Environmental Engineering Reference
In-Depth Information
where
a
2
(
2
2
μ
n
b
1
2
μ
l
c
1
2
m
+
1
)
π
mnl
=
+
+
.
ω
2
a
1
The generalized Fourier coefficients
C
mnl
can be obtained by using the initial
conditions
u
, the completeness and the orthogonality of the
eigenfunction set. Finally, we have
⎧
⎨
(
x
,
y
,
z
,
0
)=
ϕ
(
x
,
y
,
z
)
sin
μ
l
z
c
1
+
ϕ
l
+
∞
∑
mnl
t
sin
(
2
m
+
1
)
π
x
cos
μ
n
y
b
1
2
C
mnl
e
−
ω
=
W
ϕ
(
,
,
,
)=
,
u
x
y
z
t
2
a
1
m
=
0
,
n
,
l
=
1
sin
μ
l
z
c
1
+
ϕ
l
d
1
M
mnl
sin
(
2
m
+
1
)
π
x
cos
μ
n
y
b
1
⎩
C
mnl
=
ϕ
(
x
,
y
,
z
)
Ω
,
2
a
1
Ω
(3.15)
where
M
mnl
=
M
m
M
n
M
l
is the product of normal squares of three sets of eigenfunc-
μ
l
and
tions, and
ϕ
l
are referred to Table 2.1.
Therefore, the solution of PDS (3.12) subject to the boundary conditions (3.13)
is, by Eq. (3.15) and the solution structure theorem,
μ
n
,
t
u
=
W
ϕ
(
x
,
y
,
z
,
t
)+
W
f
τ
(
x
,
y
,
z
,
t
−
τ
)
d
τ
,
0
where
f
τ
=
f
(
x
,
y
,
z
,
τ
)
.
Spherical Domain
Find the solution of PDS
⎧
⎨
a
2
x
2
y
2
z
2
a
0
,
u
t
=
Δ
u
+
f
(
x
,
y
,
z
,
t
)
,
+
+
<
0
<
t
,
L
(
u
,
u
n
)
|
x
2
+
y
2
+
z
2
=
a
0
=
0
,
(3.16)
⎩
u
(
x
,
y
,
z
,
0
)=
ϕ
(
x
,
y
,
z
)
.
Search WWH ::
Custom Search