Environmental Engineering Reference
In-Depth Information
4.4 Three-Dimensional Mixed Problems
We discuss mixed problems in cubic, cylinderical and spherical domains in this
section.
4.4.1 Cuboid Domain
Consider the PDS of type
⎧
⎨
a
2
u
t
+
τ
0
u
tt
=
Δ
u
+
f
(
x
,
y
,
z
,
t
)
,
Ω
×
(
0
,
+
∞
)
,
u
z
)
∂Ω
=
L
(
u
,
u
x
,
u
y
,
0
,
(4.52)
⎩
u
(
x
,
y
,
z
,
0
)=
ϕ
(
x
,
y
,
z
)
,
u
t
(
x
,
y
,
z
,
0
)=
ψ
(
x
,
y
,
z
)
,
where
Ω
stands for a cuboid domain: 0
<
x
<
l
1
,
0
<
y
<
l
2
,
0
<
z
<
l
3
,
∂Ω
the six
. If all combinations of the boundary conditions of all three
kinds are considered, for a cuboid domain
boundary surfaces of
Ω
, there exist 729 combinations.
The method of finding solutions of PDS (4.52) is the same as that in Example 3
in Section 4.3.1. It is crucial to find the Green function and
W
ψ
(
Ω
x
,
y
,
z
,
t
)
,thesolu-
tion of (4.52) at
0. Depending on the boundary conditions, there are a total
of 729 complete and orthogonal sets of eigenfunctions. All of them can be easily
written out by using Table 2.1. Let the eigenfunction set corresponding to a com-
ϕ
=
f
=
Tabl e 4 . 1
Eigenvalues and Eigenfunctions of Bessel equations
Boundary
Conditions
Eigenvalues
Eigen-
functions
Normal square
M
mn
n
=
0
,
1
,
2
, ···
m
=
1
,
2
, ···
(
n
m
a
0
2
μ
(
n
m
are the
m
-th
positive zero-point of
J
n
(
x
)
2
J
n
+
1
(
n
m
a
0
u
|
r
=
a
0
=
0
μ
J
n
(
)
k
mn
=
μ
(
n
m
a
0
k
mn
r
μ
⎡
⎣
1
2
⎤
⎦
J
n
n
μ
(
n
)
(
n
m
a
0
2
μ
(
n
m
are the positive
zero-point of
J
n
(
x
)
μ
(
n
m
a
0
2
u
r
|
r
=
a
0
=
0
μ
J
n
)
k
mn
=
μ
(
n
m
a
0
(
k
mn
r
−
μ
m
(
0
)
1
=
0
⎡
⎣
⎤
⎦
(
n
m
a
0
2
μ
(
n
m
are the positive
zero-point of
1
J
n
(
n
m
2
a
0
2
n
2
+
(
a
0
h
)
−
(
u
r
+
hu
)
|
r
=
a
0
=
0
μ
J
n
)
k
mn
=
μ
(
k
mn
r
1
μ
μ
(
n
m
2
(
n
m
a
0
a
0
xJ
n
(
x
)+
hJ
n
(
x
)
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