численные методыА.Б. САМОХИН, В.В. ЧЕРДЫНЦЕВ, А.А. ВОРОНЦОВ
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ɇɚ ɪɢɫ ɩɨɤɚɡɚɧ ɩɨɥɭɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɪɚɫɫɱɢ ɬɚɧɧɵɣ ɩɨ ɮɨɪɦɭɥɚɦ ɩɪɢ ɫɥɟɞɭɸɳɢɯ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ ɝɪɚɧɢɱɧɵɯ ɢ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ X 1, T 1, c 1, L R 0 , f (x) sin(Sx) , g(x) 0 ɒɚɝɢ ɫɟɬɤɢ ɩɨ ɩɪɨɫɬɪɚɧɫɬ ɜɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ ɢ ɩɨ ɜɪɟɦɟɧɢ 'x 0.05, 't 0.05.
ɑɢɫɥɨ ɲɚɝɨɜ ɩɨ x ɢ ɩɨ t ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ N 20, M 20.
7.3. ɉɚɪɚɛɨɥɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ
Ⱦɚɧɧɵɣ ɬɢɩ ɭɪɚɜɧɟɧɢɣ ɪɚɫɫɦɨɬɪɢɦ ɧɚ ɩɪɢɦɟɪɟ ɨɞɧɨɦɟɪɧɨɝɨ ɧɟɫɬɚɰɢɨɧɚɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɝɪɚɧɢɱ ɧɵɦɢ ɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɨɩɢɫɵɜɚɸɳɟɝɨ ɩɪɨɰɟɫɫ ɭɫɬɚɧɨɜɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɢɡɨɥɢɪɨɜɚɧɧɨɦ ɫɬɟɪɠɧɟ ɢɦɟɸɳɟɦ ɧɚ ɤɨɧɰɚɯ ɩɨɫɬɨɹɧɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ L ɢ R ɢ ɡɚɞɚɧɧɨɟ ɧɚɱɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜɞɨɥɶ ɫɬɟɪɠɧɹ f (x) :
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(x,t) c2u |
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(x,t) , 0 x X , 0 d t T ; |
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f (x), |
x [0, X ]. |
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Ⱦɥɹ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɭɪɚɜɧɟɧɢɹ ɢɫɩɨɥɶɡɭɟɦ ɤɨɧɟɱɧɵɟ ɪɚɡɧɨɫɬɢ ɢ
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Ɉɛɨɡɧɚɱɢɦ q |
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ɉɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɚɟɦ ɹɜ |
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ɧɭɸ ɱɟɬɵɪɟɯɬɨɱɟɱɧɭɸ ɫɟɬɨɱɧɭɸ ɫɯɟɦɭ ɜ ɤɨɬɨɪɨɣ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɧɚ (i 1)-ɦ ɫɥɨɟ ɩɨ ɜɪɟɦɟɧɢ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɬɪɢ ɫɨ ɫɟɞɧɢɯ ɡɧɚɱɟɧɢɹ ɧɚ ɧɢɠɧɟɦ i -ɦ ɫɥɨɟ
ui 1, j (1 2q)ui, j q(ui, j 1 ui, j 1) . |
(7.3.4) |
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Ɏɨɪɦɭɥɚ ɩɨɡɜɨɥɹɟɬ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɧɚɣɬɢ ɜɫɟ ɡɧɚɱɟ ɧɢɹ ɫɟɬɨɱɧɨɣ ɮɭɧɤɰɢɢ ɧɚɱɢɧɚɹ ɫɨ ɫɥɨɹ i 0 ɧɚ ɤɨɬɨɪɨɦ ɡɚɞɚɧɵ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ Ɉɞɧɚɤɨ ɜɵɱɢɫɥɟɧɢɹ ɩɨ ɷɬɨɣ ɮɨɪɦɭɥɟ ɭɫɬɨɣɱɢɜɵ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ ɟɫɥɢ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ 0 d q d 0.5 ɗɬɨ ɧɚɤɥɚɞɵɜɚɟɬ ɠɟɫɬɤɢɟ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɲɚɝ ɫɟɬɤɢ ɩɨ ɜɪɟɦɟɧɢ ɨɛɹɡɵɜɚɹ ɜɵɛɢɪɚɬɶ ɷɬɨɬ ɲɚɝ ɧɚɦɧɨɝɨ ɦɟɧɶɲɢɦ ɱɟɦ ɲɚɝ ɩɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ ɱɬɨ ɫɭɳɟɫɬɜɟɧɧɨ ɭɜɟɥɢɱɢ ɜɚɟɬ ɜɪɟɦɹ ɪɚɫɱɟɬɚ ɢ ɨɝɪɚɧɢɱɢɜɚɟɬ ɩɪɢɦɟɧɢɦɨɫɬɶ ɹɜɧɨɣ ɫɯɟɦɵ
