ɎȿȾȿɊȺɅɖɇɈȿ ȺȽȿɇɌɋɌȼɈ ɉɈ ɈȻɊȺɁɈȼȺɇɂɘ ȽɈɋɍȾȺɊɋɌȼȿɇɇɈȿ ɈȻɊȺɁɈȼȺɌȿɅɖɇɈȿ ɍɑɊȿɀȾȿɇɂȿ ȼɕɋɒȿȽɈ ɉɊɈɎȿɋɋɂɈɇȺɅɖɇɈȽɈ ɈȻɊȺɁɈȼȺɇɂə «ȼɈɊɈɇȿɀɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɍɇɂȼȿɊɋɂɌȿɌ»
ɍɊȺȼɇȿɇɂə ɉȺɊȺȻɈɅɂɑȿɋɄɈȽɈ ɌɂɉȺ ɑɚɫɬɶ 3 ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɜɭɡɨɜ ɋɨɫɬɚɜɢɬɟɥɶ Ɉ.ɉ. Ɇɚɥɸɬɢɧɚ
ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ 2007
ɍɬɜɟɪɠɞɟɧɨ ɧɚɭɱɧɨ-ɦɟɬɨɞɢɱɟɫɤɢɦ ɫɨɜɟɬɨɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ 28 ɮɟɜɪɚɥɹ 2006 ɝ., ɩɪɨɬɨɤɨɥ ʋ 6
Ɋɟɰɟɧɡɟɧɬ Ɇ.ɂ. Ɂɚɣɰɟɜɚ
ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɩɨɞɝɨɬɨɜɥɟɧɨ ɧɚ ɤɚɮɟɞɪɟ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɢ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɧɢɜɟɪɫɢɬɟɬɚ. Ɋɟɤɨɦɟɧɞɭɟɬɫɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 4 ɤɭɪɫɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ ɨɱɧɨɡɚɨɱɧɨɣ ɮɨɪɦɵ ɨɛɭɱɟɧɢɹ, ɨɛɭɱɚɸɳɢɯɫɹ ɩɨ ɫɩɟɰɢɚɥɶɧɨɫɬɢ 010101 (010100) – Ɇɚɬɟɦɚɬɢɤɚ .
2
I. ȼɵɜɨɞ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ u x, y , z , t ɬɟɦɩɟɪɚɬɭɪɭ ɫɪɟɞɵ ɜ ɬɨɱɤɟ x, y , z ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t . ɉɨɞɫɱɢɬɚɟɦ ɛɚɥɚɧɫ ɬɟɩɥɚ ɜ ɩɪɨɢɡɜɨɥɶɧɨɦ ɨɛɴɟɦɟ V ɡɚ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ t1 , t2 . Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ S ɝɪɚɧɢɰɭ V , n – ɧɨɪɦɚɥɶ ɤ ɩɨɜɟɪɯɧɨɫɬɢ S ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɞɜɢɠɟɧɢɹ ɬɟɩɥɚ. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɪɟɞɚ ɢɡɨɬɪɨɩɧɚ, ɬ. ɟ. ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ ɜɧɭɬɪɟɧɧɟɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ k ɡɚɜɢɫɢɬ ɨɬ ɬɨɱɤɢ x, y , z ɬɟɥɚ ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ ɧɨɪɦɚɥɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ S ɜ ɷɬɨɣ ɬɨɱɤɟ. ɋɨɝɥɚɫɧɨ ɡɚɤɨɧɭ Ɏɭɪɶɟ, ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ S ɜ ɨɛɴɟɦɟ V ɩɨɫɬɭɩɚɟɬ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ: t2 § wu · Q1 ³ ³³ ¨ k ¸dsdt . (1) wn ¹ © t1 S ȼɵɪɚɠɟɧɢɟ, ɤɨɬɨɪɨɟ ɫɬɨɢɬ ɜ ɮɨɪɦɭɥɟ (1) ɜ ɤɪɭɝɥɵɯ ɫɤɨɛɤɚɯ, ɧɚɡɵɜɚɸɬ ɬɟɩɥɨɜɵɦ ɩɨɬɨɤɨɦ – ɷɬɨ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɩɪɨɯɨɞɹɳɟɝɨ ɱɟɪɟɡ ɟɞɢɧɢɰɭ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɢ ɨɛɨɡɧɚɱɚɸɬ wu q k . (2) wn ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɜɧɭɬɪɢ ɢɦɟɸɬɫɹ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ F x, y , z , t ɩɥɨɬɧɨɫɬɶ (ɤɨɥɢɱɟɫɬɜɨ ɩɨɝɥɨɳɺɧɧɨɝɨ ɢɥɢ ɜɵɞɟɥɹɟɦɨɝɨ ɬɟɩɥɚ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɟɞɢɧɢɰɟɣ ɨɛɴɺɦɚ) ɬɟɩɥɨɜɵɯ ɢɫɬɨɱɧɢɤɨɜ. Ɂɚ ɫɱɺɬ ɬɟɩɥɨɜɵɯ ɢɫɬɨɱɧɢɤɨɜ ɜ ɨɛɴɺɦɟ V ɜɨɡɧɢɤɚɟɬ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ: t1
Q2
³ ³ ³ ³ F ( x, y, z, t )dVdt .
t2
(3)
V
Ɍɚɤ ɤɚɤ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɨɛɴɺɦɟ V ɜɵɪɨɫɥɚ ɧɚ ɜɟɥɢɱɢɧɭ u x, y , z , t2 u x, y , z, t1 , ɬɨ ɞɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ ɡɚɬɪɚɬɢɬɶ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ Q3 ³ ³ ³ ª¬u x, y , z , t2 u x, y , z , t1 º¼ c U dV ,
(4)
V
ɝɞɟ U ( x, y , z ) - ɩɥɨɬɧɨɫɬɶ; c x, y , z – ɭɞɟɥɶɧɚɹ ɬɟɩɥɨɺɦɤɨɫɬɶ. ɋɨɫɬɚɜɢɦ ɬɟɩɟɪɶ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɬɟɩɥɚ ɞɥɹ ɜɵɞɟɥɟɧɧɨɝɨ ɨɛɴɺɦɚ V . Ɉɱɟɜɢɞɧɨ, ɱɬɨ Q3 Q1 Q2 , ɬ. ɟ. t2
t2
³ dt ³ ³ ³ cU dV t1
V
t
2 wu · § k x y z dsdt , , ¸ ³t ³S ³ ¨© ³t ³ V³ ³ F x, y, z, t dVdt , (5) wn ¹ 1 1
ɬɚɤ ɤɚɤ
3
t2
u x, y , z, t2 u x, y, z, t1
wu
³ wt dt . t1
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ ɜɨ ɜɬɨɪɨɦ ɢɧɬɟɝɪɚɥɟ, ɢɦɟɟɦ t2 wu ³t dt ³ V³ ³ [ c U w n div( kgradu ) F ( x , y , z , t )]dV 0 . (6) 1 ȼ ɫɢɥɭ ɬɨɝɨ, ɱɬɨ ɩɨɞɵɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ ɧɟɩɪɟɪɵɜɧɚ, ɩɪɨɢɡɜɨɥɶɧɨɫɬɢ ɨɛɴɺɦɚ V ɢ ɩɪɨɦɟɠɭɬɤɚ ɜɪɟɦɟɧɢ t1 , t2 , ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ:
cU
wu wn
div(k grad u ) F ( x, y, z, t ) .
