Environmental Engineering Reference
In-Depth Information
bination of boundary conditions be
X
m
(
x
)
Y
n
(
y
)
Z
l
(
z
)
. The solution of PDS (4.52) at
ϕ
=
f
=
0 can thus be written as
=
∑
u
T
mnl
(
t
)
X
m
(
x
)
Y
n
(
y
)
Z
l
(
z
)
.
Substituting it into the equation of PDS (4.52) leads to
τ
0
T
mnl
+
T
mnl
+
a
2
(
λ
m
+
λ
n
+
λ
l
)
T
mnl
=
0
,
where
λ
m
,
λ
n
and
λ
l
are the corresponding eigenvalues of
{
X
m
(
x
)
}
,
{
Y
n
(
y
)
}
and
{
Z
l
(
)
}
z
, respectively. Its general solution is
t
e
−
T
mnl
(
t
)=
2
τ
0
(
a
mnl
cos
γ
mnl
t
+
b
mnl
sin
γ
mnl
t
)
,
where
4
1
2
γ
mnl
=
τ
0
a
2
(
λ
m
+
λ
n
+
λ
l
)
−
1
.
(4.53)
τ
0
The solution of PDS (4.52) can then be obtained using a similar approach to that in
Example 3 in Section 4.3.1.
Example 1.
Solve
⎧
⎨
a
2
u
t
+
τ
0
u
tt
=
Δ
u
+
f
(
x
,
y
,
z
,
t
)
,
Ω
×
(
0
,
+
∞
)
,
u
(
0
,
y
,
z
,
t
)=
u
(
l
1
,
y
,
z
,
t
)=
0
,
u
(
x
,
0
,
z
,
t
)=
u
y
(
x
,
l
2
,
z
,
t
)+
hu
(
x
,
l
2
,
z
,
t
)=
0
,
(4.54)
⎩
u
z
(
x
,
y
,
0
,
t
)=
u
z
(
x
,
y
,
l
3
,
t
)+
hu
(
x
,
y
,
l
3
,
z
,
t
)=
0
,
u
(
x
,
y
,
z
,
0
)=
ϕ
(
x
,
y
,
z
)
,
u
t
(
x
,
y
,
z
,
0
)=
ψ
(
x
,
y
,
z
)
.
Solution.
1. We first seek the Green function
G
. Based on the given boundary conditions, the
appropriate eigenfunction sets are those in Rows 1, 3 and 6 in Table 2.1. Let
u
ijk
(
x
,
y
,
z
)(
i
,
j
,
k
=
1
,
2
, ··· ,
9
)
denote a complete and orthogonal eigenfunction
set of 729 in total. Here we use
cos
μ
l
z
sin
m
π
x
sin
μ
n
y
l
2
u
136
(
x
,
y
,
z
)=
l
3
,
l
1
m
2
μ
n
l
2
2
μ
l
l
3
2
π
l
1
where the eigenvalues
λ
m
=
,
λ
n
=
,
λ
l
=
. By following
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