Ⱦɥɹ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɭɪɚɜɧɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨ ɜɚɧɚ ɥɟɜɚɹ ɤɨɧɟɱɧɚɹ ɪɚɡɧɨɫɬɶ :
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ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɧɟɹɜɧɨɣ ɱɟɬɵɪɺɯɬɨɱɟɱɧɨɣ ɪɚɡɧɨɫɬɧɨɣ ɫɯɟɦɟ : |
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ɤɨɬɨɪɚɹ ɭɫɬɨɣɱɢɜɚ ɩɪɢ ɥɸɛɵɯ ɫɨɨɬɧɨɲɟɧɢɹɯ ɲɚɝɨɜ ɫɟɬɤɢ ɂɡ ɫɥɟɞɭɟɬ ɱɬɨ ɞɥɹ ɤɚɠɞɨɝɨ ɫɥɨɹ i ɩɨ ɜɪɟɦɟɧɢ ɡɧɚɱɟ
ɧɢɹ ɧɟɢɡɜɟɫɬɧɨɣ ɫɟɬɨɱɧɨɣ ɮɭɧɤɰɢɢ ui, j , j 1, 2,..., N 1 ɫɜɹɡɚɧɵ ɋɅȺɍ ɫ ɬɪɟɯɞɢɚɝɨɧɚɥɶɧɨɣ ɦɚɬɪɢɰɟɣ ȼ ɷɬɨɣ ɦɚɬɪɢɰɟ ɧɚ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɧɚɯɨɞɢɬɫɹ ɡɧɚɱɟɧɢɟ (2 1q ) ɚ ɧɚ ɞɜɭɯ ɫɨɫɟɞɧɢɯ ɞɢɚ
ɝɨɧɚɥɹɯ -1 Ɂɧɚɱɟɧɢɟ ɧɚ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɛɥɢɡɤɨ ɤ 2 ɬ ɤ ɡɧɚ ɱɟɧɢɟ q ɤɚɤ ɩɪɚɜɢɥɨ !!1 ȼɟɤɬɨɪ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɡɧɚɱɟɧɢɢ i const ɢɡɜɟɫɬɟɧ ɢɡ ɜɵɱɢɫɥɟɧɢɣ ɧɚ ɩɪɟ ɞɵɞɭɳɟɦ ɲɚɝɟ ɩɨ ɜɪɟɦɟɧɢ ɢ ɜɯɨɞɢɬ ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ ɋɅȺɍ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɪɟɲɚɹ ɋɅȺɍ ɧɚɱɢɧɚɹ ɫɨ ɫɥɨɹ i 1, ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɫɟɬɨɱɧɭɸ ɮɭɧɤɰɢɸ ɜɨ ɜɫɟɣ ɨɛɥɚɫɬɢ ɪɟɲɟɧɢɹ ɋɢɫɬɟɦɚ ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɚ ɤɚɤ ɫɬɚɧɞɚɪɬɧɵɦ ɦɟɬɨɞɨɦ
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ɬɚɤ ɤɚɤ ɩɨɪɹɞɨɤ ɫɢɫɬɟɦɵ ɧɟ ɫɥɢɲɤɨɦ ɜɟɥɢɤ: N 1 ɬɚɤ ɢ ɫɩɟɰɢ |
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ɚɥɶɧɵɦɢ ɦɟɬɨɞɚɦɢ ɩɪɢɦɟɧɹɟɦɵɦɢ ɞɥɹ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦ ɫ ɬɪɟɯ |
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ɞɢɚɝɨɧɚɥɶɧɵɦɢ ɦɚɬɪɢɰɚɦɢ ɧɚɩɪɢɦɟɪ ɦɟɬɨɞɨɦ ɩɪɨɝɨɧɤɢ > @ |
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Ɋɢɫ 7.2 |
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ɇɚ ɪɢɫ 7. ɩɪɟɞɫɬɚɜɥɟɧ ɪɚɫɱɟɬ ɭɫɬɚɧɨɜɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɬɟɪɠɧɟ ɩɪɨɜɟɞɟɧɧɵɣ ɩɨ ɧɟɹɜɧɨɣ ɫɯɟɦɟ ɩɪɢ ɫɥɟɞɭɸɳɢɯ
ɧɚɱɚɥɶɧɵɯ ɢ ɝɪɚɧɢɱɧɵɯ |
ɭɫɥɨɜɢɹɯ X |
1, T 1; u(0,t) |
0 , |
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u( X ,t) 0 , t [0,T ]; u(x,0) |
f (x) |
sin(Sx) , |
x [0, X ]. ɒɚɝɢ ɫɟɬ |
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ɤɢ ɩɨ ɜɪɟɦɟɧɢ ɢ ɩɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ 't |
0.1, |
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'x 0.01 ɉɪɢ ɞɚɧɧɨɦ ɡɧɚɱɟɧɢɢ q |
1000 ɪɚɫɱɟɬɵ ɩɨ ɹɜɧɨɣ ɫɯɟɦɟ |
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ɛɵɥɢ ɛɵ ɧɟɜɨɡɦɨɠɧɵ ɢɡ-ɡɚ ɛɨɥɶɲɨɣ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ ɑɢɫ ɥɨ ɲɚɝɨɜ ɩɨ t ɢ ɩɨ x ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ M = 10, N = 100.