(7)
ȿɫɥɢ ɫɪɟɞɚ ɨɞɧɨɪɨɞɧɚ, ɬ. ɟ. c, U , k – ɩɨɫɬɨɹɧɧɵɟ, ɬɨ ɭɪɚɜɧɟɧɢɟ (7) ɩɪɢɦɟɬ ɜɢɞ: wu wn
a2 (
w 2 u w 2u w 2u ) f , wx 2 wy 2 wz 2
(8)
k F , f . cg cg ɍɪɚɜɧɟɧɢɟ (8) ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. ɑɢɫɥɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɜ ɷɬɨɦ ɭɪɚɜɧɟɧɢɢ ɦɨɠɟɬ ɛɵɬɶ ɥɸɛɵɦ. ȿɫɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ x, y , t (ɜ ɬɨɧɤɨɣ ɩɥɚɫɬɢɧɟ), ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢɦɟɟɬ ɜɢɞ: ª w 2u w 2u º wu a 2 « 2 2 » f t , x, y . wt wy ¼ ¬ wx ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɬɟɪɠɧɹ: ɝɞɟ a 2
wu wt
a2
w 2u f (t , x, y ) . wx 2
Ʉɚɤ ɜ ɫɥɭɱɚɟ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ, ɞɥɹ ɩɨɥɧɨɝɨ ɨɩɢɫɚɧɢɹ ɩɪɨɰɟɫɫɚ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɧɚɱɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɪɟɞɟ (ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ) ɢ ɪɟɠɢɦ ɧɚ ɝɪɚɧɢɰɟ ɫɪɟɞɵ (ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ). ɉɪɢɦɟɪɵ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ
1. ȿɫɥɢ ɧɚ ɝɪɚɧɢɰɟ S ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɡɚɞɚɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ u , ɬɨ u S u0 . (9) 2. ȿɫɥɢ ɧɚ S ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɡɚɞɚɧɧɵɣ ɩɨɬɨɤ ɬɟɩɥɚ u1 , ɬɨ 4
wu (I0) u1 . wn S 3. ȿɫɥɢ ɧɚ S ɩɪɨɢɫɯɨɞɢɬ ɬɟɩɥɨɨɛɦɟɧ ɫɨɝɥɚɫɧɨ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ, ɬɨ wu k H u u0 S u0 , (11) wn ɝɞɟ H – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɛɦɟɧɚ ɢ u0 – ɬɟɦɩɟɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ. k
II. ɉɟɪɜɚɹ ɝɪɚɧɢɱɧɚɹ ɡɚɞɚɱɚ. Ɍɟɨɪɟɦɚ ɨ ɦɚɤɫɢɦɭɦɟ ɢ ɦɢɧɢɦɭɦɟ
Ɋɚɫɫɦɨɬɪɢɦ ɨɝɪɚɧɢɱɟɧɧɵɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɵ l . Ⱦɥɹ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ P c d x l , 0 d t d T ɩɟɪɜɚɹ ɤɪɚɟɜɚɹ ɡɚɞɚɱɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ: ɧɚɣɬɢ ɧɟɩɪɟɪɵɜɧɭɸ ɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ P ɮɭɧɤɰɢɸ u x, t , ɞɜɚɠɞɵ ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɭɸ ɩɨ x ɢ ɨɞɢɧ ɪɚɡ ɩɨ t ɜ P , ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɜ P ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ: wu w 2u a 2 2 f x, t , (1) wt wx ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ u t 0 D x , 0 d x d l (2) ɢ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɟɦ u x 0 E t , u x l J t , 0 d t d T . (3) ɉɪɢ ɷɬɨɦ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɮɭɧɤɰɢɢ f x, t , D x , E t , J t ɧɟɩɪɟɪɵɜɧɵ ɢ D 0 E 0 , a l J l . (4) Ɍɟɨɪɟɦɚ. Ɏɭɧɤɰɢɹ u x, t , ɧɟɩɪɟɪɵɜɧɚɹ ɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ P ɢ
wu w 2u a2 2 wt wx ɜɧɭɬɪɢ P , ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɢ ɧɚɢɛɨɥɶɲɟɟ ɡɧɚɱɟɧɢɟ ɧɚ Ƚ: ɛɨɤɨɜɵɯ ɫɬɨɪɨɧɚɯ ɢ ɧɢɠɧɟɦ ɨɫɧɨɜɚɧɢɢ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ P , ɬ. ɟ. ɥɢɛɨ ɩɪɢ x 0 , ɥɢɛɨ ɩɪɢ x l , ɥɢɛɨ ɩɪɢ t 0 , ɥɢɛɨ ɹɜɥɹɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ. Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ M ɧɚɢɛɨɥɶɲɟɟ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ u x, t ɜ P , ɚ m – ɧɚɢɛɨɥɶɲɟɟ ɧɚ Ƚ. ɉɪɟɞɩɨɥɨɠɢɦ ɩɪɨɬɢɜɧɨɟ, ɬ. ɟ. ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ ɪɟɲɟɧɢɟ u x, t , ɞɥɹ ɤɨɬɨɪɨɝɨ M m . ɉɭɫɬɶ ɷɬɚ ɮɭɧɤɰɢɹ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɟ M ɜ ɬɨɱɤɟ x0 , y0 M u x0 , y0 . ȼɜɟɞɟɦ ɮɭɧɤɰɢɸ ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ ɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
5
v x, t
u x, t
M m 2 x x0 , 2 6d
(5)
ɝɞɟ d – ɞɢɚɦɟɬɪ ɨɛɥɚɫɬɢ P , M m 2 5m M vȽ d m d M , v x0 , t0 M . 6d 2 6 6 ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɮɭɧɤɰɢɹ v x, t ɬɚɤ ɠɟ, ɤɚɤ ɢ u x, t ɧɟ ɩɪɢɧɢɦɚɟɬ ɧɚɢɛɨɥɶɲɟɟ ɡɧɚɱɟɧɢɟ ɧɚ Ƚ. ɉɭɫɬɶ v x, t ɩɪɢɧɢɦɚɟɬ ɧɚɢɛɨɥɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜ ɬɨɱɤɟ x, t , ɬɨɝɞɚ ɜ ɷɬɨɣ ɬɨɱɤɟ: w2 v wv d 0, t 0, wx 2 wt ɨɬɤɭɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɜ ɬɨɱɤɟ x, t ɞɨɥɠɧɨ ɛɵɬɶ: wv w2 v a2 2 t 0 . wt wx
(6)
ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, wv w 2 v wu M m M m a2 2 a2 a 2 0 , 2 wt wx wt 6d d2 ɱɬɨ ɩɪɨɬɢɜɨɪɟɱɢɬ (6); ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɬɟɨɪɟɦɚ ɨ ɦɚɤɫɢɦɭɦɟ ɢ ɦɢɧɢɦɭɦɟ ɞɨɤɚɡɚɧɚ. Ɍɚɤ ɤɚɤ ɬɟɨɪɟɦɚ ɨ ɦɢɧɢɦɭɦɟ ɫɜɨɞɢɬɫɹ ɤ ɬɟɨɪɟɦɟ ɨ ɦɚɤɫɢɦɭɦɟ ɩɟɪɟɦɟɧɧɨɣ ɡɧɚɤɚ u x, y , ɬɨ ɬɟɨɪɟɦɚ ɩɨɥɧɨɫɬɶɸ ɞɨɤɚɡɚɧɚ. I. ɋɥɟɞɫɬɜɢɟ. Ɋɟɲɟɧɢɟ ɩɟɪɜɨɣ ɡɚɞɚɱɢ (1)–(3) ɟɞɢɧɫɬɜɟɧɧɨ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɦɵ ɢɦɟɟɦ ɞɜɚ ɪɟɲɟɧɢɹ u1 , u2 ɧɚɲɟɣ ɡɚɞɚɱɢ, ɬɨɝɞɚ u
u1 u2 ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
ɢ ɧɭɥɟɜɵɦ ɭɫɥɨɜɢɹɦ
u
t 0
wu w2 u a2 2 wt wx 0, u 0, x 0
u
x l
0. ɇɨ ɬɨɝɞɚ, ɜ ɫɢɥɭ
ɬɟɨɪɟɦɵ ɨ ɦɚɤɫɢɦɭɦɟ ɢ ɦɢɧɢɦɭɦɟ, ɫɥɟɞɭɟɬ, ɱɬɨ u 0 , ɬ. ɟ. u1 u2 . 2. ɋɥɟɞɫɬɜɢɟ. Ɋɟɲɟɧɢɟ ɩɟɪɜɨɣ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ (1)–(3) ɧɟɩɪɟɪɵɜɧɨ ɡɚɜɢɫɢɬ ɨɬ ɧɚɱɚɥɶɧɵɯ ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɟɫɥɢ ɪɚɡɧɨɫɬɶ ɮɭɧɤɰɢɣ, ɜɯɨɞɹɳɢɯ ɜ ɧɚɱɚɥɶɧɨɟ ɢ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɹ, ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ ɧɟɤɨɬɨɪɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɱɢɫɥɚ H , ɬɨ ɪɚɡɧɨɫɬɶ u u1 u2 ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɪɟɲɟɧɢɣ ɤɚɤ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɦɚɥɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɢ ɝɪɚɧɢɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɬɚɤɠɟ ɧɟ ɛɭɞɟɬ ɩɪɟɜɨɫɯɨɞɢɬɶ H ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ.
6
III. Ɋɟɲɟɧɢɟ ɩɟɪɜɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Ⱦɥɹ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ Q : >0 d x d l , 0 d t d T @ ɩɟɪɜɭɸ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ ɦɨɠɧɨ
ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ
ɬɚɤ:
ɧɚɣɬɢ
u x, t C Q C x2,1,t Q
ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɜ Q ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ: wu w 2u a 2 2 f x, t , wt wx ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ u t 0 M x , 0 d x d l ɢ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ u x 0 P1 t , u x l P2 t , 0 d t d T .
