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Явная схема решения третьей смешанной задачи для квазилинейного уравнения теплопроводности.

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ɍȾɄ 519.633
əȼɇȺə ɋɏȿɆȺ Ɋȿɒȿɇɂə ɌɊȿɌɖȿɃ ɋɆȿɒȺɇɇɈɃ
ɁȺȾȺɑɂ ȾɅə ɄȼȺɁɂɅɂɇȿɃɇɈȽɈ ɍɊȺȼɇȿɇɂə
ɌȿɉɅɈɉɊɈȼɈȾɇɈɋɌɂ
Ɇ.Ɂ. ɏɚɣɪɢɫɥɚɦɨɜ1, Ⱥ.ȼ. Ƚɟɪɟɧɲɬɟɣɧ2
ɉɪɟɞɥɚɝɚɟɬɫɹ ɱɢɫɥɟɧɧɵɣ ɦɟɬɨɞ ɪɟɲɟɧɢɹ ɬɪɟɬɶɟɣ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ
ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɤɜɚɡɢɥɢɧɟɣɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɚɪɚɛɨɥɢɱɟɫɤɨɝɨ ɬɢɩɚ, ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɹɜɧɨɣ ɪɚɡɧɨɫɬɧɨɣ ɫɯɟɦɵ. Ɂɚɜɢɫɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɪɚɜɧɟɧɢɹ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɟɨɞɨɥɟɜɚɟɬɫɹ ɜɜɟɞɟɧɢɟɦ ɧɨɜɨɣ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ – ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. ɉɪɟɞɥɚɝɚɟɬɫɹ ɬɟɫɬɨɜɚɹ ɡɚɞɚɱɚ ɫ ɢɡɜɟɫɬɧɵɦ ɬɨɱɧɵɦ ɪɟɲɟɧɢɟɦ ɞɥɹ ɱɢɫɥɟɧɧɵɯ
ɪɚɫɱɟɬɨɜ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ, ɤɜɚɡɢɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɹɜɧɵɟ ɪɚɡɧɨɫɬɧɵɟ ɫɯɟɦɵ, ɚɩɩɪɨɤɫɢɦɚɰɢɹ.
ȼ ɧɚɫɬɨɹɳɟɣ ɪɚɛɨɬɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɢɞɟɢ, ɢɡɥɨɠɟɧɧɵɟ ɜ ɪɚɛɨɬɚɯ [1, 2], ɜ ɤɨɬɨɪɵɯ ɛɵɥɚ ɩɪɟɞɥɨɠɟɧɚ ɢ ɨɛɨɫɧɨɜɚɧɚ ɹɜɧɚɹ ɭɫɬɨɣɱɢɜɚɹ ɫɯɟɦɚ ɞɥɹ ɥɢɧɟɣɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
1. ɑɢɫɥɟɧɧɵɣ ɦɟɬɨɞ
Ɋɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɭɸ ɩɨɫɬɚɧɨɜɤɭ ɬɪɟɬɶɟɣ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɨɞɧɨɪɨɞɧɨɝɨ ɤɜɚɡɢɥɢɧɟɣɧɨɝɨ ɭɪɚɜɧɟɧɢɹ [3]:
­
∂u ∂ §
∂u ·
0 < t ≤ T , 0 < x < L,
°ɫ(u ) ∂t = ∂x ¨ q(u ) ∂x ¸ ,
©
¹
°
°u ( x, 0) = ϕ ( x),
°°
(1)
® − § q (u ) ∂u ·
= λl (u (0, t )) (θl − u (0, t ) ) + Ql ,
¨
¸
° ©
∂x ¹ x =0
°
°§
∂u ·
= λr (u ( L, t )) (θ r − u ( L, t ) ) + Qr ,
°¨ q (u ) ¸
∂x ¹ x = L
°̄©