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7.4. ɍɪɚɜɧɟɧɢɹ ɷɥɥɢɩɬɢɱɟɫɤɨɝɨ ɬɢɩɚ
Ⱦɜɭɦɟɪɧɵɟ ɤɪɚɟɜɵɟ ɡɚɞɚɱɢ ɞɥɹ ɭɪɚɜɧɟɧɢɣ ɞɚɧɧɨɝɨ ɬɢɩɚ ɪɚɫɫɦɨɬɪɢɦ ɧɚ ɩɪɢɦɟɪɟ ɭɪɚɜɧɟɧɢɣ Ʌɚɩɥɚɫɚ ɉɭɚɫɫɨɧɚ ɢ Ƚɟɥɶɦ ɝɨɥɶɰɚ
Ɉɛɨɡɧɚɱɢɦ ɤɚɤ ɨɛɵɱɧɨ ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ
'u uxx uyy , x, y S .
Ɍɨɝɞɚ ɭɤɚɡɚɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢɦɟɸɬ ɜɢɞ |
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ɭɪɚɜɧɟɧɢɟ Ʌɚɩɥɚɫɚ: |
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ɭɪɚɜɧɟɧɢɟ ɉɭɚɫɫɨɧɚ: |
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ɭɪɚɜɧɟɧɢɟ Ƚɟɥɶɦɝɨɥɶɰɚ: 'u g(x, y)u f (x, y) . |
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Ƚɪɚɧɢɱɧɵɟ |
ɭɫɥɨɜɢɹ |
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ɧɚ ɝɪɚɧɢɰɟ ɨɛɥɚɫɬɢ S : |
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ɱɚɫɬɧɨɫɬɢ ɧɚ |
ɝɪɚɧɢɰɟ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ |
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x [0, X ], y [o,Y ]: |
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u0 (x) , u(x,Y ) uY (x) , u(0, y) u0 ( y), |
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u( X , y) uX ( y) . |
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Ɋɚɡɧɨɫɬɧɚɹ ɫɯɟɦɚ ɭɪɚɜɧɟɧɢɣ
Ɋɚɡɧɨɫɬɧɭɸ ɫɯɟɦɭ ɪɚɫɫɦɨɬɪɢɦ ɧɚ ɩɪɢɦɟɪɟ ɭɪɚɜɧɟɧɢɹ ɉɭɚɫ ɫɨɧɚ ɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ ɢɫɩɨɥɶɡɭɹ ɞɥɹ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɤɨɧɟɱɧɵɟ ɪɚɡɧɨɫɬɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɬɨɱɧɨɫɬɢ
ȼɜɨɞɹ ɫɟɬɤɭ yi |
i 'y, |
x j |
j 'x, 0 d i d M ,0 d j d N ɩɨ |
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ɥɭɱɚɟɦ |
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ɢɥɢ ɜɜɟɞɹ ɨɛɨɡɧɚɱɟɧɢɟ q |
('x)2 / ('y)2 ɩɨɥɭɱɚɟɦ ɩɹɬɢɬɨɱɟɱ |
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ɧɭɸ ɪɚɡɧɨɫɬɧɭɸ ɫɯɟɦɭ ɞɥɹ ɜɧɭɬɪɟɧɧɢɯ ɭɡɥɨɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ:
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Ⱦɚɧɧɚɹ ɧɟɹɜɧɚɹ ɫɯɟɦɚ ɨɯɜɚɬɵɜɚɟɬ ɜɫɟ ɜɧɭɬɪɟɧɧɢɟ ɬɨɱɤɢ ɨɛ ɥɚɫɬɢ 1 d i d M 1,1 d j d N 1 ɢɯ ɤɨɥɢɱɟɫɬɜɨ n (M 1)(N 1) .
