ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ. ɇɚɲɚ ɰɟɥɶ – ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ u(x,t) ɩɨ ɜɫɟɦɭ ɫɬɟɪɠɧɸ ɜ ɩɨɫɥɟɞɭɸ-
ɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ.
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Ɉɛɳɢɟ ɩɪɢɧɰɢɩɵ ɦɟɬɨɞɚ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ
Ⱦɥɹ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɪɚɡɞɟɥɟɧɢɟ ɩɟɪɟɦɟɧɧɵɯ
– ɷɬɨ ɩɨɢɫɤ ɪɟɲɟɧɢɣ ɜɢɞɚ u(x,t) X (x)T (t) , ɝɞɟ X (x) – ɮɭɧɤɰɢɹ, ɡɚɜɢɫɹɳɚɹ ɬɨɥɶɤɨ ɨɬ ɩɟɪɟɦɟɧɧɨɣ ɯ, ɚ T (t) - ɡɚɜɢɫɹɳɚɹ ɬɨɥɶɤɨ ɨɬ t .
Ɍɚɤɨɟ ɪɟɲɟɧɢɟ ɹɜɥɹɟɬɫɹ ɜ ɤɚɤɨɦ-ɬɨ ɫɦɵɫɥɟ ɷɥɟɦɟɧɬɚɪɧɵɦ, ɩɨɫɤɨɥɶɤɭ ɬɟɦɩɟɪɚɬɭɪɚ u(x,t) , ɩɪɟɞɫɬɚɜɥɟɧɧɚɹ ɜ ɬɚɤɨɦ ɜɢɞɟ, ɛɭɞɟɬ ɫɨɯɪɚɧɹɬɶ
«ɮɨɪɦɭ» ɩɪɨɮɢɥɹ ɜ ɪɚɡɥɢɱɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ (ɫɦ. Ɋɢɫ. 5).
Ɉɛɳɚɹ ɢɞɟɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨɛɵ ɧɚɣɬɢ ɛɟɫɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɬɚɤɢɯ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ (ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ). ɗɬɢ ɮɭɧɤɰɢɢ un(x,t) X n(x)Tn(t)
(ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɟ peɲɟɧɢɹ) ɹɜɥɹɸɬɫɹ ɷɥɟɦɟɧɬɚɪɧɵɦɢ ɤɢɪɩɢɱɢɤɚɦɢ, ɢɡ ɤɨɬɨɪɵɯ ɫɬɪɨɢɬɫɹ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ. ɗɬɨ ɪɟɲɟɧɢɟ u(x,t) ɧɚ-
ɯɨɞɢɬɫɹ ɜ ɜɢɞɟ ɬɚɤɨɣ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɪɟ-
f
ɲɟɧɢɣ X n(x)Tn(t) , ɱɬɨ ɪɟɡɭɥɶɬɢɪɭɸɳɚɹ ɫɭɦɦɚ ¦ An X n(x)Tn(t) (ɪɹɞ
n 1
Ɏɭɪɶɟ ɩɨ ɞɜɭɦ ɩɟɪɟɦɟɧɧɵɦ) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ, ɧɚɱɚɥɶɧɵɦ ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ, ɢ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ. ɉɪɨɞɟɥɚɟɦ ɜɫɟ ɜɵɤɥɚɞɤɢ ɩɨɞɪɨɛɧɨ.
Рɢɫ. 5. Ƚɪɚɮɢɤ ɮɭɧɤɰɢɣ X(ɯ) Ɍ(t) ɜ ɪɚɡɥɢɱɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ
Ɋɚɡɞɟɥɟɧɢɟ ɩɟɪɟɦɟɧɧɵɯ
ɒȺȽ 1. (ɇɚɯɨɠɞɟɧɢɟ ɷɥɟɦɟɧɬɚɪɧɵɯ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ.) Ɇɵ ɯɨɬɢɦ ɧɚɣɬɢ ɮɭɧɤɰɢɸ u(x,t) ɤɨɬɨɪɚɹ ɹɜɥɹ-
ɟɬɫɹ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ:
(ɍɑɉ) ut D2uxx , 0< x <1, 0< t < f , |
|
(Ƚɍ) ^uu(1,(0,tt)) |
00 , 0< t < f , |
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