ɝɞɟ u = u ( x, t ) – ɬɟɦɩɟɪɚɬɭɪɚ ɫɬɟɪɠɧɹ; 0 ≤ x ≤ L – ɤɨɨɪɞɢɧɚɬɚ; 0 ≤ t ≤ T – ɜɪɟɦɹ; L – ɞɥɢɧɚ
ɫɬɟɪɠɧɹ; T – ɤɨɧɟɱɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ; ɫ(u ) – ɨɛɴɟɦɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɫɬɟɪɠɧɹ;
q (u ) – ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɫɬɟɪɠɧɹ; ϕ ( x) – ɮɭɧɤɰɢɹ ɧɚɱɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɫɬɟɪɠɧɹ; λl (u ) – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɧɚ ɥɟɜɨɦ ɤɨɧɰɟ ɫɬɟɪɠɧɹ; λr (u ) – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɧɚ ɩɪɚɜɨɦ ɤɨɧɰɟ ɫɬɟɪɠɧɹ; θl – ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɟɲɧɟɣ ɫɪɟɞɵ ɧɚ ɥɟɜɨɦ ɤɨɧɰɟ
ɫɬɟɪɠɧɹ; θ r – ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɟɲɧɟɣ ɫɪɟɞɵ ɧɚ ɩɪɚɜɨɦ ɤɨɧɰɟ ɫɬɟɪɠɧɹ; Ql = Ql (t ) – ɦɨɳɧɨɫɬɶ ɩɨɬɨɤɚ ɬɟɩɥɚ ɧɚ ɥɟɜɨɦ ɤɨɧɰɟ ɫɬɟɪɠɧɹ; Qr = Qr (t ) – ɦɨɳɧɨɫɬɶ ɩɨɬɨɤɚ ɬɟɩɥɚ ɧɚ ɩɪɚɜɨɦ ɤɨɧɰɟ ɫɬɟɪɠɧɹ. Ɏɭɧɤɰɢɢ c = ɫ (u ) , q = q (u ) , λl = λl (u ) ɢ λr = λr (u ) ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɧɟɩɪɟɪɵɜɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɬɟɦɩɟɪɚɬɭɪɵ, ɡɚɞɚɧɧɵɦɢ ɞɥɹ ɜɫɟɯ ɡɧɚɱɟɧɢɣ ɬɟɦɩɟɪɚɬɭɪɵ.
Ɂɚɦɟɧɚ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ
∂u
ɉɨɫɤɨɥɶɤɭ ɜ ɭɪɚɜɧɟɧɢɢ ɩɪɢɫɭɬɫɬɜɭɟɬ ɱɥɟɧ q (u ) , ɬɨ ɭɞɨɛɧɨ ɫɞɟɥɚɬɶ ɡɚɦɟɧɭ
∂x
u
∂G
∂ 2G
q (u )
= a 2 (u ) 2 , ɝɞɟ a (u ) =
G (u ) = ³ q (ξ )d ξ . Ɍɨɝɞɚ ɞɥɹ ɮɭɧɤɰɢɢ G ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
–
∂
t
c
(
u
)
∂
x
0
1
ɏɚɣɪɢɫɥɚɦɨɜ Ɇɢɯɚɢɥ Ɂɢɧɚɬɭɥɥɚɟɜɢɱ – ɚɫɩɢɪɚɧɬ, ɤɚɮɟɞɪɚ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ, ɘɠɧɨ-ɍɪɚɥɶɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ.
E-mail: zinatmk@gmail.com
2
Ƚɟɪɟɧɲɬɟɣɧ Ⱥɪɤɚɞɢɣ ȼɚɫɢɥɶɟɜɢɱ – ɤɚɧɞɢɞɚɬ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ, ɤɚɮɟɞɪɚ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ, ɘɠɧɨɍɪɚɥɶɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ.
