Environmental Engineering Reference
In-Depth Information
Thus
ψ
(
ω
)
β
(
ω
)
¯
e
α
(
ω
)
t
sin
u
(
ω
,
t
)=
β
(
ω
)
t
.
Thus its inverse Fourier transformation yields the solution of PDS (6.164)
1
e
i
(
ω
1
x
+
ω
2
y
+
ω
3
z
)
d
u
(
M
,
t
)=
u
(
ω
,
t
)
ω
1
d
ω
2
d
ω
3
.
3
(
2
π
)
R
3
The one-dimensional
W
ψ
(
M
,
t
)
satisfies
⎧
⎨
u
t
τ
0
+
A
2
u
xx
+
B
2
u
txx
,
R
1
u
tt
=
×
(
0
,
+
∞
)
,
(6.166)
⎩
u
(
x
,
0
)=
0
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
Its Laplace transformation with respect to
t
yields
s
τ
0
u
s
2
u
A
2
u
xx
B
2
s u
xx
(
,
)+
(
,
)
−
ψ
(
)=
(
,
)+
x
s
x
s
x
x
s
or
s
2
u
B
2
s
A
2
u
xx
(
s
τ
0
+
x
,
s
)+
+
(
x
,
s
)=
−
ψ
(
x
)
,
so that
s
τ
0
s
2
+
)=
−
ψ
(
x
)
u
xx
(
x
,
s
)+
A
2
u
(
x
,
s
A
2
.
B
2
s
+
B
2
s
+
An integral expression of
W
ψ
(
M
,
t
)
can thus be found by inverse Laplace transfor-
mation (see Section 5.2.2).
=
α
τ
0
=
α
Remark 3
. In PDS (6.164),
A
2
and
B
2
τ
0
·
τ
T
.Since
α
,
τ
0
and
τ
T
are
all normally very small, we have
B
2
A
2
and 0
B
2
<
1. Also the solution of
PDS (6.166) is known at
B
0 (see Section 5.2). Hence we can obtain an approxi-
mate analytical solution of PDS (6.166) by using a perturbationmethod with respect
to
B
2
.
=
6.9 Perturbation Method for Cauchy Problems
While PDS (6.164) can be solved by methods of integral transformation, the inverse
transformations are normally quite involved. The perturbation method is an effec-
tive method of obtaining approximate analytical solutions and can also be used to
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