Environmental Engineering Reference
In-Depth Information
by the solution structure theorem,
1
W
ϕ
(
t
τ
0
+
∂
(
,
)=
,
)+
W
ψ
−
B
2
ϕ
(
,
)+
W
f
τ
(
,
−
τ
)
u
x
t
x
t
x
t
x
t
d
τ
∂
t
0
Remark 4.
1. The structure of
W
ψ
(
is invariant with the combination of boundary con-
ditions. The detailed equations of
x
,
t
)
λ
m
,
X
m
(
x
)
,and
β
m
depend, however, on the
boundary conditions. The
μ
m
also have different meanings in Rows 3, 6 and 9 of
Tab l e 2 . 1 .
2. We may follow the process in Section 6.2.2 to obtain the final detailed series
solution if so required. Take the solution in Example 3 as an example. We can
obtain detailed expressions of
W
ϕ
(
,
)
,
W
−
B
2
ϕ
(
,
)
and
W
f
τ
(
,
)
x
t
x
t
x
t
simply through
B
2
ϕ
(
replacing
ψ
(
x
)
in coefficients
B
m
of (6.51) by
ϕ
(
x
)
,−
x
)
and
f
(
x
,
τ
)
,re-
B
2
W
ϕ
(
.
3. The solution structure theorem developed in Section 6.1 is valid only for well-
posed CDS. The
spectively. Also,
W
ϕ
(
x
,
t
)=
W
ψ
(
x
,
t
)
−
x
,
t
)
ψ
−
B
2
must thus satisfy consistency conditions. In
PDS (6.50), for example, the boundary conditions must be satisfied for all
t
ϕ
(
x
)
and
ψ
(
x
)
≥
0.
ϕ
(
)=
ϕ
(
At
t
0. By taking the deriva-
tive of boundary conditions with respect to
t
, we arrive at
u
xt
(
=
0, we have
0
)
−
h
ϕ
(
0
l
)+
h
ϕ
(
l
)=
0
,
t
)
−
hu
t
(
0
,
t
)=
ψ
(
)=
ψ
(
0
,
u
xt
(
l
,
t
)+
hu
t
(
l
,
t
)=
0, so
ψ
(
x
)
must satisfy
0
)
−
h
ψ
(
0
l
)+
h
ψ
(
l
)=
0.
4. If
β
m
t
into
e
iβ
m
t
e
−
iβ
m
t
−
β
m
is purely imaginary for some
m
, we can change sin
2i
e
−
i
z
2i
for any imaginary
z
.
5. The general term of the series solution decays very quickly toward zero. This
facilitates its applications of taking only the first few terms and is deduced from
the following observations: (a) the appearance of
e
i
z
−
by using the formula sin
z
=
β
m
in the denominator
B
m
of
m
2
which increases very quickly with
m
(
β
m
=
O
(
)
as
m
→
+
∞
); (b) the integral
l
0
ψ
(
x
)
X
m
(
x
)
d
x
in Eq. (6.48) tends to zero, by the Riemann Lemma, as
m
→
+
∞
are either the sin or cos functions; (c) the appearance of e
α
m
t
(
)
because the
X
m
x
1
β
m
m
2
e
α
m
t
sin
where
α
m
<
0and
α
m
=
O
(
)
as
m
→
+
∞
, and (d) the
β
m
t
in the
general term. Depending on the characteristic roots of Eq. (6.41), we have
⎧
⎨
β
m
e
α
m
t
sin
1
β
m
t
,
when
Δ
<
0
,
1
β
m
t
e
α
m
t
e
α
m
t
sin
=
,
when
Δ
=
0
,
β
m
t
⎩
1
e
−
r
1
t
e
−
r
2
t
γ
m
(
−
)
,
when
Δ
>
0
.
2
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