174
ȼɟɫɬɧɢɤ ɘɍɪȽɍ. ɋɟɪɢɹ «Ɇɚɬɟɦɚɬɢɤɚ. Ɇɟɯɚɧɢɤɚ. Ɏɢɡɢɤɚ»
ɏɚɣɪɢɫɥɚɦɨɜ Ɇ.Ɂ.,
Ƚɟɪɟɧɲɬɟɣɧ Ⱥ.ȼ.
əɜɧɚɹ ɫɯɟɦɚ ɪɟɲɟɧɢɹ ɬɪɟɬɶɟɣ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ
ɞɥɹ ɤɜɚɡɢɥɢɧɟɣɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ. Ɏɭɧɤɰɢɹ G (u ) ɹɜɥɹɟɬɫɹ ɫɬɪɨɝɨ ɦɨɧɨɬɨɧɧɨɣ ɮɭɧɤɰɢɟɣ
ɬɟɦɩɟɪɚɬɭɪɵ, ɩɨɷɬɨɦɭ ɨɛɪɚɬɧɚɹ ɮɭɧɤɰɢɹ G −1 ɫɭɳɟɫɬɜɭɟɬ ɢ ɦɨɠɟɬ ɛɵɬɶ ɜɵɱɢɫɥɟɧɚ ɜ ɤɨɧɤɪɟɬɧɨɣ
ɬɨɱɤɟ, ɧɚɩɪɢɦɟɪ, ɦɟɬɨɞɨɦ ɞɢɯɨɬɨɦɢɢ.
ɒɚɛɥɨɧ ɫɯɟɦɵ. Ɋɚɫɱɟɬɧɵɟ ɮɨɪɦɭɥɵ
ɇɚ ɩɥɨɫɤɨɫɬɢ ( x, t ) ɢɫɩɨɥɶɡɭɟɬɫɹ ɪɚɜɧɨɦɟɪɧɚɹ ɫɟɬɤɚ [2]
­
½
§ 1·
©
¹
¯
¿
ɝɞɟ h = L N – ɲɚɝ ɩɨ ɩɟɪɟɦɟɧɧɨɣ x , τ – ɲɚɝ ɩɨ ɩɟɪɟɦɟɧɧɨɣ t . ɒɚɛɥɨɧ ɩɪɟɞɥɚɝɚɟɦɨɣ ɫɯɟɦɵ
ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɪɢɫ. 1. Ⱦɥɹ ɨɛɨɡɧɚɱɟɧɢɹ ɡɧɚɱɟɧɢɣ ɫɟɬɨɱɧɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɮɭɧɤɰɢɢ G ɧɚ ɫɥɟɞɭɸɳɟɦ ɜɪɟɦɟɧɧɨɦ ɫɥɨɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜɟɪɯɧɢɣ ɢɧɞɟɤɫ (+1), ɧɚ ɫɥɟɞɭɸɳɟɦ ɩɨɥɭɰɟɥɨɦ ɜɪɟɦɟɧ§ 1·
§ 1·
ɧɨɦ ɫɥɨɟ – ¨ + ¸ , ɚ ɧɚ ɩɪɟɞɵɞɭɳɟɦ ɩɨɥɭɰɟɥɨɦ ɜɪɟɦɟɧɧɨɦ ɫɥɨɟ – ¨ − ¸ .
© 2¹
© 2¹
ɂɫɩɨɥɶɡɭɟɦɚɹ ɪɚɫɱɟɬɧɚɹ ɮɨɪɦɭɥɚ ɢɦɟɟɬ ɜɢɞ
ωhτ = ωh × ωτ , ωh = ® xi = ¨ i − ¸ h, i = 1, 2, ..., N ¾ , ωτ = {t j = jτ , j = 0, 1, ...},
2
t
Gi( +1)
t0 + τ
§ 1·
¨+ ¸
§ 1·
¨+ ¸
Gi©−12 ¹
Gi −1
t0
Gi©+12 ¹
Gi
Gi +1
§ 1·
¨− ¸
§ 1·
¨− ¸
Gi©−12 ¹
t0 − τ
0
1
xi −1
2
h
Gi©+12 ¹
xi
2h
xi+1
N
N +1
Nh = L
ih
x
Ɋɢɫ. 1. ɒɚɛɥɨɧ ɪɚɡɧɨɫɬɧɨɣ ɫɯɟɦɵ
Gi( +1)
= (Gi − B )e
−
2 a 2 (ui )
h2
τ
+ Aτ + B ,
(2)
§ 1·
§ 1·
§ 1· ·
§ § 1·
¨− ¸
¨+ ¸
¨− ¸
G + Gi +1
h2
1 ¨ ¨© + 2 ¹¸
2¹
2¹
©
©
− A⋅ 2
.
