Environmental Engineering Reference
In-Depth Information
Substituting Eqs. (5.23) and (5.25) into (5.22) leads to
⎧
⎨
1
z
v
(
v
(
c
2
v
z
)+
z
)
−
(
z
)=
0
,
⎩
v
(
0
)=
1
.
ξ
,
Therefore,
v
depends
on
τ
,
ξ
and
t
only through
z
. By another variable transfor-
i
cz
i
=
√
−
1
, it becomes
⎧
⎨
mation
η
=
1
η
v
(
η
)+
v
(
η
)+
v
(
η
)=
0
,
⎩
v
(
0
)=
1
,
where the equation is the Bessel equation of order zero. Its solution is (see Ap-
pendix A)
J
0
i
c
2
I
0
c
2
v
ξ
,
ξ
,
τ
=
2
2
−
(
ξ
−
ξ
)
−
(
ξ
−
ξ
)
t
;
(
t
−
τ
)
=
(
t
−
τ
)
,
where
J
0
is the Bessel equation of order zero of the first kind.
I
0
is the modified
Bessel function of order zero of the first kind.
Find solutions of PDS (5.18) and (5.16)
The solution of PDS (
5.1
8) follows from Eq. (5.9) in Section 5.
1,
by noting that
f
ξ
√
τ
0
)
(
ξ
,
0and
u
τ
(
ξ
,
(
x
,
t
)=
0, d
τ
=
0on
PQ
,
u
0
)=
0
)=
ψ
(
a
,
I
0
c
t
2
2
ξ
+
t
ψ
a
ξ
√
τ
0
d
1
2
ξ
.
U
(
ξ
,
t
)=
−
(
ξ
−
ξ
)
ξ
−
t
The solution of PDS (5.17) is
t
2τ
0
U
e
−
u
(
ξ
,
t
)=
(
ξ
,
t
)
.
ξ
=
√
τ
/
Since
0
x
a
, the solution of PDS (5.16) is
u
=
W
ψ
(
x
,
t
)
⎛
a
−
ξ
2
⎞
√
τ
0
a
√
τ
0
x
a
d
(5.26)
ξ
√
τ
0
1
2
x
+
t
t
2τ
0
e
−
⎝
c
⎠
ψ
ξ
.
=
I
0
t
2
−
√
τ
0
a
x
−
t
Finally, the solution of PDS (5.15) is, by the solution structure theorem,
1
W
ϕ
(
t
τ
0
+
∂
u
=
x
,
t
)+
W
ψ
(
x
,
t
)+
W
f
τ
(
x
,
t
−
τ
)
d
τ
∂
t
0
,
τ
)
τ
0
.
where
f
τ
=
f
(
x
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