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A
n
B
n
B
n
= c
0
m
A
n
.
c
0
n
=
h
0
λ
n
k
0
, c
l
n
=
h
l
λ
n
k
l
.
v(x, t) =
∞
X
n=1
A
n
(cos λ
n
x + c
0
n
sin λ
n
x)e
−λ
2
n
a
2
t
.
λ
n
n
A
n
v(x, 0) =
∞
X
n=1
A
n
φ
n
(x) = F (x),
φ
n
(x) = cos λ
n
x + c
0
n
sin λ
n
x
φ
m
(x) x
∞
X
n=1
A
n
Z
l
0
φ
n
(x)φ
m
(x)dx =
Z
l
0
φ
m
(x)F (x)dx.
Z
l
0
φ
n
(x)φ
m
(x)dx = δ
nm
Φ
m
,
Φ
m
=
(1 + c
l 2
m
)c
0
m
+ (1 + c
0 2
m
)(c
l
m
+ lλ
m
(1 + c
l 2
m
))
2λ
m
(1 + c
l 2
m
)
.
A
m
=
1
Φ
m
Z
l
0
φ
m
(x)F (x)dx,
u(x, t) = v(x, t) + α + βx,
v(x, t) =
∞
X
n=1
A
n
(cos λ
n
x + c
0
m
sin λ
n
x)e
−λ
2
n
a
2
t
,
A
n
=
1
Φ
n
Z
l
0
φ
n
(x)F (x)dx,
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
.
T
l
h
l
→ ∞ h
0
→ 0
α = T
l
β = 0
tg λ
m
l → ∞
cos λ
m
l = 0.
λ
m
=
π(2m − 1)
2l
, m = 1, 2, · · · .
φ
m
= cos λ
m
x, Φ
m
=
l
2
,
ãäå êîý èöèåíòû An è Bn ñâÿçàíû ñîîòíîøåíèåì Òàêèì îáðàçîì,
Bn = c0m An . 1 l
Z
Am = φm (x)F (x)dx,
Φm
Çäåñü äëÿ óïðîùåíèÿ çàïèñè ïîëîæåíî: 0
è çàäà÷à ðåøåíà ïîëíîñòüþ äëÿ ïðîèçâîëüíûõ ãðàíè÷íûõ óñëîâèé.
h0 hl
c0n = , cln = . åøåíèå èìååò ñëåäóþùèé âèä:
λn k0 λn kl
Îòìåòèì ñðàçó, ÷òî ñïåêòð çàäà÷è íå çàâèñèò îò òåìïåðàòóð îêðó- u(x, t) = v(x, t) + α + βx,
æàþùåé ñðåäû ó êîíöîâ ñòåðæíÿ.
ãäå
Ó÷èòûâàÿ îäíîðîäíîñòü ãðàíè÷íûõ óñëîâèé ïîëó÷àåì îáùåå ðå-
øåíèå óðàâíåíèÿ òåïëîïðîâîäíîñòè, óäîâëåòâîðÿþùåå ãðàíè÷íûì ∞
X 2 2
óñëîâèÿì v(x, t) = An (cos λn x + c0m sin λn x)e−λn a t ,
n=1
∞ Z l
v(x, t) =
X
An (cos λn x + c0n sin λn x)e −λ2n a2 t
. (6) 1
An = φn (x)F (x)dx,
n=1 Φn 0
h0 (kl + lhl )T0 + hl k0 Tl
Çäåñü ñóììèðîâàíèå ïðîèçâîäèòñÿ ïî âñåì λn , óäîâëåòâîðÿþùèì α = ,
h0 (lhl + kl ) + hl k0
óñëîâèþ (5). Èíäåêñ n íóìåðóåò ðåøåíèÿ.
Äëÿ íàõîæäåíèÿ êîý èöèåíòîâ An èñïîëüçóåì íà÷àëüíîå óñëî- h0 hl (Tl − T0 )
β = .
âèå (3). Ïîëó÷àåì ñîîòíîøåíèå h0 (lhl + kl ) + hl k0
∞
X ×àñòíûå ñëó÷àè ãðàíè÷íûõ óñëîâèé (1):
v(x, 0) = An φn (x) = F (x), Ïðèìåð 1. Íà ïðàâîì êîíöå ñòåðæíÿ ïîääåðæèâàåòñÿ ïîñòîÿí-
n=1 íàÿ òåìïåðàòóðà, ðàâíàÿ òåìïåðàòóðå îêðóæàþùåé ñðåäû Tl , à íà
ãäå φn (x) = cos λn x + c0n sin λn x. Óìíîæèì, äàëåå, ýòî ñîîòíîøåíèå ëåâîì êîíöå òåìïåðàòóðà ìåíÿåòñÿ, íî ñ ïîñòîÿííûì íóëåâûì ãðàäè-
íà óíêöèþ φm (x) è ïðîèíòåãðèðóåì ïî x. Èìååì åíòîì. Äëÿ âûïîëíåíèÿ ýòèõ óñëîâèé ïîëîæèì hl → ∞ è h0 → 0. Â
ýòîì ñëó÷àå ïîëó÷àåì, ÷òî α = Tl è β = 0. Èç óðàâíåíèÿ (5) ïîëó÷àåì
∞
X Z l Z l ÷òî tg λm l → ∞, ò.å.
An φn (x)φm (x)dx = φm (x)F (x)dx.
n=1 0 0 cos λm l = 0. (7)
Íåòðóäíî ïîêàçàòü, ÷òî Òàêèì îáðàçîì, ñïåêòð íàõîäèòñÿ â ÿâíîì âèäå:
Z l
φn (x)φm (x)dx = δnm Φm , π(2m − 1)
λm = , m = 1, 2, · · · .
0 2l
ãäå Ïðè ýòîì
(1 + clm2 )c0m + (1 + c0m2 )(clm + lλm (1 + clm2 )) l
Φm = . φm = cos λm x, Φm = ,
2λm (1 + clm2 ) 2
5 6
