Environmental Engineering Reference
In-Depth Information
Remark.
Solutions and Green functions can also be readily obtained for the other
kinds of boundary conditions based on Table 2.1.
3.2.2 Two-Dimensional Mixed Problems
Rectangular Domain
The solutions can be readily obtained, based on Table 2.1, for mixed problems sub-
jected to various boundary conditions. Here, we illustrate this by finding the solution
of
⎧
⎨
a
2
u
t
=
Δ
u
+
f
(
x
,
y
,
t
)
,
0
<
x
<
l
1
,
0
<
y
<
l
2
,
0
<
t
,
u
(
0
,
y
,
t
)=
u
(
l
1
,
y
,
t
)+
h
2
u
x
(
l
1
,
y
,
t
)=
0
,
⎩
u
y
(
x
,
0
,
t
)
−
h
1
u
(
x
,
0
,
t
)=
u
y
(
x
,
l
2
,
t
)=
0
,
u
(
x
,
y
,
0
)=
ϕ
(
x
,
y
)
.
Solution.
By the solution structure theorem, we first develop
W
ϕ
(
x
,
y
,
t
)
, the solution
for the case of
f
0. Based on the given boundary conditions, we should
use the eigenfunctions in Rows 3 and 8 in Table 2.1 to expand the solution. From
Section 2.5.1, we have
⎧
⎨
⎩
(
x
,
y
,
t
)=
sin
μ
n
y
l
2
+
ϕ
n
+
∞
m
,
n
=
1
b
mn
e
−
(
ω
mn
a
)
2
t
sin
μ
m
x
u
=
W
ϕ
(
x
,
y
,
t
)=
,
l
1
sin
μ
n
y
n
d
x
d
y
l
2
l
1
1
M
mn
sin
μ
m
x
l
1
=
ϕ
(
,
)
l
2
+
ϕ
,
b
mn
x
y
0
0
x
l
1
h
2
μ
n
are the positive zero points of
f
where
μ
m
,
(
x
)=
tan
x
+
and
g
(
x
)=
x
l
2
h
1
, respectively;
M
mn
is the product of normal squares of two sets of eigen-
functions. Also,
cot
x
−
a
2
μ
2
2
μ
n
l
2
ϕ
n
=
μ
n
m
l
1
mn
tan
l
2
h
1
,
ω
=
+
.
Therefore, the solution is, by the solution structure theorem,
t
u
=
W
ϕ
(
x
,
y
,
t
)+
W
f
τ
(
x
,
y
,
t
−
τ
)
d
τ
,
0
where
f
τ
=
f
(
x
,
y
,
t
−
τ
)
.
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