ɢ
(1) (2) (3)
ɉɪɢ ɷɬɨɦ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɮɭɧɤɰɢɢ f x, t , M x , P1 x , P2 x ɧɟɩɪɟɪɵɜɧɵ ɢ M 0 P1 , M l P2 0 . ɂɡɭɱɟɧɢɟ ɨɛɳɟɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ (1)–(3) ɧɚɱɧɟɦ ɫ ɪɟɲɟɧɢɹ ɩɪɨɫɬɟɣɲɟɣ ɡɚɞɚɱɢ I / : ɧɚɣɬɢ ɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ wu w 2u a2 2 , (4) wt wx ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ u t 0 M x , 0 d x d l (5) ɢ ɨɞɧɨɪɨɞɧɵɦ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ u x 0 0 , u x l 0 , 0 d t d T , (6) ɝɞɟ M x ɢɦɟɟɬ ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɭɸ ɩɟɪɜɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɢ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɩɪɢ x 0 ɢ x l . Ⱦɨɤɚɠɟɦ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɪɟɲɟɧɢɹ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ I / ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ Q ɦɟɬɨɞɨɦ Ɏɭɪɶɟ wu w 2u Ȼɭɞɟɦ ɢɫɤɚɬɶ ɱɚɫɬɧɵɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ a 2 2 ɜ ɜɢɞɟ wt wx u x, t T t X x . (7) ɉɨɞɫɬɚɜɥɹɹ (7) ɜ (4), ɢɦɟɟɦ X x T / t a 2T t X // x ɢɥɢ, T // t X // x O , a 2T t X x ɨɬɤɭɞɚ ɩɨɥɭɱɚɟɦ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɜ ɫɢɥɭ ɬɨɝɨ, ɱɬɨ ɥɟɜɨɟ ɜɵɪɚɠɟɧɢɟ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ t , ɚ ɩɪɚɜɨɟ ɬɨɥɶɤɨ ɨɬ x 7
T / t a 2 OT t
0,
X x O X x 0 . (8) ɑɬɨɛɵ ɩɨɥɭɱɢɬɶ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ u x, t ɜɢɞɚ (7), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ (6), ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (8), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ X 0 0 , X l 0. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɢ X ( x ) ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɡɚɞɚɱɟ ɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɹɯ: X // ( x ) O X ( x ) , X 0 0 , X l 0 , (9) ɢɫɫɥɟɞɨɜɚɧɧɵɯ ɜ ɡɚɞɚɱɟ ɨ ɤɨɥɟɛɚɧɢɢ ɨɝɪɚɧɢɱɟɧɧɨɣ ɨɞɧɨɪɨɞɧɨɣ ɫɬɪɭɧɵ, ɝɞɟ ɛɵɥɨ ɩɨɤɚɡɚɧɨ, ɱɬɨ ɬɨɥɶɤɨ ɞɥɹ ɡɧɚɱɟɧɢɣ ɩɚɪɚɦɟɬɪɚ O , ɪɚɜɧɵɯ 2 § nS · (10) On ¨ ¸ , ɝɞɟ n 1, 2 ,3,... , © l ¹ ɫɭɳɟɫɬɜɭɸɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (9): nS x X n ( x ) sin . (11) l Ɂɧɚɱɟɧɢɹɦ ɩɚɪɚɦɟɬɪɚ O On ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (7): //
2
Tn (t ) ɝɞɟ an – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ. ɂɬɚɤ, ɜɫɟ ɮɭɧɤɰɢɢ
an e
§ nS a · ¨ ¸ t © l ¹
,
(12)
2
§ nS a · ¨ ¸ t © l ¹
nS x l ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɪɚɜɧɟɧɢɸ (4) ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ (6). ɋɨɫɬɚɜɢɦ ɪɹɞ un x, t Tn (t ) X n ( x )
an e
sin
(13)
2
f
§ nS a · ¨ ¸ t © l ¹
nS x . (14) l n 1 Ɍɪɟɛɭɹ ɜɵɩɨɥɧɟɧɢɹ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ (5), ɩɨɥɭɱɚɟɦ f nS x . ( 15) u x,0 M ( x ) ¦ an sin l n 1 ɇɚɩɢɫɚɧɧɵɣ ɪɹɞ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɚɡɥɨɠɟɧɢɟ ɡɚɞɚɧɧɨɣ ɮɭɧɤɰɢɢ M x ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɧɭɫɚɦ ɜ ɩɪɨɦɟɠɭɬɤɟ 0 ,1 . Ʉɨɷɮɮɢɰɢɟɧɬɵ an ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɢɡɜɟɫɬɧɨɣ ɮɨɪɦɭɥɟ 1 2 nS x an M x sin dx . (16) ³ l 0 l u x, t
¦a e n
sin
Ɍɚɤ ɤɚɤ ɦɵ ɩɪɟɞɩɨɥɨɠɢɥɢ, ɱɬɨ ɮɭɧɤɰɢɹ M x ɧɟɩɪɟɪɵɜɧɚ, ɢɦɟɟɬ ɤɭɫɨɱɧɨɧɟɩɪɟɪɵɜɧɭɸ ɩɟɪɜɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɢ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɩɪɢ x 0 ɢ x l , 8
ɬɨ ɪɹɞ (15) ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ an , ɨɩɪɟɞɟɥɹɟɦɵɦɢ ɩɨ ɮɨɪɦɭɥɚɦ (16), ɪɚɜɧɨɦɟɪɧɨ ɢ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɢɬɫɹ ɤ M x , ɱɬɨ ɢɡɜɟɫɬɧɨ ɢɡ ɬɟɨɪɢɢ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ ɪɹɞɨɜ. Ɍɚɤ ɤɚɤ ɩɪɢ t t 0 2
§ nS a · ¨ ¸ t © l ¹
0 e d 1, ɬɨ ɪɹɞ (14) ɩɪɢ t t 0 ɬɚɤɠɟ ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ. ɉɨɷɬɨɦɭ ɮɭɧɤɰɢɹ u x, t , ɨɩɪɟɞɟɥɹɟɦɚɹ ɪɹɞɨɦ (14), ɧɟɩɪɟɪɵɜɧɚ ɜ ɨɛɥɚɫɬɢ 0 d x d l , t t 0 ɢ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɧɚɱɚɥɶɧɨɦɭ ɢ ɝɪɚɧɢɱɧɨɦɭ ɭɫɥɨɜɢɹɦ. Ɉɫɬɚɟɬɫɹ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɹ u x, t ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ (4) ɜ ɨɛɥɚɫɬɢ 0 d x d l , t t 0 . Ⱦɥɹ ɷɬɨɝɨ ɞɨɫɬɚɬɨɱɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɪɹɞɵ, ɩɨɥɭɱɟɧɧɵɟ ɢɡ (14) ɩɨɱɥɟɧɧɵɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ ɩɨ t ɨɞɢɧ ɪɚɡ ɢ ɩɨɱɥɟɧɧɵɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ ɩɨ x ɞɜɚ ɪɚɡɚ, ɬɚɤɠɟ ɚɛɫɨɥɸɬɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɫɯɨɞɹɬɫɹ ɜ ɨɛɥɚɫɬɢ 0 d x d l , t t 0 . Ⱥ ɷɬɨ ɩɨɫɥɟɞɧɟɟ ɭɬɜɟɪɠɞɟɧɢɟ ɫɥɟɞɭɟɬ ɢɡ ɬɨɝɨ, ɱɬɨ ɩɪɢ ɥɸɛɨɦ t t t0 ! 0 § nS a ·
2
§ nS a ·
2
n 2S 2 a 2 ¨© l ¸¹ t n 2S 2 ¨© l ¸¹ t e e 0 1 , 0 1, l2 l2 ɟɫɥɢ n ɞɨɫɬɚɬɨɱɧɨ ɜɟɥɢɤɨ. ɋɨɜɟɪɲɟɧɧɨ ɬɚɤ ɠɟ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɭ ɮɭɧɤɰɢɢ u x, t ɧɟɩɪɟɪɵɜɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɥɸɛɨɝɨ ɩɨɪɹɞɤɚ ɩɨ x ɢ t ɜ ɨɛɥɚɫɬɢ 0 d x d l , t t t0 ! 0 . ȿɞɢɧɫɬɜɟɧɧɨɫɬɶ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ I / ɢ ɧɟɩɪɟɪɵɜɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɧɚɱɚɥɶɧɨɣ ɮɭɧɤɰɢɢ M x ɛɵɥɚ ɭɫɬɚɧɨɜɥɟɧɚ ɤɚɤ ɫɥɟɞɫɬɜɢɟ ɬɟɨɪɟɦɵ ɨ
ɦɚɤɫɢɦɭɦɟ ɢ ɦɢɧɢɦɭɦɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɞɚɱɚ I / ɩɨɫɬɚɜɥɟɧɚ ɤɨɪɪɟɤɬɧɨ ɞɥɹ t t t0 ! 0 . Ɂɚɞɚɱɚ I // . ɭɪɚɜɧɟɧɢɹ
ɇɚɣɬɢ ɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ Q ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ
wu w 2u a 2 2 f x, t , wt wx ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ u t 0 0; 0 d x d l ɢ ɨɞɧɨɪɨɞɧɵɦ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ u x 0 0; u x l 0; 0 d t d T .
(17) (18) (19)
ɉɪɢ ɷɬɨɦ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ f x, t ɢɦɟɟɬ ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɩɨ x ɢ ɱɬɨ ɩɪɢ ɜɫɟɯ t ! 0 ɜɵɩɨɥɧɹɸɬɫɹ ɭɫɥɨɜɢɹ f 0, t f l , t 0 . Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ u x, t ɡɚɞɚɱɢ I // ɜ ɜɢɞɟ ɪɹɞɚ Ɏɭɪɶɟ 9
f
nS x (20) l ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ ɡɚɞɚɱɢ (9). Ɋɚɡɥɚɝɚɹ ɮɭɧɤɰɢɸ f x, t ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɧɭɫɚɦ, ɛɭɞɟɦ ɢɦɟɬɶ f nS x f x, t ¦ f n (t ) sin , (21) l n 1 ɝɞɟ l 2 nS x f n (t ) f x, t sin dx . (22) ³ l 0 l ɉɨɞɫɬɚɜɥɹɹ ɪɹɞ (20) ɜ ɭɪɚɜɧɟɧɢɟ (17), ɩɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ (21), ɩɨɥɭɱɚɟɦ 2 f ª º nS x § anS · / T t ( ) 0. « ¦ ¨ ¸ Tn (t ) f n (t ) » sin n l © l ¹ n 1¬ ¼ Ɉɬɫɸɞɚ 2 § nS a · Tn/ (t ) ¨ (23) ¸ Tn (t ) f n (t ) , n 1, 2 ,3,... . © l ¹ ɉɨɥɶɡɭɹɫɶ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ ɞɥɹ u x, t u x, t
¦ T (t ) sin n
n 1
u x,0
f
¦ T (0) sin n
n 1
nS x l
0,
ɩɨɥɭɱɚɟɦ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɞɥɹ Tn (t ) : Tn (0) 0 , n 1, 2 ,3, ... . Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (23) ɩɪɢ ɧɚɱɚɥɶɧɨɦ ɭɫɥɨɜɢɢ (24) ɢɦɟɟɬ ɜɢɞ 2
t
Tn t
³e
§ nS a · ¨ ¸ t W © l ¹
f n W dW .
