Environmental Engineering Reference
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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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