Environmental Engineering Reference
In-Depth Information
The integrand of (6.199) contains power series that are quickly convergent. The
integral (6.199) can thus be obtained very easily by using integration term by term.
The
g
2
k
(
t
)
in Eq. (6.200) can also be obtained quite easily.
Polynomials of Zero- and First-Order
1
τ
0
. Thus
t
e
−
By Eq. (6.176), for
P
1
(
x
)=
a
0
+
a
1
x
,
u
0
(
x
,
t
)=
τ
0
P
1
(
x
)
−
u
1
=
S
(
u
0
)=
0
.
The solution of PDS (6.198) is thus
1
τ
0
t
e
−
u
=
W
P
1
(
x
,
t
)=
u
0
=
τ
0
(
a
0
+
a
1
x
)
−
.
(6.201)
This shows that
u
txx
-term in (6.198) has no effect on the solution for an initial value
of type
P
1
(
x
)
. Its solution is the same as that of PDS (6.169).
Polynomials of Second- and Third-Order
a
2
x
2
a
3
x
3
For polynomials of second- or third- order,
P
3
(
x
)=
a
0
+
a
1
x
+
+
(
P
3
(
x
)
a
2
x
2
when
a
3
=
reduces to
P
2
(
x
)=
a
0
+
a
1
x
+
0). By Eq. (6.176), we have
1
τ
0
P
3
(
1
2
A
x
)
t
t
e
−
e
−
u
0
(
x
,
t
)=
τ
0
P
3
(
x
)
−
+
G
2
(
t
)
2
τ
0
,
2!
t
x
+
A
(
t
−
τ
)
1
2
A
t
−
τ
2
e
−
τ
0
d
e
−
u
1
=
S
(
u
0
)=
τ
0
d
τ
I
0
·
(
2
a
2
+
6
a
3
ξ
)
ξ
0
x
−
A
(
t
−
τ
)
τ
0
t
e
−
=
τ
0
P
3
(
x
)
τ
0
−
(
τ
0
+
t
)
τ
0
t
e
−
=
τ
0
(
2
a
2
+
6
a
3
x
)
τ
0
−
(
τ
0
+
t
)
.
Thus, the solution of (6.198) is
u
=
u
0
+
S
(
u
0
)
ε
a
3
x
3
1
τ
0
0
a
0
1
2
A
2
a
2
+
6
a
3
x
2!
t
t
e
−
e
−
a
2
x
2
=
τ
+
a
1
x
+
+
−
+
G
2
(
t
)
2
τ
0
τ
0
−
(
τ
0
+
τ
0
t
e
−
+
τ
0
(
2
a
2
+
6
a
3
x
)
t
)
ε
.
(6.202)
This shows that the effect of
u
txx
-term is
x
-independent for the case of
a
3
=
0and
2
increases towards 2
a
2
τ
0
as
t
→
∞
.
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