(25)
0
ɉɨɞɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɟ (25) ɞɥɹ Tn (t ) ɜ ɪɹɞ (20), ɩɨɥɭɱɚɟɦ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ I // ɜ ɜɢɞɟ 2 f ª 1 § nS a · t W º ¨ ¸ nS x © l ¹ u x, t ¦ « ³ e f n W dW » sin . l »¼ n 1 «0 ¬ Ɂɚɦɟɱɚɧɢɟ. ȿɫɥɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɧɟɨɞɧɨɪɨɞɧɵ, ɬɨ ɤ ɪɟɲɟɧɢɸ (26) ɫɥɟɞɭɟɬ ɩɪɢɛɚɜɢɬɶ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɡɚɞɚɧɧɵɦ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ u x,0 M x ɢ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ
ɭɫɥɨɜɢɹɦɢ u 0, t . ȼɟɪɧɟɦɫɹ ɬɟɩɟɪɶ ɤ ɨɛɳɟɣ ɩɟɪɜɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɟ (1)–(3). ȼɜɟɞɟɦ ɧɨɜɭɸ ɧɟɢɡɜɟɫɬɧɭɸ ɮɭɧɤɰɢɸ X x, t , ɩɨɥɨɠɢɜ u x, t X x, t Z x, t , 10
ɝɞɟ x l Ɏɭɧɤɰɢɹ X x, t ɛɭɞɟɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɤɚɤ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ
Z x, t P1 (t ) ª¬ P2 t P1 t º¼ . wX wt
w 2X f x, t , wx 2
a2
(27)
ɝɞɟ f x, t
f x, t
wZ x, t wt
ɫ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ
X x,0 M x Z x,0
(28)
ɢ ɤɪɚɟɜɵɦɢ ɭɫɥɨɜɢɹɦɢ
X 0, t u 0, t Z 0, t 0 , X l, t u l , t Z l, t 0 .
(29) Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (1)–(3) ɫɜɟɞɟɧɨ ɤ ɪɟɲɟɧɢɸ (27)–(29), ɤɨɬɨɪɚɹ ɧɚɦɢ ɪɟɲɟɧɚ ɜɵɲɟ. IV. ɉɪɢɦɟɪ ɪɟɲɟɧɢɹ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ ɬɢɩɚ
Ɋɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ wu w 2 u wt wx 2 ɩɪɢ ɧɚɱɚɥɶɧɨɦ ɭɫɥɨɜɢɢ
t x 1 , 0 x 1, t ! 0 ut
0
0
(1) (2)
ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹɯ ux
0
t2 , u x
2
l
t2 .
Ɋɟɲɟɧɢɟ. Ɏɭɧɤɰɢɹ Z xt ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ wZ w 2Z ɭɪɚɜɧɟɧɢɸ 2xt ɢ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ Z t 0 0 . w t wx 2 ɉɨɷɬɨɦɭ ɮɭɧɤɰɢɹ X u xt 2 ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ wX w 2X 1 x t wt wx 2 ɢ ɭɫɥɨɜɢɹɦ wX 0, Xxl 0. Xt 0 0, wx x 0 11
(3) (3),
(4) (5)
(6)
ɉɪɢɦɟɧɹɹ ɦɟɬɨɞ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɞɥɹ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɝɨ wX w 2X ɭɪɚɜɧɟɧɢɹ 0 ɩɪɢ ɭɫɥɨɜɢɹɯ (6), ɩɨɥɨɠɢɦ X X x T t , wt w x 2 ɩɨɥɭɱɢɦ ɡɚɞɚɱɭ ɒɬɭɪɦɚ–Ʌɢɭɜɢɥɥɹ X // ( x ) O 2 X ( x ) 0 , X / (0) 0 , X (l ) 0 , ɫɨɛɫɬɜɟɧɧɵɦɢ n
ɡɧɚɱɟɧɢɹɦɢ
ɤɨɬɨɪɨɣ
ɹɜɥɹɸɬɫɹ
ɱɢɫɥɚ
0 ,1, 2 , ... , ɚ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ – ɮɭɧɤɰɢɢ X n ( x ) cos On x . Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (5), (6) ɢɳɟɦ ɜ ɨɛɳɟɦ ɜɢɞɟ
On
S 2
Sn
, (7)
f
X x, t
¦ T (t ) cos O x . n
(8)
n
n 0
ɉɨɞɫɬɚɜɥɹɹ X x, t ɢɡ (8) ɜ ɭɪɚɜɧɟɧɢɟ (5), ɩɨɥɭɱɚɟɦ f
¦ T t O T t cos O x 1 x t . / n
2 n n
n
(9)
n 0
Ɋɚɡɥɨɠɢɦ ɮɭɧɤɰɢɸ 1 x ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ ɮɭɧɤɰɢɣ (7) ɧɚ ɢɧɬɟɪɜɚɥɟ 0 ,1 : f
1 x
¦a
n
cos On x .
(10)
n 0
Ɍɚɤ ɤɚɤ 1
an
2
2 ³ 1 x cos On xdx
On2
0
,
ɬɨ ɢɡ (9) ɢ (10) ɧɚɯɨɞɢɦ 2t
Tn/ (t ) On2T (t )
On2
.
(11)
Ɋɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (11) ɩɪɢ ɭɫɥɨɜɢɢ Tn (0) 0 ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
2
2 O n 6 e O n t O n2 t 1 .
Tn ( t )
(12)
ɂɡ (4), (8) ɢ (12) ɧɚɯɨɞɢɦ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (1)–(3): f
u
n 0
ɝɞɟ On
S 2
2
xt 2 2¦ On6 e On t On2t 1 cos On2 x ,
Sn. V. Ɂɚɞɚɱɚ Ʉɨɲɢ
1. ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ Ʉɨɲɢ. ɇɚɣɬɢ ɮɭɧɤɰɢɸ u( x, t ) t ! 0 , f x f , ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ 12
wu wt
a2
w 2u wx 2
(1)
ɢ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ ut
0
M x ,
f x f
,
ɝɞɟ M x - ɧɟɩɪɟɪɵɜɧɚɹ ɢ ɨɝɪɚɧɢɱɟɧɧɚɹ ɮɭɧɤɰɢɹ. 2. ȿɞɢɧɫɬɜɟɧɧɨɫɬɶ ɪɟɲɟɧɢɹ. Ⱦɨɤɚɠɟɦ ɟɞɢɧɫɬɜɟɧɧɨɫɬɶ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ, ɩɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ɪɟɲɟɧɢɟ u( x, t ) ɨɝɪɚɧɢɱɟɧɨ ɜɨ ɜɫɟɣ ɨɛɥɚɫɬɢ, ɬ. ɟ. ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ ɱɢɫɥɨ M , ɱɬɨ u( x, t ) M ɞɥɹ ɜɫɟɯ f x f ɢ ɥɸɛɨɦ t t 0 . ɉɭɫɬɶ u1 ( x, t ) ɢ u2 ( x, t ) – ɞɜɚ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (1), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɨɞɧɨɦɭ ɢ ɬɨɦɭ ɠɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ (2). Ɍɨɝɞɚ ɪɚɡɧɨɫɬɶ Z x, t u1 ( x, t ) u2 ( x, t ) , ɛɭɞɟɬ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɭɪɚɜɧɟɧɢɸ (1) ɢ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ Z t
0
0.