75
Ɍɚɤɨɜɨ ɠɟ ɱɢɫɥɨ ɭɪɚɜɧɟɧɢɣ ɢ ɧɟɢɡɜɟɫɬɧɵɯ ɜ ɋɅȺɍ ɩɨɫɬɪɨɟɧɧɨɣ ɧɚ ɨɫɧɨɜɟ
ɉɭɫɬɶ ɞɥɹ ɩɪɨɫɬɨɬɵ f (x, y) 1 ɞɥɹ |
ɜɫɟɯ |
x, y S ɬ ɟ |
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fi, j { 1 ɞɥɹ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɬɨɱɟɤ |
1 d i d M 1 , |
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ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɬɚɤɨɜɵ ɜɧɢɡɭ |
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0 ɫɥɟɜɚ ɢ ɫɩɪɚɜɚ |
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u(0, y) 0 , u( X , y) 0 ɢ ɬɨɥɶɤɨ ɧɚɜɟɪɯɭ ɡɚɞɚɧɚ ɨɬɥɢɱɧɚɹ ɨɬ ɧɭɥɹ ɮɭɧɤɰɢɹ u(x,Y ) sin(Sx) Ɂɚɞɚɞɢɦ 'x 'y 0.2 , X Y 1 Ɍɨ ɝɞɚ M 5, N 5 ɚ ɱɢɫɥɨ ɜɧɭɬɪɟɧɧɢɯ ɬɨɱɟɤ ɢ ɭɪɚɜɧɟɧɢɣ n 16 . Ɇɚɬɪɢɰɚ ɋɅȺɍ ɞɥɹ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɡɚɞɚɟɬɫɹ ɩɨ ɫɥɟɞɭɸɳɟɦɭ ɡɚ ɤɨɧɭ ɧɚ ɹɡɵɤɟ ɩɚɤɟɬɚ Mathcad):
A : matrix(n, n, a)
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Ɂɞɟɫɶ 0 d i d n 1, 0 d j d n 1 – ɷɬɨ ɢɧɞɟɤɫɵ ɦɚɬɪɢɰɵ A ɨɧɢ ɫɜɹɡɚɧɵ ɫ ɞɪɭɝɢɦɢ ɪɚɧɟɟ ɜɜɟɞɟɧɧɵɦɢ ɢɧɞɟɤɫɚɦɢ i, j ɞɥɹ ɭɡɥɨɜ ɫɟɬɤɢ ɋɟɬɨɱɧɚɹ ɮɭɧɤɰɢɹ ui, j , 1 d i d M 1, 1 d j d N 1 ɜɵɪɚɠɚ
ɟɬɫɹ ɱɟɪɟɡ ɧɚɣɞɟɧɧɵɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ ɋɅȺɍ ɜɟɤɬɨɪ ɪɟɲɟ
ɧɢɹ v |
, 1 d l d n 1 ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ui, j |
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ɉɹɬɢɞɢɚɝɨɧɚɥɶɧɚɹ ɦɚɬɪɢɰɚ A ɢɦɟɟɬ ɫɥɟɞɭɸɳɟɟ ɫɬɪɨɟɧɢɟ
76
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b ɡɚɞɚɟɬɫɹ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ |
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ȼɟɤɬɨɪ ɩɪɚɜɨɣ ɱɚɫɬɢ ɋɅȺɍ Av |
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ɩɨ ɡɚɤɨɧɭ |
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ɝɞɟ IK |
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(M 2)N M 2 – ɱɢɫɥɨ ɨɩɪɟɞɟɥɹɸɳɟɟ ɜ ɢɧɞɟɤɫɚɯ ɜɟɤ |
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ɬɨɪɚ ɪɟɲɟɧɢɹ ɧɚɱɚɥɨ ɩɨɫɥɟɞɧɟɝɨ ɫɥɨɹ ɜɧɭɬɪɟɧɧɢɯ ɭɡɥɨɜ ɩɨ ɨɫɢ y , ɧɚ ɤɨɬɨɪɵɯ ɭɱɢɬɵɜɚɟɬɫɹ ɡɚɞɚɧɧɨɟ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ Ɂɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɢɡ ɭɪɚɜɧɟɧɢɹ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɹɜɥɹɸɳɚɹɫɹ ɤɨɧɫɬɚɧɬɨɣ f (x, y) { 1 ɜɯɨɞɢɬ ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ ɋɅȺɍ ɜ ɜɢɞɟ ɫɥɚɝɚɟɦɨɝɨ
1 ('x)2 Ɂɧɚɱɟɧɢɹ ɜɟɤɬɨɪɚ ɩɪɚɜɨɣ ɱɚɫɬɢ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ
bT= |
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–0.04 |
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–0.04 |