ɝɞɟ A =
Gi −1 − Gi −1 + Gi +1 − Gi©+12 ¹ ¸ , B = i −1
¸
2τ ¨
2
2a (ui )
©
¹
Ⱦɥɹ ɪɚɫɱɟɬɚ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ G ɧɚ ɜɪɟɦɟɧɧɨɦ ɫɥɨɟ t = τ , ɚ ɬɚɤɠɟ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ ɜ ɩɨɥɭɰɟɥɵɯ ɫɥɨɹɯ ɩɨ ɜɪɟɦɟɧɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɮɨɪɦɭɥɵ:
2 a 2 (ui )
2 a 2 (ui ) ·
a 2 (u )
a 2 (u ) ·
§ 1·
§
− 2 i ⋅τ
− 2 i ⋅τ G
−
τ §
−
τ G
¨+ ¸
G
+
2
2
2
¨
¸
¨
¸ i −1 + Gi +1 .(3)
i
1
i
1
−
+
©
¹
Gi (τ ) = Gi e h
+ 1− e h
, Gi
= Gi e h
+ 1− e h
¨¨
¸¸
¨¨
¸¸
2
2
©
¹
©
¹
Ⱦɥɹ ɜɵɩɨɥɧɟɧɢɹ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ ɜɜɟɞɟɧɵ ɮɢɤɬɢɜɧɵɟ ɭɡɥɵ ɫ ɧɨɦɟɪɚɦɢ 0 ɢ N + 1 (ɫɦ.
ɪɢɫ. 1): ɫɧɚɱɚɥɚ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɡɧɚɱɟɧɢɹ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ ɜɨ ɜɧɭɬɪɟɧɧɢɯ ɬɨɱɤɚɯ, ɩɨɫɥɟ ɱɟɝɨ,
ɢɫɯɨɞɹ ɢɡ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ, ɡɚɞɚɸɬɫɹ ɟɟ ɡɧɚɱɟɧɢɹ ɜ ɮɢɤɬɢɜɧɵɯ ɭɡɥɚɯ.
ɂɫɩɨɥɶɡɭɹ ɫɥɟɞɭɸɳɢɟ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɬɨɱɧɨɫɬɢ
3(λl )1 − (λl ) 2
u +u
3q − q ∂u (0, t ) u1 − u0
, q (u (0, t )) = 1 2 ,
λl (u (0, t )) =
=
, u (0, t ) = 0 1 ,
2
h
2
2
∂x
ɧɟɬɪɭɞɧɨ ɩɨɥɭɱɢɬɶ ɮɨɪɦɭɥɭ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɮɢɤɬɢɜɧɨɦ ɭɡɥɟ 0:
3(λl )1 − (λl )2 ·
§ 3q − q
u1 ¨ 1 2 −
¸ + (3(λl )1 − (λl ) 2 )θl + 2Ql
2
h
©
¹
u0 =
.
(4)
3q1 − q2 3(λl )1 − (λl ) 2
+
2
h
2013, ɬɨɦ 5, ʋ 2
175
Ʉɪɚɬɤɢɟ ɫɨɨɛɳɟɧɢɹ
Ⱥɧɚɥɨɝɢɱɧɵɟ ɪɚɫɫɭɠɞɟɧɢɹ ɞɥɹ ɩɪɚɜɨɝɨ ɤɨɧɰɚ ɩɪɢɜɨɞɹɬ ɤ ɪɚɫɱɟɬɧɨɣ ɮɨɪɦɭɥɟ ɞɥɹ ɭɡɥɚ N + 1
§ 3q − q N −1 3(λr ) N − (λr ) N −1 ·
uN ¨ N
−
¸ + (3(λr ) N − (λr ) N −1 )θ r + 2Qr
h
2
©
¹
u N +1 =
.