Ʉɪɨɦɟ ɬɨɝɨ M x ɨɝɪɚɧɢɱɟɧɚ ɜɨ ɜɫɟɣ ɨɛɥɚɫɬɢ
Z x, t d u1 x, t u2 x, t d 2 M . Ɍɟɨɪɟɦɭ ɨ ɦɚɤɫɢɦɭɦɟ ɢ ɦɢɧɢɦɭɦɟ ɞɥɹ ɧɟɨɝɪɚɧɢɱɟɧɧɨɣ ɨɛɥɚɫɬɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ, ɬɚɤ ɤɚɤ ɮɭɧɤɰɢɹ Z x, t ɦɨɠɟɬ ɧɢɝɞɟ ɧɟ ɞɨɫɬɢɝɚɬɶ ɧɚɢɛɨɥɶɲɟɝɨ ɢ ɧɚɢɦɟɧɶɲɟɝɨ ɡɧɚɱɟɧɢɣ. ɑɬɨɛɵ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɷɬɨɣ ɬɟɨɪɟɦɨɣ ɪɚɫɫɦɨɬɪɢɦ ɤɨɧɟɱɧɭɸ ɨɛɥɚɫɬɶ x d L , 0dt dT . ( 3) ȼɨɡɶɦɟɦ ɮɭɧɤɰɢɸ · 4M § x2 X x, t a 2t ¸ , 2 ¨ L © 2 ¹ ɤɨɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (1). Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ X x,0 t Z x,0 0 ,
X rL , t t 2M t Z rL , t . ɉɪɢɦɟɧɹɹ ɬɟɨɪɟɦɭ ɨ ɦɚɤɫɢɦɭɦɟ ɢ ɦɢɧɢɦɭɦɟ ɤ ɪɚɡɧɨɫɬɢ ɦɟɠɞɭ ɮɭɧɤɰɢɹɦɢ X x, t ɢ rZ x, t ɜ ɨɛɥɚɫɬɢ (3) , ɛɭɞɟɦ ɢɦɟɬɶ X x, t Z x, t t 0 , X x, t Z x, t t 0 , ɨɬɤɭɞɚ X x, t d Z x, t d X x, t ɢɥɢ · 4M § x2 Z x, t d X x, t a 2t ¸ . 2 ¨ L © 2 ¹ 13
Ɏɢɤɫɢɪɭɹ ɧɟɤɨɬɨɪɨɟ ɡɧɚɱɟɧɢɟ x0 , t0 ɢ ɜɵɛɢɪɚɹ L ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɦ, ɩɨɥɭɱɚɟɦ Z x0 , t0 H , ɨɬɤɭɞɚ ɜɜɢɞɭ ɩɪɨɢɡɜɨɥɶɧɨɫɬɢ H ɢ ɬɨɱɤɢ x0 , t0 ɫɥɟɞɭɟɬ, ɱɬɨ Z x, t { 0 , ɬ. ɟ. u1 ( x, t ) u2 ( x, t ) . 3. ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ. ɇɚɣɞɟɦ ɫɧɚɱɚɥɚ ɱɚɫɬɧɵɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (1) ɜɢɞɚ u x , t T (t ) X ( x ) . (4) ɉɨɞɫɬɚɜɥɹɹ (4) ɜ ɭɪɚɜɧɟɧɢɟ (1) ɢ ɪɚɡɞɟɥɹɹ ɩɟɪɟɦɟɧɧɵɟ, ɩɨɥɭɱɚɟɦ T / (t ) X // ( x ) O 2 , a 2T (t ) X ( x) ɝɞɟ O 2 – ɩɨɫɬɨɹɧɧɚɹ. Ɉɬɫɸɞɚ T / (t ) a 2 O 2T (t ) 0 , X // ( x ) O 2 X ( x ) 0 . ɂɧɬɟɝɪɢɪɭɹ ɷɬɢ ɭɪɚɜɧɟɧɢɹ ɢ ɩɨɥɚɝɚɹ ɩɨɫɬɨɹɧɧɵɣ ɦɧɨɠɢɬɟɥɶ ɜ ɜɵɪɚɠɟɧɢɢ T (t ) ɪɚɜɧɵɦ ɟɞɢɧɢɰɟ, ɩɨɥɭɱɚɟɦ 2 2
T (t ) e a O t , X ( x ) A cos O x B sin O x , ɩɨɫɬɨɹɧɧɵɟ A ɢ B ɦɨɝɭɬ ɡɚɜɢɫɟɬɶ ɨɬ O . Ɍɚɤ ɤɚɤ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɨɬɫɭɬɫɬɜɭɸɬ, ɬɨ ɩɚɪɚɦɟɬɪ O ɨɫɬɚɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɵɦ. ɋɨɝɥɚɫɧɨ (4) ɩɨɥɭɱɢɦ, ɱɬɨ 2 2 uO x, t e a O t ª¬ A O cos O x B O sin O x º¼ (5)
ɟɫɬɶ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) ɩɪɢ ɥɸɛɵɯ A O ɢ B O . ɂɧɬɟɝɪɢɪɭɹ (5) ɩɨ ɩɚɪɚɦɟɬɪɭ O , ɩɨɥɭɱɚɟɦ ɬɚɤɠɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) f
u x, t
³e
a 2 O 2t
ª¬ A O cos O x B O sin O x º¼dx ,
(6)
f
ɟɫɥɢ ɷɬɨɬ ɢɧɬɟɝɪɚɥ ɪɚɜɧɨɦɟɪɧɨ ɫɯɨɞɢɬɫɹ ɢ ɟɝɨ ɦɨɠɧɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɨɞɢɧ ɪɚɡ ɩɨ t ɢ ɞɜɚɠɞɵ ɩɨ x . ȼɵɛɟɪɟɦ ɮɭɧɤɰɢɢ A O ɢ B O ɬɚɤ, ɱɬɨɛɵ ɜɵɩɨɥɧɹɥɨɫɶ ɢ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ (2). ɉɨɥɚɝɚɹ ɜ (6) t 0 , ɩɨɥɭɱɢɦ ɜ ɫɢɥɭ (2) f
M x
³ ª¬ A O cos O x B O sin O x º¼d O
.
(7)
-f
ɋɪɚɜɧɢɜɚɹ ɢɧɬɟɝɪɚɥ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɫ ɢɧɬɟɝɪɚɥɨɦ Ɏɭɪɶɟ ɞɥɹ ɮɭɧɤɰɢɢ
M x , ɢɦɟɟɦ
14
1 2S
M x 1 2S
f
f
f
³ d O ³ M [ cos O [ x d[
f
f
f
ª
f
º
³ «¬cos O x ³ M [ cos O[ d[ sin O x ³ M [ sin O[ d[ »¼d O.
f
f
f
Ɇɵ ɜɢɞɢɦ, ɱɬɨ ɦɨɠɧɨ ɭɞɨɜɥɟɬɜɨɪɢɬɶ ɪɚɜɟɧɫɬɜɭ (7), ɩɨɥɨɠɢɜ f 1 A O ³ M [ cos O[ d [ , 2S f f
1 B O ³ M [ sin O[ d [ . 2S f ɉɨɞɫɬɚɜɥɹɹ (8) ɜ (6), ɩɨɥɭɱɚɟɦ f f 2 2 1 u x, t e a O t d O ³ M [ cos O [ x d [ ³ 2S f f 1
f
e S³
(8)
f a 2 O 2t
d O ³ M [ cos O [ x d [ . f
0
Ɇɟɧɹɹ ɩɨɪɹɞɨɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢ ɩɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɨɣ f
³e
a 2O 2
S
cos EO d O
2a
0
e
E2 4 a2
,
ɥɟɝɤɨ ɧɚɯɨɞɢɦ f
u x, t
³ M H 2a
1
f
St
e
[ x 2 4 a 2t
d[ .
(9)
ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɮɭɧɤɰɢɹ F x, t , [
1
e
[ x 2 4 a 2t
, (10) 2a S t ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɤɚɤ ɮɭɧɤɰɢɹ ɨɬ x, t , ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (1). Ɏɭɧɤɰɢɸ (10) ɧɚɡɵɜɚɸɬ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɦ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (1). Ⱦɨɤɚɠɟɦ, ɱɬɨ ɞɥɹ ɥɸɛɨɣ ɧɟɩɪɟɪɵɜɧɨɣ ɢ ɨɝɪɚɧɢɱɟɧɧɨɣ ɮɭɧɤɰɢɢ M x ɮɭɧɤɰɢɹ (9) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (1). Ⱦɥɹ ɷɬɨɝɨ ɧɚɦ ɞɨɫɬɚɬɨɱɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɢɧɬɟɝɪɚɥ (9), ɚ ɬɚɤɠɟ ɢɧɬɟɝɪɚɥɵ, ɩɨɥɭɱɟɧɧɵɟ ɟɝɨ ɮɨɪɦɚɥɶɧɵɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɩɨ x ɢ t ɫɤɨɥɶɤɨ ɭɝɨɞɧɨ ɪɚɡ, ɪɚɜɧɨɦɟɪɧɨ ɫɯɨɞɹɬɫɹ ɜ ɥɸɛɨɦ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ > 1 d x d 1 , t0 d t d T @ , ɝɞɟ t0 ! 0 . Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɞɢɮɮɟɪɟɧɰɢɪɭɹ (9) ɧɟɫɤɨɥɶɤɨ ɪɚɡ ɩɨ x ɢ t , ɦɵ ɩɨɥɭɱɚɟɦ ɫɭɦɦɭ ɢɧɬɟɝɪɚɥɨɜ, ɢ ɧɭɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɤɚɠɞɵɣ ɢɧɬɟɝɪɚɥ ɪɚɜɧɨɦɟɪɧɨ ɫɯɨɞɢɬɫɹ. 15
ɉɨɫɥɟ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɜɵɞɟɥɹɟɬɫɹ ɦɧɨɠɢɬɟɥɶ [ x ɜ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɫɬɟɩɟɧɢ, ɤɨɬɨɪɵɣ ɨɫɬɚɟɬɫɹ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ, ɢ ɦɧɨɠɢɬɟɥɶ t ɜ ɧɟɤɨɬɨɪɨɣ ɫɬɟɩɟɧɢ, ɤɨɬɨɪɵɣ ɦɨɠɧɨ ɜɵɧɟɫɬɢ ɢɡ ɩɨɞ ɡɧɚɤɚ ɢɧɬɟɝɪɚɥɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɦɵ ɩɨɥɭɱɢɦ ɫɭɦɦɭ ɢɧɬɟɝɪɚɥɨɜ ɜɢɞɚ f
1 m I M [ [ x e k ³ t f ɉɪɨɢɡɜɨɞɹ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ [x D , t ! 0, 2a t ɩɪɟɨɛɪɚɡɭɟɦ ɢɧɬɟɝɪɚɥ (11) ɤ ɜɢɞɭ I
2a
m1
t
f m1 k 2
[ x 2 4 a 2t
³ M x 2aD t D
d[ .