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0.628 |
0.991 |
0.991 |
0.628 |
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77
Ɋɢɫ 7.3
ɇɚ ɪɢɫ ɩɨɤɚɡɚɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɮɭɧɤɰɢɢ ɪɟɲɟɧɢɹ ɚɧɚɥɨ ɝɢɱɧɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɜ ɞɜɭɦɟɪɧɨɣ ɨɛɥɚɫɬɢ ɩɪɢ ɩɨɪɹɞɤɟ ɋɅȺɍ n 361 Ɋɟɲɟɧɢɟ ɩɨɥɭɱɟɧɨ ɤɨɦɛɢɧɢɪɨɜɚɧɧɵɦ ɦɟɬɨɞɨɦ Ɂɟɣɞɟɥɹ-
Ɉɋɉ ɩɪɢ ɨɩɬɢɦɚɥɶɧɨɦ ɩɚɪɚɦɟɬɪɟ k0 |
0.425 ɡɚ m 41 ɢɬɟɪɚɰɢɣ |
ɫ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɬɨɱɧɨɫɬɶɸ ɪɟɲɟɧɢɹ ɜ H 10 5 Ɉɛɵɱɧɵɣ ɦɟɬɨɞ |
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Ɂɟɣɞɟɥɹ ɫɯɨɞɢɬɫɹ ɡɞɟɫɶ ɥɢɲɶ ɡɚ m |
305 ɢɬɟɪɚɰɢɣ ɢ ɫɨɩɨɫɬɚɜɢɦ |
ɩɨ ɜɪɟɦɟɧɢ ɪɟɲɟɧɢɹ ɫ ɩɪɹɦɵɦ ɦɟɬɨɞɨɦ ȿɳɟ ɛɨɥɶɲɟɟ ɱɢɫɥɨ ɬɪɟ ɛɭɟɦɵɯ ɢɬɟɪɚɰɢɣ ɩɨɤɚɡɵɜɚɸɬ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɦɟɬɨɞ Ɉɋɉ ɫ ɦɚɬ ɪɢɰɟɣ - m 546 .
Ɉɬɦɟɬɢɦ ɱɬɨ ɦɚɬɪɢɰɚ ɡɚɞɚɱɢ ɩɪɢ 'x 'y ɢ ɡɚɞɚɧɧɨɦ n ɩɨ ɫɬɨɹɧɧɚ ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ ɢ ɢɫɬɨɱɧɢɤɨɜ f (x, y) , ɤɨɬɨɪɵɟ ɜɯɨɞɹɬ ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ ɋɅȺɍ ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɡɚɞɚɱɢ ɫ ɪɚɡɥɢɱɧɵɦɢ ɤɪɚɟɜɵɦɢ ɭɫɥɨɜɢɹɦɢ ɢ ɢɫɬɨɱɧɢɤɚɦɢ ɦɨɝɭɬ ɪɟɲɚɬɶɫɹ ɫ ɬɟɦ ɠɟ ɫɚɦɵɦ ɨɩɬɢɦɚɥɶɧɵɦ ɩɚɪɚɦɟɬɪɨɦ ɧɚɣɞɟɧɧɵɦ ɨɞɢɧ ɪɚɡ ɞɥɹ ɞɚɧɧɨɣ ɫɟɬɤɢ
7.5. Ʌɚɛɨɪɚɬɨɪɧɵɟ ɡɚɞɚɧɢɹ ɤ ɬɟɦɟ ©ɑɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯª
Ʌɚɛɨɪɚɬɨɪɧɵɟ ɪɚɛɨɬɵ ɩɨ ɬɟɦɟ ɦɨɝɭɬ ɛɵɬɶ ɜɵɩɨɥɧɟɧɵ ɫ ɩɨ ɦɨɳɶɸ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɩɚɤɟɬɨɜ ɩɪɨɝɪɚɦɦ Mathcad ɢɥɢ Matlab. ȼ ɪɟɡɭɥɶɬɚɬɟ ɪɚɛɨɬɵ ɞɨɥɠɧɚ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɢɫɤɨɦɚɹ ɫɟɬɨɱɧɚɹ ɮɭɧɤɰɢɹ ɜ ɜɢɞɟ ɦɚɬɪɢɰɵ ɡɧɚɱɟɧɢɣ ɜ ɭɡɥɚɯ ɫɟɬɤɢ ɥɢɛɨ ɜ ɜɢɞɟ ɩɨ
78
ɫɥɨɣɧɨɝɨ ɩɨ ɜɪɟɦɟɧɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɡɧɚɱɟɧɢɣ ɫɟɬɨɱɧɵɯ ɜɟɤɬɨɪɨɜ ɋɥɟɞɭɟɬ ɬɚɤɠɟ ɩɪɢɜɟɫɬɢ ɝɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɪɟɡɭɥɶɬɚɬɨɜ
Ƚɢɩɟɪɛɨɥɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ
ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɣ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɝɪɚɧɢɱɧɵɦɢ ɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɫɦ 7.2).
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ɉɚɪɚɛɨɥɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ
ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɣ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞ ɧɨɫɬɢ ɫ ɝɪɚɧɢɱɧɵɦɢ ɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɫɦ
79
ʋ |
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T |
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f (x) |
