Environmental Engineering Reference
In-Depth Information
Finally, the solution of PDS (6.76) is, by the solution structure theorem,
1
W
ϕ
(
τ
0
+
∂
B
2
W
(
λ
m
+
λ
n
)
ϕ
(
(
,
,
)=
,
,
)+
,
,
)
u
x
y
t
x
y
t
x
y
t
∂
t
t
+
W
ψ
(
,
,
)+
W
f
τ
(
,
,
−
τ
)
τ
,
x
y
t
x
y
t
d
0
where
f
τ
=
f
(
x
,
y
,
τ
)
.
6.4.3 Three-Dimensional Mixed Problems
Let
∂Ω
thus consists of six boundary surfaces. 729 combinations of linear homogeneous
boundary conditions can be written in a general form as
Ω
be the cubic region: 0
<
x
<
l
1
,
0
<
y
<
l
2
,
0
<
z
<
l
3
. Its boundary
(
,
)
|
∂Ω
=
.
L
u
u
n
0
where the normal derivative
u
n
are
u
z
, respectively. The correspond-
ing eigenvalues and eigenfunctions are available in Table 2.1 and are denoted by
±
u
x
,
±
u
y
and
±
λ
m
,
X
m
(
x
)
;
λ
n
,
Y
n
(
y
)
;
λ
k
,
Z
k
(
z
)
.
Theorem 3
.Let
u
=
W
ψ
(
x
,
y
,
z
,
t
)
be the solution of
⎧
⎨
⎩
u
t
τ
0
+
B
2
∂
∂
A
2
=
+
,
u
tt
Δ
u
t
Δ
u
Ω
×
(
0
,
+
∞
)
,
(6.77)
L
(
u
,
u
n
)
|
∂Ω
=
0
,
u
(
x
,
y
,
z
,
0
)=
0
,
u
t
(
x
,
y
,
z
,
0
)=
ψ
(
x
,
y
,
z
)
.
The solution of
⎧
⎨
u
t
τ
0
+
B
2
∂
∂
A
2
u
tt
=
Δ
u
+
t
Δ
u
+
f
(
x
,
y
,
z
,
t
)
,
Ω
×
(
0
,
+
∞
)
,
(6.78)
⎩
L
(
u
,
u
n
)
|
∂Ω
=
0
,
u
(
x
,
y
,
z
,
0
)=
ϕ
(
x
,
y
,
z
)
,
u
t
(
x
,
y
,
z
,
0
)=
ψ
(
x
,
y
,
z
)
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