Environmental Engineering Reference
In-Depth Information
where
1
τ
0
+
λ
m
B
2
1
τ
0
+
λ
m
B
2
2
1
1
2
4
λ
m
A
2
λ
m
A
2
Δ
=
−
4
,
γ
m
=
−
m
B
2
,
1
τ
0
+
λ
r
1
=
α
m
+
γ
m
<
0
,
r
2
=
α
m
−
γ
m
<
0
.
m
2
.The
1
β
m
e
α
m
t
sin
Note that
|
sin
β
m
t
|≤
1and
γ
m
=
O
(
)
as
m
→
+
∞
β
m
t
thus
decays quickly for all cases.
6.4 Solution Structure Theorem: Another Form and Application
The solution structure theorem developed in Section 6.1 is valid only for well-posed
PDS. It requires that initial values
must satisfy consistency condi-
tions. This limits its applications because we only need nominal solutions for ap-
plications so that the given
ϕ
(
x
)
and
ψ
(
x
)
do not normally satisfy the consistency
conditions. In this section we improve the solution structure theorem in Cartesian
coordinates to find the structural relations among nominal solutions due to
ϕ
(
x
)
and
ψ
(
x
)
and
f
. We also use examples to show some applications of the modified solution struc-
ture theorem.
ϕ
,
ψ
6.4.1 One-Dimensional Mixed Problems
<
<
Let
Ω
be an one-dimensional region: 0
x
l
. Its boundary
∂Ω
thus consists of
=
=
the two ends point
x
0and
x
l
. Nine combinations of boundary conditions can
be written in a general form as
L
(
u
,
u
n
)
|
∂Ω
=
0
,
where the
u
n
are
±
u
x
. The corresponding eigenvalues and the eigenfunction set are
denoted by
λ
m
and
{
X
m
(
x
)
}
, respectively.
(
,
)=
W
ψ
(
,
)
Theorem 1
.Let
u
x
t
x
t
be the solution of
⎧
⎨
A
2
u
xx
+
B
2
u
txx
,
u
t
/
τ
0
+
u
tt
=
(
0
,
l
)
×
(
0
,
+
∞
)
,
L
(
u
,
u
n
)
|
∂Ω
=
0
,
(6.52)
⎩
u
(
x
,
0
)=
0
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
The solution of
⎧
⎨
A
2
u
xx
+
B
2
u
txx
+
u
t
/
τ
0
+
u
tt
=
f
(
x
,
t
)
,
(
0
,
l
)
×
(
0
,
+
∞
)
,
L
(
u
,
u
n
)
|
∂Ω
=
0
,
(6.53)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
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