Environmental Engineering Reference
In-Depth Information
and
c
k
are constant, the approximation error
√
Δ
n
will reach its minimum
As
(
f
,
f
)
at
a
k
=
c
k
,
n
k
=
1
c
k
≥
0
,
2
=
−
min
Δ
f
(2.25)
n
which is called the
Bessel inequality
.
Therefore, the generalized polynomial constructed by the Fourier coefficient
c
k
n
k
=
1
c
k
e
k
(
x
)
,
c
k
=(
f
,
e
k
(
x
))
f
n
(
x
)=
(
)
{
e
k
(
)
}
is the best approximation of
f
x
for fixed
n
and
x
,i.e.
f
2
l
n
k
=
1
c
k
e
k
(
x
)
=
(
)
−
,
c
k
=(
,
e
k
(
))
.
min
Δ
x
d
x
f
x
n
0
When lim
n
→
∞
min
Δ
n
=
0,
x
∈
[
0
,
l
]
, in particular, let
∞
k
=
1
c
k
e
k
(
x
)
,
f
(
x
)=
(2.26)
which is called the
generalized Fourier expansion
. The right-hand side of Eq. (2.26)
converges to
f
(
x
)
in an average sense. Also
∞
k
=
1
c
k
=
f
2
.
(2.27)
It is this elegant convergence as well as its weak demand for the smoothness of
ϕ
(
x
)
and
ψ
(
x
)
that popularize the nominal solution and its application.
2.4.5 PDS with Variable Coefficients
When the coefficients in Eq. (1.2) are variable, it is very difficult to find analytical
solutions of its PDS. Numerical methods are normally used to obtain numerical so-
lutions. Here we attempt to find the analytical solution of a mixed problem of hyper-
bolic equations with variable coefficients by transforming the PDS to an eigenvalue
problem of integral equations. We detail the procedure by considering the following
example.
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