m a2
e
(11)
dD .
f
Ɉɬɫɸɞɚ ɥɟɝɤɨ ɭɜɢɞɟɬɶ, ɱɬɨ ɷɬɨɬ ɢɧɬɟɝɪɚɥ ɪɚɜɧɨɦɟɪɧɨ ɫɯɨɞɢɬɫɹ ɩɪɢ t t t0 ! 0 , ɬɚɤ ɤɚɤ ɩɨɞɵɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ ɦɚɠɨɪɢɪɭɟɬɫɹ ɮɭɧɤɰɢɟɣ 2
m
M D e a ,
ɤɨɬɨɪɚɹ ɢɧɬɟɝɪɢɪɭɟɦɚ ɜ ɩɪɨɦɟɠɭɬɤɟ f, f , ɝɞɟ M x 2aD t M . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɭɧɤɰɢɹ u x, t , ɨɩɪɟɞɟɥɹɟɦɚɹ ɮɨɪɦɭɥɨɣ (9), ɧɟɩɪɟɪɵɜɧɚ ɢ ɢɦɟɟɬ ɩɪɨɢɡɜɨɞɧɵɟ ɥɸɛɨɝɨ ɩɨɪɹɞɤɚ ɩɨ x ɢ t ɩɪɢ t ! 0 . Ɍɚɤ 1
e
[ x 2 2
4a t ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ (1) 2a S t ɩɪɢ t ! 0 , ɬɨ ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɢ ɮɭɧɤɰɢɹ u x, t ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɷɬɨɦɭ ɭɪɚɜɧɟɧɢɸ ɩɪɢ t ! 0 . Ⱦɨɤɚɠɟɦ ɬɟɩɟɪɶ, ɱɬɨ ɮɭɧɤɰɢɹ (9) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ (2), ɬ. ɟ. lim u x, t M x
ɤɚɤ ɩɨɞɵɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ
t o0
ɩɪɢ ɥɸɛɨɦ x ɢɡ ɮɨɪɦɭɥɟ
f, f .
ȼɜɟɞɟɦ ɜɦɟɫɬɨ [ ɧɨɜɭɸ ɩɟɪɟɦɟɧɧɭɸ D ɩɨ
[x , t ! 0. 2a t Ɍɨɝɞɚ ɢɧɬɟɝɪɚɥ (9) ɩɪɢɦɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ f 1 a2 u x, t ³ M x 2aD t e dD . D
S
f
Ɉɬɫɸɞɚ ɥɟɝɤɨ ɜɵɬɟɤɚɟɬ ɨɝɪɚɧɢɱɟɧɧɨɫɬɶ ɪɟɲɟɧɢɹ
(12) u x, t
f x f ɢ t ! 0 , ɟɫɥɢ M x M ɞɥɹ ɜɫɟɯ x . Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ,
16
ɩɪɢ
u x, t d
1
S
f
³ M x 2aD t e
a2
dD d
M
S
f
f
³e
a2
dD
M
f
ɬɚɤ ɤɚɤ f
1
S
³e
a2
dD
1.
(13)
f
ɍɦɧɨɠɚɹ (13) ɧɚ M x ɢ ɜɵɱɢɬɚɹ ɢɡ (12), ɩɨɥɭɱɚɟɦ f
1
u x, t M x
S
³ ª¬M x 2aD t M x º¼ e
a2
dD ,
f
ɨɬɤɭɞɚ u x, t M x d
1
S
f
³ ª¬M x 2aD t M x º¼ e
a2
dD .
(14)
f
ȼ ɫɢɥɭ ɨɝɪɚɧɢɱɟɧɧɨɫɬɢ ɮɭɧɤɰɢɢ M x ɩɪɢ ɥɸɛɵɯ x , t ɢ D ɢɦɟɟɦ
M x 2aD t M x d 2 M .
(15)
ɉɭɫɬɶ H ! 0 – ɫɤɨɥɶ ɭɝɨɞɧɨ ɦɚɥɨɟ ɱɢɫɥɨ. Ɍɚɤ ɤɚɤ ɢɧɬɟɝɪɚɥ (13) ɫɯɨɞɢɬɫɹ, ɬɨ ɦɨɠɧɨ ɮɢɤɫɢɪɨɜɚɬɶ ɫɬɨɥɶ ɛɨɥɶɲɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ N , ɱɬɨ N 2 2M H e a dD d , ³ 3 S f (16) f 2 M a2 H e dD d . 3 S N³ Ɋɚɡɛɢɜɚɹ ɩɪɨɦɟɠɭɬɨɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɧɚ ɬɪɢ f , N , N , N , N , f ɢ ɩɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ (15) ɢ (16) ɢɡ (14) ɛɭɞɟɦ ɢɦɟɬɶ N 2 2H 1 u x, t M x d M x 2aD t M x e a dD . (17) ³ 3 S N
ȼ ɫɢɥɭ ɧɟɩɪɟɪɵɜɧɨɫɬɢ M x ɩɪɢ ɜɫɟɯ t , ɞɨɫɬɚɬɨɱɧɨ ɛɥɢɡɤɢɯ ɤ ɧɭɥɸ, ɢ ɩɪɢ a d N ɢɦɟɟɦ
M x 2aD t M x d
H
. 3 ɂɡ ɩɨɫɥɟɞɧɟɣ ɨɰɟɧɤɢ ɢ ɢɡ ɧɟɪɚɜɟɧɫɬɜɚ (17) ɩɨɥɭɱɚɟɦ ɧɟɪɚɜɟɧɫɬɜɨ N 2 2H H 1 u x, t M x d e a dD . 3 3 S ³N Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɨɰɟɧɤɚ
17
u x, t M x d
2H H 1 3 3 S
f
³e
a2
dD ,
f
ɬ. ɟ., ɜ ɫɢɥɭ (13), ɦɵ ɢɦɟɟɦ u x, t M x d H ɩɪɢ ɜɫɟɯ t , ɞɨɫɬɚɬɨɱɧɨ ɛɥɢɡɤɢɯ ɤ ɧɭɥɸ, ɢ ɩɪɢ ɜɫɟɯ x , ɨɬɤɭɞɚ ɜ ɜɢɞɭ ɩɪɨɢɡɜɨɥɶɧɨɫɬɢ H ! 0 ɢ ɫɥɟɞɭɟɬ lim u x, t M x , t o0
ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ. 4. ɇɟɩɪɟɪɵɜɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ ɨɬ ɧɚɱɚɥɶɧɨɣ ɮɭɧɤɰɢɢ. ɉɭɫɬɶ u x, t – ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ
ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ (2) ɢ u x, t – ɪɟɲɟɧɢɟ ɬɨɝɨ ɠɟ ɭɪɚɜɧɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ u t 0 M x . (18) Ɍɨɝɞɚ,
ɟɫɥɢ
u x, t u x, t H
M x M x H ɩɪɢ ɥɸɛɵɯ
ɞɥɹ
ɜɫɟɯ
ɢɡ
x
f , f ,
ɬɨ
x ɢ t ! 0 . Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɪɟɲɟɧɢɹ
ɭɪɚɜɧɟɧɢɹ (1), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ (2) ɢ (18) , ɜɵɪɚɠɚɸɬɫɹ ɮɨɪɦɭɥɨɣ (9). ȼɡɹɜ ɢɯ ɪɚɡɧɨɫɬɶ, ɛɭɞɟɦ ɢɦɟɬɶ f
u x, t u x, t
1 ³f ª¬M [ M [ º¼ 2a S t e
x [ 2 4 a 2t
d[ ,
ɨɬɤɭɞɚ u x, t u x, t d ɉɨɥɚɝɚɹ D
f
H
³
2a S t
e
x [ 2 4 a 2t
d[ .
