Example
−
∂
2
u
∂
x
2
=
−
∂
u
∂
t
,
(1)
u
= 0
at
x
= 0,1
(2)
u
=1
at
t
= 0
(3)
From the general discussion of the Sturm-Liouville approach, one obtains
u
(
x
,
t
)
=
∞
∑
n
=1
(
f
,
e
n
) exp(−λ
n
t
)
e
n
(
x
)
=
∞
∑
n
=1
(1, √2 sin
n
π
x
) exp(−
n
2
π
2
t
) √2 sin
n
π
x
=
∞
∑
n
=1
2 (1−(−1)
n
)
n
π
exp(−
n
2
π
2
t
) sin
n
π
x
.
(4)
Here is how the temperature decays.
Laplace Equations
∂
2
u
∂
x
2
+
∂
2
u
∂
y
2
=
0
in
D
,
(5)
u
(0,
y
)
=
0 (0 <
y
< 1)
(6)
u
(1,
y
)
=
f
(
y
) (0 <
y
< 1)
(7)
u
(
x
,0)
=
u
(
x
,1) = 0 (0 <
x
< 1)
(8)
Here is an example of the profile of
u
(
x
,
y
) when
f
(
y
) = 1.
File translated from T
E
X by
T
T
H
, version 4.03.
On 18 Jan 2021, 22:17.
