Environmental Engineering Reference
In-Depth Information
the rule of writing the Green function, we obtain
+
∞
∑
τ
0
γ
mnl
M
mnl
sin
m
1
πξ
l
1
sin
m
π
x
t
−
τ
2τ
0
e
−
G
(
x
,
ξ
;
y
,
η
;
z
,
ζ
;
t
−
τ
)=
l
1
m
,
n
,
l
=
1
(4.55)
cos
μ
l
ζ
l
3
cos
μ
l
z
l
3
sin
μ
η
l
2
sin
μ
n
y
l
2
n
·
sin
γ
mnl
(
t
−
τ
)
where
M
mnl
=
γ
mnl
are determined by Eq. 4.53. Therefore, we have the
solution of PDS (4.54) at
M
m
M
n
M
l
,
ϕ
=
ψ
=
0.
t
u
3
=
d
τ
G
(
x
,
ξ
;
y
,
η
;
z
,
ζ
;
t
−
τ
)
f
(
ξ
,
η
,
ζ
,
τ
)
d
v
.
(4.56)
0
Ω
2. By the rule for writing
W
ψ
(
from the
G
in Section 4.3.1 (Remark 1), we
can obtain the solution of PDS (4.54) for the case
f
x
,
y
,
z
,
t
)
=
ϕ
=
0,
⎧
⎨
u
2
=
W
ψ
(
x
,
y
,
z
,
t
)
+
∞
∑
m
,
n
,
l
=
1
cos
μ
l
z
l
3
b
mnl
sin
m
π
x
sin
μ
n
y
l
2
t
e
−
=
2
τ
0
sin
γ
mnl
t
,
l
1
(4.57)
⎩
cos
μ
l
z
l
3
1
γ
mnl
M
mnl
sin
m
π
x
sin
μ
n
y
l
2
b
mnl
=
ψ
(
x
,
y
,
z
)
d
v
.
l
1
Ω
3. We can obtain
W
ϕ
(
x
,
y
,
z
,
t
)
by replacing
ψ
in Eq. (4.57) by
ϕ
. The solution of
PDS (4.54) at
f
=
ψ
=
0 reads, by the solution structure theorem,
1
τ
W
ϕ
(
0
+
∂
u
1
=
x
,
y
,
z
,
t
)
.
(4.58)
∂
t
4. The solution of PDS (4.54) is, by the principle of superposition
u
(
x
,
y
,
t
)=
u
1
(
x
,
y
,
t
)+
u
2
(
x
,
y
,
t
)+
u
3
(
x
,
y
,
t
)
.
Remark.
By the multiple separation of variables for solving PDS (4.52), the first
separation of variables separates
t
from
x
,
y
and
z
and yields
0
T
+
T
+
λ
a
2
T
τ
=
0
,
(4.59)
is the separation constant. Another two separations of variables lead to
three eigenvalue groups
where
−
λ
λ
m
,
λ
n
,
λ
l
and three eigenfunction sets
X
m
(
x
)
,
Y
n
(
y
)
and
Z
l
(
z
)
.The
λ
in Eq. 4.59 is
λ
=
λ
m
+
λ
n
+
λ
l
, which is true only for some special
cases and is not universal.
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