1.
u
t
=
k
(
u
xx
+
u
yy
)
u
(
x, y,
0) =
f
(
x, y
)
u
=
X
(
x
)
Y
(
y
)
T
(
t
)
xY
˙
T
=
kY X
0
T
+
kXTY
0
˙
T
kT
=
X
0
X
+
Y
0
Y
=
−
λ
˙
T
+
λkT
=0
X
0
X
=
−
Y
0
Y
−
λ
=
−
µ
X
0
+
µX
=0
Y
0
+(
λ
−
µ
)
Y
=0
a.
X
(0) =
X
0
(
L
)=0
Y
(0) =
Y
(
H
)=0
⇒
X
n
=s
in
n
−
1
2
π
L
xµ
n
=
n
−
1
2
π
L
2
n
=1
,
2
,
···
⇒
Y
nm
=s
in
mπ
H
yλ
nm
−
µ
n
=
mπ
H
±
2
n
=1
,
2
,
···
m
=1
,
2
,
···
λ
nm
=
n
−
1
2
π
L
2
+
mπ
H
±
2
n, m
=1
,
2
,
···
T
nm
=
e
−
λ
nm
kt
u
(
x, y, t
)=
∞
X
n
=1
∞
X
n
=1
a
nm
e
−
λ
nm
kt
sin
n
−
1
2
±
π
L
x
sin
mπ
H
y
f
(
x, y
)=
u
(
x, y,
0) =
∞
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.
 Fall '08
 BELL,D

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