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u
′
t
(x, t) − a
2
u
′′
xx
(x, t) = U(x, t).
[u] = [ ] ,
[U] =
,
a
2
=
k
cρ
=
2
.
k, c, ρ
/ · /
/
3
u(x, 0) = f(x).
h
0
(u(0, t) − T
0
) = k
0
u
′
x
(0, t),
h
l
(u(l, t) − T
l
) = −k
l
u
′
x
(l, t).
u(x, t) → v(x, t)
u(x, t) = v(x, t) + α + βx
α β
h
0
(α − T
0
) = k
0
β,
h
l
(α + βl − T
l
) = −k
l
β.
α =
h
0
(k
l
+ lh
l
)T
0
+ h
l
k
0
T
l
h
0
(lh
l
+ k
l
) + h
l
k
0
,
β =
h
0
h
l
(T
l
− T
0
)
h
0
(lh
l
+ k
l
) + h
l
k
0
.
α β v(x, t)
v
′
t
(x, t) − a
2
v
′′
xx
(x, t) = V (x, t) = U(x, t) − α
′
t
− β
′
t
x,
v(x, 0) = F (x) = f(x) − α(0) − β(0)x,
h
0
v(0, t) = k
0
v
′
x
(0, t),
h
l
v(l, t) = −k
l
v
′
x
(l, t).
V (x, t) = 0
U(x, t)−α
′
t
−β
′
t
x = 0
v
λ
(x, t) = e
−λ
2
a
2
t
(A
λ
cos λx + B
λ
sin λx),
λ ≥ 0 λ
A
λ
B
λ
v
λ
λ = λ
n
tg λ
n
l =
c
0
n
+ c
l
n
1 − c
0
n
c
l
n
,
v
n
(x, t) = A
n
e
−λ
2
n
a
2
t
(cos λ
n
x + c
0
n
sin λ
n
x),
1 Óðàâíåíèå òåïëîïðîâîäíîñòè β =
h0 hl (Tl − T0 )
.
h0 (lhl + kl ) + hl k0
Îäíîìåðíîå óðàâíåíèå òåïëîïðîâîäíîñòè
Äëÿ òàêèõ çíà÷åíèé α è β ïîëó÷àåì, ÷òî óíêöèÿ v(x, t) óäîâëå-
u′t (x, t) − a2 u′′xx (x, t) = U (x, t). òâîðÿåò óðàâíåíèþ òåïëîïðîâîäíîñòè ñ èçìåíåííîé ïðàâîé ÷àñòüþ
àçìåðíîñòè: vt′ (x, t) − a2 vxx
′′
(x, t) = V (x, t) = U (x, t) − α′t − βt′ x, (2)
[u] = [Äæ] Òåìïåðàòóðà, íà÷àëüíîìó óñëîâèþ
Äæ
[U ] = Ìîùíîñòü èñòî÷íèêîâ, v(x, 0) = F (x) = f (x) − α(0) − β(0)x, (3)
ñåê
ì
2
2 k è îäíîðîäíûì ãðàíè÷íûì óñëîâèÿì
a = = .
cρ ñåê
h0 v(0, t) = k0 vx′ (0, t),
Çäåñü k, c, ρ êîý èöèåíò âíóòðåííåé òåïëîïðîâîäíîñòè ñ ðàçìåð-
íîñòüþ Äæ/ì · ñåê, óäåëüíàÿ òåïëîåìêîñòü ñ ðàçìåðíîñòüþ Äæ/êã, hl v(l, t) = −kl vx′ (l, t). (4)
îáúåìíàÿ ïëîòíîñòü ñ ðàçìåðíîñòüþ êã/ì3 , ñîîòâåòñòâåííî.
Íà÷àëüíîå óñëîâèå (íà÷àëüíîå ðàñïðåäåëåíèå òåìïåðàòóðû): 1.1 Îòñóòñòâèå âíåøíèõ èñòî÷íèêîâ: V (x, t) = 0.
u(x, 0) = f (x). Òàêàÿ ñèòóàöèÿ ðåàëèçóåòñÿ, åñëè U (x, t)−α′t −βt′ x = 0.  ýòîì ñëó÷àå
óðàâíåíèå (2) ÿâëÿåòñÿ îäíîðîäíûì è åãî ÷àñòíîå ðåøåíèå ìåòîäîì
ðàíè÷íûå óñëîâèÿ Ôóðüå èìååò ñëåäóþùèé âèä:
h0 (u(0, t) − T0 ) = k0 u′x (0, t), (1a) vλ (x, t) = e−λ
2 2
a t
(Aλ cos λx + Bλ sin λx),
hl (u(l, t) − Tl ) = −kl u′x (l, t). (1b)
ñ λ ≥ 0. åøåíèå íóìåðóåòñÿ ÷èñëîì λ è çàâèñÿò îò äâóõ ïðîèç-
Ýòè óñëîâèÿ ÿâëÿþòñÿ íåîäíîðîäíûìè. ×òîáû ñäåëàòü óñëîâèÿ îä- âîëüíûõ ïîñòîÿííûõ Aλ è Bλ . Ïîñêîëüêó óíêöèÿ v óäîâëåòâîðÿåò
íîðîäíûìè ââåäåì íîâóþ óíêöèþ u(x, t) → v(x, t) ïî ïðàâèëó îäíîðîäíûì ãðàíè÷íûì óñëîâèÿì, òî îáùåå ðåøåíèå ìîæíî ïðåä-
ñòàâèòü â âèäå ñóïåðïîçèöèè âûøåïðèâåäåííûõ ðåøåíèé. Âíà÷àëå
u(x, t) = v(x, t) + α + βx íåîáõîäèìî íàéòè ñïåêòð âîçìîæíûõ çíà÷åíèé λ.
è ïîä÷èíèì óíêöèè α è β óñëîâèÿì Èç ãðàíè÷íûõ óñëîâèé (4) ïîëó÷àåì, ÷òî ñïåêòð ñîáñòâåííûõ çíà-
÷åíèé, λ = λn , ïîä÷èíÿåòñÿ òðàíñöåíäåíòíîìó óðàâíåíèþ
h0 (α − T0 ) = k0 β,
c0n + cln
hl (α + βl − Tl ) = −kl β. tg λn l = , (5)
1 − c0n cln
Îòñþäà
à ðåøåíèÿ èìåþò ñëåäóþùèé âèä
h0 (kl + lhl )T0 + hl k0 Tl 2 2
α = , vn (x, t) = An e−λn a t (cos λn x + c0n sin λn x),
h0 (lhl + kl ) + hl k0
3 4