f
[x , ɩɨɥɭɱɚɟɦ 2a t u x, t u x, t d H
1
S
f
³e
a2
dD
H ,
f
ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ. ɂɡ ɮɨɪɦɭɥɵ (9) ɫɥɟɞɭɟɬ, ɱɬɨ ɬɟɩɥɨ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɜɞɨɥɶ ɫɬɟɪɠɧɹ ɧɟ ɫ ɤɚɤɨɣ-ɥɢɛɨ ɤɨɧɟɱɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɚ ɦɝɧɨɜɟɧɧɨ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɭɫɬɶ ɧɚɱɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ M x ɩɨɥɨɠɢɬɟɥɶɧɚ ɞɥɹ ɜɫɟɯ D d x d E ɢ ɪɚɜɧɚ ɧɭɥɸ ɜɧɟ ɷɬɨɝɨ ɨɬɪɟɡɤɚ. Ɍɨɝɞɚ ɞɥɹ ɩɨɫɥɟɞɭɸɳɟɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪ ɩɨɥɭɱɚɟɦ E
u x, t
³ M [
D
1 2a S t 18
e
[ x 2 4 a 2t
d[ ,
ɨɬɤɭɞɚ ɹɫɧɨ, ɱɬɨ u x, t ! 0 . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɚɤ ɛɵ ɧɢ ɛɵɥɨ ɦɚɥɨ t ! 0 ɢ ɤɚɤ ɛɵ ɧɢ ɛɵɥɚ ɞɚɥɟɤɚ ɬɨɱɤɚ x ɨɬ ɨɬɪɟɡɤɚ >D , E @ , ɬɟɩɥɨ ɢɡ ɷɬɨɝɨ ɨɬɪɟɡɤɚ ɡɚ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ t ɭɫɩɟɜɚɟɬ ɞɨɣɬɢ ɞɨ ɬɨɱɤɢ x . ɗɬɨ ɢ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɬɟɩɥɨ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɫ ɛɟɫɤɨɧɟɱɧɨɣ ɫɤɨɪɨɫɬɶɸ. Ɉɬɦɟɬɢɦ ɟɳɟ ɨɞɧɨ ɜɚɠɧɨɟ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ. Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (1), (2) (ɡɚɞɚɱɢ Ʉɨɲɢ) ɟɫɬɶ ɮɭɧɤɰɢɹ, ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɫɤɨɥɶ ɭɝɨɞɧɨ ɪɚɡ ɩɨ x ɢ ɩɨ t ɜɧɟ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ, ɛɭɞɟɬ ɥɢ ɢɦɟɬɶ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɹ M x ɢɥɢ ɧɟɬ. ɗɬɚ ɝɥɚɞɤɨɫɬɶ ɪɟɲɟɧɢɣ ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɚɟɬ ɨɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɧɚɩɪɢɦɟɪ, ɨɬ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɹ ɫɬɪɭɧɵ. Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɮɭɧɞɚɦɟɧɬɚɥɶɧɨɝɨ ɪɟɲɟɧɢɹ (10) ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (1). ȼɵɞɟɥɢɦ ɦɚɥɵɣ ɷɥɟɦɟɧɬ ɫɬɟɪɠɧɹ x0 h , x0 h ɨɤɨɥɨ ɬɨɱɤɢ x0 ɢ ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɮɭɧɤɰɢɹ M x , ɞɚɸɳɚɹ ɧɚɱɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɪɚɜɧɚ ɧɭɥɸ ɜɧɟ ɩɪɨɦɟɠɭɬɤɚ x0 h , x0 h ɢ ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ U 0 ɜɧɭɬɪɢ ɧɟɝɨ. Ɏɢɡɢɱɟɫɤɢ ɷɬɨ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɬɚɤ: ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɵ ɫɨɨɛɳɢɥɢ ɷɬɨɦɭ ɷɥɟɦɟɧɬɭ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ Q 2hc UU 0 , ɤɨɬɨɪɨɟ ɜɵɡɜɚɥɨ ɩɨɜɵɲɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ U 0 ɜ ɷɬɨɦ ɭɱɚɫɬɤɟ ɫɬɟɪɠɧɹ. ȼ ɩɨɫɥɟɞɭɸɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɬɟɪɠɧɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ (9), ɤɨɬɨɪɚɹ ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ x0 h
u x, t
³U
x0 h
1 0
2a S t
e
[ x 2 4 a 2t
x h
Q 1 0 e c U 2a S t 2h x0³h
d[
[ x 2 4 a 2t
d[ .
ȿɫɥɢ ɦɵ ɛɭɞɟɦ ɭɦɟɧɶɲɚɬɶ h ɞɨ ɧɭɥɹ, ɬ. ɟ. ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɬɨ ɠɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ Q ɪɚɫɩɪɟɞɟɥɹɟɬɫɹ ɧɚ ɜɫɟ ɦɟɧɶɲɟɦ ɭɱɚɫɬɤɟ ɢ ɜ ɩɪɟɞɟɥɟ ɫɨɨɛɳɚɟɬɫɹ ɫɬɟɪɠɧɸ ɜ ɬɨɱɤɟ x x0 , ɬɨ ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɩɨɧɹɬɢɸ ɦɝɧɨɜɟɧɧɨɝɨ ɬɨɱɟɱɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ ɧɚɩɪɹɠɟɧɢɹ Q , ɩɨɦɟɳɟɧɧɨɝɨ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t 0 ɜ ɬɨɱɤɭ x x0 . Ɍɟɦɩɟɪɚɬɭɪɚ ɜ ɫɬɟɪɠɧɟ ɩɪɢ ɞɟɣɫɬɜɢɢ ɬɚɤɨɝɨ ɦɝɧɨɜɟɧɧɨɝɨ ɬɨɱɟɱɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ ɪɚɫɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ x h
Q 1 0 lim e h o0 c U 2 a S t 2 h ³ x0 h
[ x 2 4 a 2t
d[ .
(19)
ɉɪɢɦɟɧɹɹ ɬɟɨɪɟɦɭ ɨ ɫɪɟɞɧɟɦ, ɛɭɞɟɦ ɢɦɟɬɶ x h
1 0 e 2h x0³h
[ x 2 4 a 2t
d[
e
[ x 2 4 a 2t
,
ɝɞɟ x0 h [0 x0 h , ɢ ɬɚɤ ɤɚɤ [0 o x0 ɩɪɢ h o 0 , ɬɨ ɜɵɪɚɠɟɧɢɟ (19) ɩɪɢɧɢɦɚɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ 19
[ x 2
Q 2 e 4a t . c U 2a S t Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɭɧɞɚɦɟɧɬɚɥɶɧɨɟ ɪɟɲɟɧɢɟ (19) ɞɚɟɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɤɨɬɨɪɨɟ ɜɵɡɵɜɚɟɬɫɹ ɦɝɧɨɜɟɧɧɵɦ ɬɨɱɟɱɧɵɦ ɢɫɬɨɱɧɢɤɨɦ ɬɟɩɥɚ ɧɚɩɪɹɠɟɧɢɹ Q c U , ɩɨɦɟɳɟɧɧɵɦ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ t 0 ɜ ɬɨɱɤɭ x [ ɫɬɟɪɠɧɹ. Ƚɪɚɮɢɤɢ ɮɭɧɞɚɦɟɧɬɚɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
1
F x, t , [
e
[ x 2 4 a 2t
(20) 2a S t ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ [ ɤɚɤ ɮɭɧɤɰɢɹ ɨɬ x ɜ ɨɬɞɟɥɶɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ 0 t1 t2 t3 ... ɩɪɟɞɫɬɚɜɥɟɧɵ ɧɚ ɪɢɫ. 1. ɉɥɨɳɚɞɶ ɤɚɠɞɨɣ ɢɡ ɷɬɢɯ ɤɪɢɜɵɯ ɪɚɜɧɚ f
³ 2a
f
1
St
e
[ x 2 4 a 2t
d[
1
S
f
³e
a2
dD
1.
