Environmental Engineering Reference
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which is well defined and indeed meaningful. It is therefore necessary to extend the
concept of solutions from the classical one to the generalized one. Such an extension
is not arbitrary and must follow a few rules. First, the classical solution must be, if
it exists, the generalized solution. Second, the generalized solution must be unique
and stable. Under these rules, we can define the generalized solution using different
methods, for example by introducing conjugate operators. In solving PDS by using
the generalized Fourier method of expansion,
appear only in the in-
tegrand, so the demand for their smoothness is very weak. The solution obtained
by this method is thus a
generalized solution
,the
solution
for short. All solutions
of PDS in this topic refer to this kind of solutions, unless otherwise stated. There
is no essential distinction between the Fourier method and the method of separa-
tion of variables. The generalized solutions from these methods are called
nominal
solutions
.
An analysis of the convergence of such nominal solutions will benefit our ap-
preciation of their significance and relevance in applications. By normalizing eigen-
function set
ϕ
(
x
)
and
ψ
(
x
)
{
X
k
(
x
)
}
, we have a complete orthonormal basis
{
e
k
(
x
)
}
in
[
0
,
l
]
,i.e.
X
k
(
x
)
L
2
e
k
(
x
)=
X
k
,
X
k
(
x
)
∈
[
0
,
l
]
.
L
2
Consider an approximation of function
f
(
x
)
∈
[
0
,
l
]
by a generalized polynomial
n
k
=
1
a
k
e
k
(
x
)
,
f
n
(
x
)=
(2.24)
where
a
k
are constant coefficients. The error square of the approximation is
l
0
[
2
d
x
Δ
n
=
f
(
x
)
−
f
n
(
x
)]
=(
f
−
f
n
,
f
−
f
n
)
=(
f
,
f
)
−
2
(
f
,
f
n
)+(
f
n
,
f
n
)
,
also,
f
n
k
=
1
a
k
e
k
(
x
)
n
k
=
1
a
k
c
k
,
(
f
,
f
n
)=
,
=
n
k
=
1
a
k
e
k
(
x
)
,
n
k
=
1
a
k
e
k
(
x
)
(
f
n
,
f
n
)=
n
i
=
1
n
j
=
1
a
i
a
j
(
e
i
(
x
)
,
e
j
(
x
)) =
n
k
=
1
a
k
,
=
l
where
c
k
=
f
(
x
)
e
k
(
x
)
d
x
are the Fourier coefficients. Therefore,
0
n
k
=
1
a
k
c
k
+
n
k
=
1
a
k
=(
f
,
f
)+
n
k
=
1
(
c
k
−
a
k
)
n
k
=
1
c
k
.
2
=(
,
)
−
−
Δ
f
f
2
n
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