3qN − qN −1 3(λr ) N − (λr ) N −1
+
h
2
(5)
2. Ɍɟɫɬɨɜɚɹ ɡɚɞɚɱɚ
ɋ ɭɱɟɬɨɦ ɤɨɧɟɱɧɨɫɬɢ ɫɤɨɪɨɫɬɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ ɜ [4] ɛɵɥɢ ɩɨɥɭɱɟɧɵ ɩɪɢɛɥɢɠɟɧɧɵɟ
ɪɟɲɟɧɢɹ ɨɞɧɨɦɟɪɧɨɣ ɡɚɞɚɱɢ ɧɟɥɢɧɟɣɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɧɚ ɩɨɥɭɛɟɫɤɨɧɟɱɧɨɣ ɩɪɹɦɨɣ ɩɪɢ ɡɚɞɚɧɧɨɦ ɩɨɬɨɤɟ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ ɜ ɜɢɞɟ ɫɬɟɩɟɧɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ. ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɢɞɟɣ, ɢɡɥɨɠɟɧɧɵɯ ɜ [4, 5], ɩɨɥɭɱɟɧɨ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɫɥɟɞɭɸɳɟɣ ɬɪɟɬɶɟɣ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ ɧɚ ɩɨɥɭɛɟɫɤɨɧɟɱɧɨɣ ɩɪɹɦɨɣ ɞɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɤɜɚɡɢɥɢɧɟɣɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ:
∂u ∂ § n ∂u ·
t > 0, x > 0 ,
= ¨u
(6)
¸,
∂t ∂x © ∂x ¹
u ( x, 0) = 0,
x ≥ 0,
(7)
1
§ n ∂u ·
(8)
= λ u x =0 − Qt n , t > 0 ,
¨u
¸
© ∂x ¹ x =0
ɝɞɟ n > 0 – ɩɨɤɚɡɚɬɟɥɶ ɫɬɟɩɟɧɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɥɭɱɢɫɬɭɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ, λ > 0 – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ, Q > 0 – ɤɨɷɮɮɢɰɢɟɧɬ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɦɨɳɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ.
ȼ ɱɚɫɬɧɨɫɬɢ, ɩɪɢ n = 2 ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (6)–(8) ɛɭɞɟɬ ɬɚɤɢɦ:
­° 2α (α t − x), x < α t ,
(9)
u ( x, t ) = ®
x ≥ αt,
°̄0,
ɝɞɟ α =
−λ + λ 2 + 2 2Q
.
2
3. Ɋɟɡɭɥɶɬɚɬɵ ɱɢɫɥɟɧɧɵɯ ɪɚɫɱɟɬɨɜ
Ɋɟɲɚɥɚɫɶ ɡɚɞɚɱɚ (6)–(8) ɩɪɢ n = 2 , λ = 1 Ⱦɠ/(ɦ 2 × ɫ× ɨ ɋ) , Q = 5 Ⱦɠ/(ɦ 2 × ɫ) . ɒɚɝ ɩɨ ɤɨɨɪɞɢɧɚɬɟ h ɛɪɚɥɫɹ ɪɚɜɧɵɦ 1, ɲɚɝ ɩɨ ɜɪɟɦɟɧɢ τ ɛɪɚɥɫɹ ɪɚɜɧɵɦ 0,05. ɋɪɚɜɧɟɧɢɟ ɱɢɫɥɟɧɧɨɝɨ ɢ ɬɨɱɧɨɝɨ
ɪɟɲɟɧɢɣ ɜ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ t = 10, 20, 30 c ɩɪɢɜɟɞɟɧɨ ɧɚ ɪɢɫ. 2. ɋɩɥɨɲɧɚɹ ɤɪɢɜɚɹ ɩɪɟɞɫɬɚɜɥɹɟɬ
ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ.
