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February 1, 2008
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ã
∂2u
∂t2
= a2 ∂
2u
∂x2
+ f (x , t), 0 < x < `, t > 0
u|x=0 = ν(t), u|x=` = µ(t),
u(x , t)|t=0 = φ(x), ∂u∂t |t=0 = ψ(x), 0 < x < `.
�w(x , t) = `−x` ν(t) +
x
`µ(t), w(x , t) ÷v
w |x=0 = ν(t), w |x=` = µ(t)
P U(x , t) = u(x , t)− w(x , t),
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U(x , t) ÷v
∂2U
∂t2
= a2 ∂
2U
∂x2
+ f (x , t)− ( `−x` ν ′′(t) + x`µ′′(t)), 0 < x < `, t > 0
U|x=0 = 0, U|x=` = 0,⇔àg>.^
U(x , t)|t=0 = φ(x)− ( `−x` ν(0) + x`µ(0)), 0 < x < `
∂U
∂t |t=0 = ψ(x)− ( `−x` ν ′(0) + x`µ′(0)), 0 < x < `.
÷v¦ff¼ê w(x , t) kéõ§X
w(x , t) = ν(t) + sin(
x
`
)[µ(t)− ν(t)].
¯¢þ§ w(x , t) ÷v
w |x=0 = ν(t), w |x=` = µ(t)
ff®¼êÑ´#Nff"
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@oUÄé�Ü·ffC,r§Ú>.^zàg§Úà
g>.^? ùw(x , t)I÷v{
∂2w
∂t2
= a2 ∂
2w
∂x2
+ f (x , t),
u|x=0 = ν(t), u|x=` = µ(t),
�J��w(x , t). ���´´´ �f (x , t) = f (x),ν(t) = AÚµ(t) = B(tÃ
'), ±é�==xk'ff¼êw(x){
∂2w
∂t2
= a2 ∂
2w
∂x2
+ f (x , t),
u|x=0 = ν(t), u|x=` = µ(t), ⇒
{
∂2w
∂x2
= − 1
a2
f (x),
u|x=0 = A, u|x=` = B,
§>.^⇒àg§Úàg>.^
úúúôôôÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg>>>...^^^ffffff½½½)))¯¯¯KKK
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∂2u
∂t2
= a2 ∂
2u
∂x2
+ A, 0 < x < `, t > 0
u|x=0 = 0, u|x=` = B,
u(x , t)|t=0 = 0, ∂u∂t |t=0 = 0, 0 < x < `.
Ù¥ A Ú B ½ff~ê"{
∂2w
∂x2
= − A
a2
,
u|x=0 = 0, u|x=` = B, ⇒ w(x , t) = −
A
2a2
x2 + (
A`
2a2
+
B
`
)x
P U(x , t) = u(x , t)− w(x , t), @o
∂2U
∂t2
= a2 ∂
2U
∂x2
, ⇔àg§
U|x=0 = 0, U|x=` = 0, ⇔àg>.^
U(x , t)|t=0 = A2a2 x2 − ( A`2a2 + B` )x , 0 < x < `
∂U
∂t |t=0 = 0, 0 < x < `.
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àààggg>>>...^^^ffffff½½½)))¯¯¯KKK
àg>.^ff½)¯K
©lCþ{⇒
U(x , t) =
∞∑
n=1
(
Cn cos(
npiat
`
) + Dn sin(
npiat
`
)
)
sin(
npix
`
)
U(x , t)|t=0 = A
2a2
x2 − ( A`
2a2
+
B
`
)x =
∞∑
n=1
Cn sin(
npix
`
)
∂U
∂t
|t=0 = 0 =
∞∑
n=1
Dn
npi
`
sin(
npix
`
)
⇒ Dn = 0,
Cn =
2
`
∫ `
0
[
A
2a2
x2 − ( A`
2a2
+ B` )x
]
sin(npix` )dx
= − 2A`2
a2n3pi3
+ (−1)n 2npi
(
A`2
a2n2pi2
+ B
)
Ïd�5Ð>Ł¯Kff)
u(x , t) = − A
2a2
x2 + (
A`
2a2
+
B
`
)x +
∞∑
n=1
Cn cos(
npiat
`
) sin(
npix
`
).
úúúôôôÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg>>>...^^^ffffff½½½)))¯¯¯KKK
非齐次边界条件的定解问题