g(x) |
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Ɇɟɬɨɞ |
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2.2 |
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2.3 |
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2.4 |
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sin(Sx) sin(2Sx) |
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2.7 |
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2.8 |
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2.9 |
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x x3 |
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2.10 |
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2.11 |
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0.01.. |
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2.12 |
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Ɇɟɬɨɞɵ - ɹɜɧɵɣ - ɧɟɹɜɧɵɣ ɫɜɟɞɟɧɢɟɦ ɤ ɋɅȺɍ ɢ ɩɨɫɥɟ ɞɭɸɳɢɦ ɪɟɲɟɧɢɟɦ ɫɬɚɧɞɚɪɬɧɵɦ ɦɟɬɨɞɨɦ ɋɪɚɜɧɢɬɶ ɫɨ ɫɬɪɨɝɢɦ ɪɟɲɟɧɢɟɦ
80
ɗɥɥɢɩɬɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ
Ɋɟɲɢɬɶ ɡɚɞɚɧɧɭɸ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ ɦɟɬɨɞɨɦ ɫɟɬɨɤ ɫɜɟɞɟɧɢɟɦ ɟɺ ɤ ɋɅȺɍ ɢ ɩɨɫɥɟɞɭɸɳɢɦ ɪɟɲɟɧɢɟɦ ɩɪɹɦɵɦ ɫɬɚɧɞɚɪɬɧɵɦ ɢ ɢɬɟɪɚɰɢɨɧɧɵɦ Ɂɟɣɞɟɥɹ–Ɉɋɉ ɦɟɬɨɞɚɦɢ ɋɪɚɜɧɢɬɶ ɫ ɫɭɳɟɫɬ ɜɭɸɳɢɦ ɫɬɪɨɝɢɦ ɪɟɲɟɧɢɟɦ.
ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɣ ɞɥɹ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɫ ɭɪɚɜɧɟɧɢɹɦɢ ɷɥɥɢɩ ɬɢɱɟɫɤɨɝɨ ɬɢɩɚ ɫɦ
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ʋɩɩ |
ɍɪɚɜɧɟɧɢɟ |
X |
Y |
u0 ( x) |
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uY ( x) |
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u0 ( y) |
u X ( y) |
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f ( x, y) |
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3.1 |
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sin(Sy) |
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3.2 |
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3.4 |
3 |
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uȽ |
cos(2x) sin(2 y) |
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3.6 |
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120 |
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3.8 |
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1 |
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3.10 |
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0.1 |
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ɍɪɚɜɧɟɧɢɹ – Ʌɚɩɥɚɫɚ – ɉɭɚɫɫɨɧɚ – Ƚɟɥɶɦɝɨɥɶɰɚ.