Environmental Engineering Reference
In-Depth Information
t
t
0
τ
1
τ
0
d
t
+
τ
2τ
0
G
2
t
−
τ
e
−
e
−
·
(
−
τ
)
τ
+
(
τ
)
−
t
d
0
g
2
τ
ε
0
e
−
τ
0
t
2
2
t
2
0
2
+
τ
(
24
a
4
+
120
a
5
x
)
τ
0
−
τ
0
+
t
+
ε
.
(6.203)
τ
0
The solution due to
P
4
(
x
)
can be obtained by letting
a
5
=
0 in Eq. (6.203).
6.9.5 Perturbation Method for Two-
and Three-dimensional Problems
The perturbation method can also be applied to solve two- and three-dimensional
Cauchy problems. By the solution structure theorem, we can focus our attention
only on
⎧
⎨
∂
∂
A
2
R
2
or
R
3
u
t
/
τ
0
+
u
tt
=
Δ
u
+
ε
t
Δ
u
,
×
(
0
,
+
∞
)
×
(
0
,
+
∞
)
,
(6.204)
⎩
u
(
M
,
0
)=
0
,
u
t
(
M
,
0
)=
ψ
(
M
)
,
0
<
ε
1
.
Consider the perturbation solution
∞
n
=
0
u
n
(
M
,
t
)
ε
n
u
(
M
,
t
,
ε
)=
,
(6.205)
where
u
n
(
are undetermined functions.
Substituting Eq. (6.205) into the equation of PDS (6.204) and comparing coeffi-
cients of
M
,
t
)
n
(
n
ε
=
0
,
1
,
2
, ···
) yield
u
0
t
/
τ
0
+
A
2
u
0
tt
=
Δ
u
0
,
0
:
ε
(6.206)
u
0
(
M
,
0
)=
0
,
u
0
t
(
M
,
0
)=
ψ
(
M
)
.
⎧
⎨
u
1
+
∂
∂
A
2
u
1
t
/
τ
0
+
u
1
tt
=
Δ
t
Δ
u
0
,
1
:
ε
(6.207)
⎩
u
1
(
M
,
0
)=
0
,
u
1
t
(
M
,
0
)=
ψ
(
M
)
.
⎧
⎨
+
∂
∂
A
2
u
2
t
/
τ
+
u
2
tt
=
Δ
u
2
t
Δ
u
1
,
0
2
:
ε
(6.208)
⎩
(
,
)=
,
u
2
t
(
,
)=
ψ
(
)
.
u
2
M
0
0
M
0
M
···
···
···
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