Environmental Engineering Reference
In-Depth Information
size. Therefore
{
X
k
(
x
)
}
forms a complete and orthogonal set in
[
a
,
b
]
,and
L
2
N
k
=
1
c
k
X
k
(
x
)
lim
N
→
∞
f
(
x
)
−
=
0
.
[
a
,
b
]
2.4 Well-Posedness and Generalized Solution
In this section, we first attempt to establish the existence, uniqueness and stability
of the solution with respect to the initial conditions and followed by a discussion of
the necessity, importance and relevance of introducing generalized solutions. In dis-
cussing well-posedness, we take the initial condition-driven free vibration of a string
of two fixed ends as an example. Therefore, we will address the well-posedness of
⎧
⎨
=
∂
W
ϕ
∂
u
t
+
W
ψ
(
x
,
t
)
a
k
cos
k
π
at
l
+
b
k
sin
k
π
at
sin
k
π
x
∞
k
=
1
(
=
)
,
l
l
(2.20)
⎩
2
l
0
ϕ
(
sin
k
π
x
a
k
=
x
)
d
x
,
l
2
a
0
ψ
(
sin
k
π
x
b
k
=
x
)
d
x
k
π
l
for
⎧
⎨
a
2
u
xx
,
u
tt
=
(
0
,
l
)
×
(
0
,
+
∞
)
u
(
0
,
t
)=
0
,
u
(
l
,
t
)=
0
,
(2.21)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
2.4.1 Existence
Although we have obtained the solution (2.20) for PDS (2.21), we still need to prove
its existence. In developing Eq. (2.20), we have taken the limit of the series solution
and its first and second-order derivatives with respect to
t
and
x
term by term. This
requires the uniform convergence of series (2.20) and the resulting series (from
taking derivatives term by term) in
[
0
,
l
]
×
[
0
,
T
]
,
t
∈
[
0
,
T
]
,were
T
is any positive
number. Therefore, we must impose some constraints on
ϕ
(
x
)
and
ψ
(
x
)
and some
consistency conditions at two ends.
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