12
10
ɨ
Ɍɟɦɩɟɪɚɬɭɪɚ, ɋ
8
t=30c
6
t=20c
t=10c
4
2
0
0
5
10
15
20
25
30
35
40
45
50
Ʉɨɨɪɞɢɧɚɬɚ x, ɦ
Ɋɢɫ. 2. ɑɢɫɥɟɧɧɨɟ ɢ ɬɨɱɧɨɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜ ɪɚɡɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ
ɉɨɥɭɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɩɨɡɜɨɥɹɸɬ ɝɨɜɨɪɢɬɶ ɨ ɯɨɪɨɲɢɯ ɫɜɨɣɫɬɜɚɯ ɩɪɟɞɥɨɠɟɧɧɨɝɨ ɱɢɫɥɟɧɧɨɝɨ ɦɟɬɨɞɚ.
176
ȼɟɫɬɧɢɤ ɘɍɪȽɍ. ɋɟɪɢɹ «Ɇɚɬɟɦɚɬɢɤɚ. Ɇɟɯɚɧɢɤɚ. Ɏɢɡɢɤɚ»
ɏɚɣɪɢɫɥɚɦɨɜ Ɇ.Ɂ.,
Ƚɟɪɟɧɲɬɟɣɧ Ⱥ.ȼ.
əɜɧɚɹ ɫɯɟɦɚ ɪɟɲɟɧɢɹ ɬɪɟɬɶɟɣ ɫɦɟɲɚɧɧɨɣ ɡɚɞɚɱɢ
ɞɥɹ ɤɜɚɡɢɥɢɧɟɣɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Ʌɢɬɟɪɚɬɭɪɚ
1. Ƚɟɪɟɧɲɬɟɣɧ, Ⱥ.ȼ. ɇɚɝɪɟɜɚɧɢɟ ɤɪɭɝɚ ɞɜɢɠɭɳɢɦɫɹ ɬɟɩɥɨɢɫɬɨɱɧɢɤɨɦ / Ⱥ.ȼ. Ƚɟɪɟɧɲɬɟɣɧ,
ɇ. Ɇɚɲɪɚɛɨɜ // Ɉɛɨɡɪɟɧɢɟ ɩɪɢɤɥɚɞɧɨɣ ɢ ɩɪɨɦɵɲɥɟɧɧɨɣ ɦɚɬɟɦɚɬɢɤɢ. – 2008. – Ɍ. 15, ʋ 5. –
ɋ. 870–871.
2. Ƚɟɪɟɧɲɬɟɣɧ, Ⱥ.ȼ. ɍɫɬɨɣɱɢɜɵɟ ɹɜɧɵɟ ɫɯɟɦɵ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ /Ⱥ.ȼ. Ƚɟɪɟɧɲɬɟɣɧ, ȿ.Ⱥ. Ƚɟɪɟɧɲɬɟɣɧ, ɇ. Ɇɚɲɪɚɛɨɜ // ȼɟɫɬɧɢɤ ɘɍɪȽɍ. ɋɟɪɢɹ «Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɢ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ». – 2008. – ȼɵɩ. 1. – ʋ 15(115). – ɋ. 9–11.
3. ɋɚɦɚɪɫɤɢɣ, Ⱥ.Ⱥ. Ɍɟɨɪɢɹ ɪɚɡɧɨɫɬɧɵɯ ɫɯɟɦ / Ⱥ.Ⱥ. ɋɚɦɚɪɫɤɢɣ. – Ɇ.: ɇɚɭɤɚ, 1989. – 616 ɫ.