f
ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɤɨɥɢɱɟɫɬɜɨ ɜ ɫɬɟɪɠɧɟ ɨɫɬɚɟɬɫɹ Q cU ɧɟɢɡɦɟɧɧɵɦ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ. ɂɡ t t1 ɱɟɪɬɟɠɚ ɜɢɞɧɨ, ɱɬɨ ɩɨɱɬɢ ɜɫɹ ɩɥɨɳɚɞɶ, ɨɝɪɚɧɢɱɟɧɧɚɹ ɤɪɢɜɨɣ (20) ɬ ɨɫɶɸ ɚɛɫɰɢɫɫ, ɧɚɯɨɞɢɬɫɹ ɧɚɞ t t2 ɩɪɨɦɟɠɭɬɤɨɦ [ H , [ H , ɝɞɟ H – t t3 ɫɤɨɥɶ ɭɝɨɞɧɨ ɦɚɥɨɟ ɱɢɫɥɨ, ɟɫɥɢ ɬɨɥɶɤɨ t ! 0 – ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɟ 0 [ x ɱɢɫɥɨ. ȼɟɥɢɱɢɧɚ ɷɬɨɣ ɩɥɨɳɚɞɢ, Ɋɢɫ. 1. ɭɦɧɨɠɟɧɧɚɹ ɧɚ c U , ɪɚɜɧɚ ɤɨɥɢɱɟɫɬɜɭ ɬɟɩɥɚ, ɩɨɦɟɳɟɧɧɨɦɭ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɦɚɥɵɯ ɡɧɚɱɟɧɢɣ t ! 0 ɩɨɱɬɢ ɜɫɟ ɬɟɩɥɨ ɫɨɫɪɟɞɨɬɨɱɟɧɨ ɜ ɦɚɥɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ x [ . ɂɡ ɫɤɚɡɚɧɧɨɝɨ ɜɵɲɟ ɫɥɟɞɭɟɬ, ɱɬɨ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t 0 ɜɫɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ ɩɨɦɟɳɚɟɬɫɹ ɜ ɬɨɱɤɟ x [ , ɬ. ɟ. ɦɵ ɢɦɟɟɦ ɦɝɧɨɜɟɧɧɵɣ ɬɨɱɟɱɧɵɣ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ. Ɍɟɩɟɪɶ ɧɟɬɪɭɞɧɨ ɞɚɬɶ ɮɢɡɢɱɟɫɤɨɟ ɬɨɥɤɨɜɚɧɢɟ ɢ ɪɟɲɟɧɢɸ (9). Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɞɥɹ ɬɨɝɨ ɱɬɨɛɵ ɩɪɢɞɚɬɶ ɫɟɱɟɧɢɸ x [ ɫɬɟɪɠɧɹ ɬɟɦɩɟɪɚɬɭɪɭ M [ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ, ɦɵ ɞɨɥɠɧɵ ɪɚɫɩɪɟɞɟɥɢɬɶ ɧɚ y
ɦɚɥɨɦ ɷɥɟɦɟɧɬɟ d [ ɨɤɨɥɨ ɷɬɨɣ ɬɨɱɤɢ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ dQ c UM [ d [ ɢɥɢ, ɱɬɨ ɬɨ ɠɟ ɫɚɦɨɟ, ɩɨɦɟɫɬɢɬɶ ɜ ɬɨɱɤɟ [ ɦɝɧɨɜɟɧɧɵɣ ɬɨɱɟɱɧɵɣ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ ɧɚɩɪɹɠɟɧɢɹ dQ ; ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɜɵɡɵɜɚɟɦɨɟ ɷɬɢɦ ɦɝɧɨɜɟɧɧɵɦ ɬɨɱɟɱɧɵɦ ɢɫɬɨɱɧɢɤɨɦ, ɫɨɝɥɚɫɧɨ ɮɨɪɦɭɥɟ (10), ɛɭɞɟɬ 20
1
M [ d [
[ x 2 2
e 4a t . 2 St Ɉɛɳɟɟ ɠɟ ɞɟɣɫɬɜɢɟ ɨɬ ɧɚɱɚɥɶɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ M [ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɫɬɟɪɠɧɹ ɫɭɦɦɢɪɭɟɬɫɹ ɢɡ ɷɬɢɯ ɨɬɞɟɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ, ɱɬɨ ɢ ɞɚɟɬ ɧɚɦ ɩɨɥɭɱɟɧɧɨɟ ɜɵɲɟ ɪɟɲɟɧɢɟ (9) f
u x, t
³ M [ d [ 2
f
1
St
e
[ x 2 4 a 2t
d[ .
Ⱥɧɚɥɨɝɢɱɧɨ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ wu 2 § w 2 u w 2 u w 2 u · (21) a . wt ¨© wx 2 wy 2 wz 2 ¹¸ Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (21), ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ u t 0 M x, y , z , ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ f f f
u x, y , z , t
³ ³ ³ M [ ,K , ]
f f f
1
2a
St
3
e
[ x 2 K y 2 ] z 2 4 a 2t
d [ dK d ] .
VI. ɉɪɢɦɟɪ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ
ɇɚɣɬɢ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ wu w 2u 0, 25 2 xt , wt wx ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ u x,0 x . Ɋɟɲɟɧɢɟ. u x, t
ɂɳɟɦ
ɪɟɲɟɧɢɟ
ɡɚɞɚɱɢ
(1) u x, t
ɜ
ɜɢɞɟ
P x, t Q x, t , ɝɞɟ P ɢ Q – ɪɟɲɟɧɢɟ ɡɚɞɚɱ: wP w2 P 0, 25 2 ° wx ; ® wt ° P x,0 x ¯
w 2Q °Q 0, 25 2 xt . wx ® °Q x,0 0 ¯
ɋɨɝɥɚɫɧɨ ɮɨɪɦɭɥɚɦ ɮɭɧɤɰɢɢ P x, t ɢ Q x, t , ɢɦɟɸɬ ɜɢɞ P x, t
f
1
St
³ ze
z x 2 t
dz ,
§ z x 2 · exp ³ S t W ¨¨ t W ¸¸ dz . 0 f © ¹ zx zx Ⱦɟɥɚɟɦ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɨɣ Q ɜ (2 ) ɢ Q ɜ (3). Ɍɨɝɞɚ t t W t
Q x, t
³ dW
f
zW
(2)
f
1
21
(3)
P x, t Q x, t Ɉɬɜɟɬ: u x, t
f
1
³ xe
S
Q 2
t Q eQ
2
f t
1
S
³ W dW 0
f
³ x
f
dQ
x ,
2
t W Q eQ dQ
xt 2 . 2
xt 2 . x 2 Ɉɫɧɨɜɧɚɹ ɥɢɬɟɪɚɬɭɪɚ
1.
2.
3. 4.
Ɇɚɪɬɢɧɫɨɧ Ʌ.Ʉ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ : ɭɱɟɛ. ɞɥɹ ɫɬɭɞ. ɜɬɭɡɨɜ / Ʌ.Ʉ. Ɇɚɪɬɢɧɫɨɧ, ɘ.ɂ. Ɇɚɥɨɜ. – Ɇ. : ɂɡɞ-ɜɨ ɆȽɌɍ ɢɦ. ɇ.ɗ. Ȼɚɭɦɚɧɚ, 2002. – 367 ɫ. – (Ɇɚɬɟɦɚɬɢɤɚ ɜ ɬɟɯɧɢɱɟɫɤɨɦ ɭɧɢɜɟɪɫɢɬɟɬɟ ; ɜɵɩ 12). ɋɚɛɢɬɨɜ Ʉ.Ȼ. ɍɪɚɜɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ : ɭɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɫɬɭɞ., ɨɛɭɱ. ɩɨ ɫɩɟɰɢɚɥɶɧɨɫɬɹɦ «Ɇɚɬɟɦɚɬɢɤɚ», «ɉɪɢɤɥɚɞɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɢ ɢɧɮɨɪɦɚɬɢɤɚ» ɢ «Ɏɢɡɢɤɚ» / Ʉ.Ȼ. ɋɚɛɢɬɨɜ. – Ɇ. : ȼɵɫɲ. ɲɤ., 2003. – 254 ɫ. ɋɦɢɪɧɨɜ Ɇ.Ɇ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ : ɭɱɟɛ. ɩɨɫɨɛɢɟ / Ɇ.Ɇ. ɋɦɢɪɧɨɜ. – Ɇɢɧɫɤ : ɂɡɞ-ɜɨ ȻȽɍ, 1974. –232 ɫ. ɋɛɨɪɧɢɤ ɡɚɞɚɱ ɩɨ ɭɪɚɜɧɟɧɢɹɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ / ȼ.ɋ. ȼɥɚɞɢɦɢɪɨɜ [ɢ ɞɪ.]; – Ɇ. : Ɏɢɡɦɚɬɥɢɬ, 2003. – 288 ɫ. Ⱦɨɩɨɥɧɢɬɟɥɶɧɚɹ ɥɢɬɟɪɚɬɭɪɚ
1.
ɍɪɚɜɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ. Ɍɟɨɪɢɹ ɮɭɧɤɰɢɣ ɤɨɦɩɥɟɤɫɧɨɝɨ ɩɟɪɟɦɟɧɧɨɝɨ: ɭɱɟɛ. ɩɨɫɨɛɢɟ / ȼ.Ⱥ. ɉɨɝɨɪɟɥɟɧɤɨ [ɢ ɞɪ.]. – ȼɨɪɨɧɟɠ: ȼȽɍ, 1975. – 66 ɫ. ɋɨɞɟɪɠɚɧɢɟ
I. ȼɵɜɨɞ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ . . . . . . . . . . . . . . . . . . . . . . . II. ɉɟɪɜɚɹ ɝɪɚɧɢɱɧɚɹ ɡɚɞɚɱɚ. Ɍɟɨɪɟɦɚ ɨ ɦɚɤɫɢɦɭɦɟ ɢ ɦɢɧɢɦɭɦɟ . . III. Ɋɟɲɟɧɢɟ ɩɟɪɜɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. ɉɪɢɦɟɪ ɪɟɲɟɧɢɹ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ ɬɢɩɚ . . . . . . . . . . . . . . V. Ɂɚɞɚɱɚ Ʉɨɲɢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. ɉɪɢɦɟɪ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ Ʉɨɲɢ . . . . . . . . . . . . . . . . . . . . . . . . . . Ʌɢɬɟɪɚɬɭɪɚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 5 7 11 12 21 22
ɍɱɟɛɧɨɟ ɢɡɞɚɧɢɟ ɍɊȺȼɇȿɇɂə ɉȺɊȺȻɈɅɂɑȿɋɄɈȽɈ ɌɂɉȺ ɑɚɫɬɶ 3 ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɜɭɡɨɜ
ɋɨɫɬɚɜɢɬɟɥɶ Ɇɚɥɸɬɢɧɚ Ɉɤɫɚɧɚ ɉɟɬɪɨɜɧɚ Ɋɟɞɚɤɬɨɪ Ɍ.Ⱦ. Ȼɭɧɢɧɚ
ɉɨɞɩɢɫɚɧɨ ɜ ɩɟɱɚɬɶ 23.05.07. Ɏɨɪɦɚɬ 60×84/16. ɍɫɥ. ɩɟɱ. ɥ. 1,4. Ɍɢɪɚɠ 50 ɷɤɡ. Ɂɚɤɚɡ 969. 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. 23
