Environmental Engineering Reference
In-Depth Information
By following the function transformation from Eq. (5.95) to Eq. (5.96) in Sec-
tion 5.7, we obtain
⎧
⎨
2
2
∂
(
r v
)
A
2
∂
(
r v
)
c
2
=
+
(
r v
)
,
0
<
r
<
+
∞
,
0
<
t
,
t
2
r
2
∂
∂
(5.118)
⎩
r v
|
t
=
0
=
0
,
(
r v
)
t
|
t
=
0
=
r
¯
ψ
(
r
)
,
r
2
S
r
1
M
)
where
¯
ψ
(
)=
ψ
(
ψ
(
¯
)=
ψ
(
)
.
r
d
S
,
lim
r
→
+
r
M
4
π
0
PDS (5.118) is one-dimensional; hence the method of spherical means reduces the
dimensions of the problem. To apply the results for one-dimensional Cauchy prob-
lems, we should make a continuation of initial values. Since
r
¯
ψ
(
r
)=
0at
r
=
0, we
should make an odd continuation, i.e.
r
¯
ψ
(
r
)
,
r
≥
0
(
r v
)
t
|
t
=
0
=
−
∞
<
r
<
+
∞
r
¯
ψ
(
−
r
)
,
r
<
0
,
By Eq. (5.30), we obtain the solution of PDS (5.118)
I
0
b
2
r
¯
r
+
At
1
2
A
r
)
d
r
2
r
)
r v
=
(
At
)
−
(
r
−
ψ
(
r
−
At
or
I
0
b
2
r
r
+
At
1
2
Ar
r
)
d
r
,
2
r
)
¯
v
=
(
At
)
−
(
r
−
ψ
(
r
−
At
1
2
a
√
τ
1
2
A
c
A
. By L'Hôpital's rule, we obtain
where
b
=
=
τ
0
=
0
v
(
M
,
t
)=
lim
0
v
r
→
+
I
0
b
2
r
¯
r
+
At
1
2
A
d
dr
r
)
d
r
2
r
)
=
lim
(
At
)
−
(
r
−
ψ
(
r
→
+
0
r
−
At
I
1
b
2
⎡
r
)
r
+
At
2
r
)
b
(
r
−
(
At
)
−
(
r
−
1
2
A
lim
⎣
r
¯
r
)
d
r
=
ψ
(
(
)
2
−
(
−
r
)
2
r
→
+
0
At
r
r
−
At
⎤
⎦
+(
+
)
ψ
(
¯
+
)
−
(
−
)
ψ
(
¯
−
)
r
At
r
At
r
At
r
At
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