Environmental Engineering Reference
In-Depth Information
Applying the boundary condition
u
(
x
,
y
,
0
)=
ϕ
(
x
,
y
)
leads to
m
,
n
=
1
b
mn
sh
λ
mn
a
·
sin
m
π
x
∞
sin
n
π
y
a
=
ϕ
(
x
,
y
)
,
a
so that the
b
mn
can be determined. Finally, we obtain the solution of PDS (7.5)
⎧
⎨
m
,
n
=
1
b
mn
sh
m
a
2
n
2
∞
a
sin
m
π
x
sin
n
π
y
a
,
u
(
x
,
y
,
z
)=
+
(
a
−
z
)
a
a
2
a
d
x
a
1
4
sin
m
π
x
sin
n
π
y
b
mn
=
sh
a
m
2
0
ϕ
(
x
,
y
)
d
y
.
⎩
2
n
a
a
a
a
0
+
Example 3.
Solve
⎧
⎨
Δ
T
=
0
,
0
<
x
,
y
<
a
,
T
(
0
,
y
)=
T
(
a
,
y
)=
0
,
(7.8)
⎩
100sin
π
a
T
(
x
,
0
)=
0
,
T
(
x
,
a
)=
.
Solution.
Based on the given boundary conditions of
x
-side, consider
∞
m
=
1
Y
m
(
y
)
sin
m
π
x
T
(
x
,
y
)=
a
,
where the
Y
m
(
y
)
is a undetermined function. Substituting it into the equation in
PDS (7.8) yields
m
2
a
Y
m
(
y
)
−
Y
m
(
y
)=
0
,
Y
m
(
0
)=
0
.
Its general solution can be written as
a
m
ch
m
y
a
+
π
b
m
sh
m
π
y
a
,
Y
m
(
y
)=
where the
a
m
and the
b
m
are constants. Applying
Y
m
(
0
)=
0 yields
a
m
=
0. Thus
∞
m
=
1
b
m
sh
m
π
y
sin
m
π
x
a
.
T
(
x
,
y
)=
a
100sin
π
a
yields
Applying the boundary condition
T
(
x
,
a
)=
∞
m
=
1
(
b
m
sh
m
π
)
sin
m
π
x
100sin
π
x
a
.
a
=
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