4. Ʉɭɞɪɹɲɨɜ, ɇ.Ⱥ. ɉɪɢɛɥɢɠɟɧɧɵɟ ɪɟɲɟɧɢɹ ɨɞɧɨɦɟɪɧɵɯ ɡɚɞɚɱ ɧɟɥɢɧɟɣɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢ ɡɚɞɚɧɧɨɦ ɩɨɬɨɤɟ / ɇ.Ⱥ. Ʉɭɞɪɹɲɨɜ, Ɇ.Ⱥ. ɑɦɵɯɨɜ // ɀɭɪɧɚɥ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɦɚɬɟɦɚɬɢɤɢ
ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ. – 2007. – Ɍ. 47, ʋ 1. – ɋ. 110–120.
5. Ɂɟɥɶɞɨɜɢɱ, ə.Ȼ. Ʉ ɬɟɨɪɢɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ ɩɪɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɡɚɜɢɫɹɳɟɣ ɨɬ
ɬɟɦɩɟɪɚɬɭɪɵ / ə.Ȼ. Ɂɟɥɶɞɨɜɢɱ, Ⱥ.ɋ. Ʉɨɦɩɚɧɟɟɰ // Ʉ 70-ɥɟɬɢɸ Ⱥ.Ɏ. ɂɨɮɮɟ: ɫɛ. ɧɚɭɱ. ɬɪ. – Ɇ.: ɂɡɞɜɨ Ⱥɇ ɋɋɋɊ, 1950. – ɋ. 61–71.
EXPLICIT SCHEME FOR THE SOLUTION OF THIRD BOUNDARY VALUE
PROBLEM FOR QUASI-LINEAR HEAT EQUATION
1
M.Z. Khayrislamov , A.W. Herreinstein
2
In produced paper numerical method for the solution of third boundary value problem for onedimensional quasi-linear heat equation grounded on the use of explicit finite-difference scheme is offered. The coefficients’ dependence on temperature is overcome by introducing the new unknown function – a primitive integral of conduction. Test problem with known exact solution for numerical calculations is proposed.
Keywords: thermal conductivity, quasi-linear heat equation, explicit finite-difference schemes, approximation.
References
1. Gerenshteyn A.V., Mashrabov N. Obozrenie prikladnoy i promyshlennoy matematiki. 2008.
Vol. 15, no. 5. pp. 870–871.
2. Herreinstein A.W., Herreinstein E.A., Mashrabov N. Ustoychivye yavnye skhemy dlya uravneniya teploprovodnosti (Steady Obvious Schemes for Equation of Heat Conductivity). Vestnik
YuUrGU. Seriya «Matematicheskoe modelirovanie i programmirovanie». 2008. Issue 1. no. 15(115).
pp. 9–11. (in Russ.).
3. Samarskiy A.A. Teoriya raznostnykh skhem (The theory of difference schemes). Moscow:
Nauka, 1989. 616 p. (in Russ.).
4. Kudryashov N.A., Chmykhov M.A. Approximate solutions to one-dimensional nonlinear heat
conduction problems with a given flux. Computational Mathematics and Mathematical Physics. 2007.
Vol. 47, no. 1. pp. 107–117
5. Zel'dovich Ya.B., Kompaneets A.S. K 70-letiyu A.F. Ioffe: sb. nauch. tr. (On the 70th anniversary
of the A.F. Ioffe: collection of scientific papers). Moscow: Izd-vo AN SSSR, 1950. pp. 61–71. (in
Russ.).
ɉɨɫɬɭɩɢɥɚ ɜ ɪɟɞɚɤɰɢɸ 6 ɦɚɪɬɚ 2013 ɝ.
1
Khayrislamov Mikhail Zinatullaevich is Post-Graduate student, Applied Mathematics Department, South Ural State University.
E-mail: zinatmk@gmail.com
2
Herreinstein Arcady Wasilevich is Cand. Sc. (Physics and Mathematics), Associate Professor, Applied Mathematics Department, South Ural
State University.
2013, ɬɨɦ 5, ʋ 2
177
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