Environmental Engineering Reference
In-Depth Information
Its inverse transformation thus yields the solution of PDS (7.37)
∞
m
,
n
=
0
16
u
0
π
sh
ly
sh
l
sin
(
2
n
+
1
)
x
sin
(
2
m
+
1
)
z
u
(
x
,
y
,
z
)=
.
(7.38)
2
π
2
n
+
1
2
m
+
1
Remark 1
. If the region is
[
0
,
l
]
, the Fourier sine/cosine transformation of
f
(
x
)
be-
comes
⎧
⎨
l
sin
k
π
x
f
(
)=
(
)
,
k
f
x
d
x
l
0
∞
k
=
1
2
l
sin
k
π
x
⎩
f
f
(
x
)=
(
k
)
.
l
⎧
⎨
l
cos
k
π
x
f
(
k
)=
f
(
x
)
d
x
,
l
0
f
∞
k
=
1
⎩
(
0
)
2
l
cos
k
π
x
f
f
(
x
)=
+
(
k
)
.
π
l
Such a Fourier transformation works only for the case that the boundary conditions
at the two ends along one spatial direction are of the same type (either the first or
the second kind). Its application can remove the terms involving
u
xx
and thus reduce
the problem to one of ordinary differential equations. Such a transformation fails,
however, to work if the equation involves both a
u
xx
-term and a
u
x
-term.
Remark 2
. Consider a function
v
=
f
(
x
)
that must satisfy the conditions
v
x
|
x
=
0
=
0
,
(
v
x
+
hv
)
|
x
=
l
=
0
,
h
>
0
.
Based on the condition
v
x
|
x
=
0
=
0, we consider a cosine expansion of
f
(
x
)
)=
n
a
n
cos
p
n
x
,
f
(
x
(7.39)
where the
a
n
are constants and the
p
n
are determined to satisfy
(
v
x
+
hv
)
|
x
=
l
=
0so
that
n
(
−
p
n
sin
p
n
l
+
h
cos
p
n
l
)
a
n
=
0
.
Thus the
p
n
are the positive zero-roots of
p
tan
pl
−
h
=
0. The function set
{
cos
p
n
x
}
is complete and orthogonal in
[
0
,
l
]
. Also note that
⎧
⎨
0
,
p
m
=
p
n
,
l
cos
p
m
x
cos
p
n
x
d
x
=
p
n
+
h
2
l
(
)+
h
⎩
,
p
m
=
p
n
.
0
p
n
+
h
2
2
(
)
Thus the
a
n
can be determined by Eq. (7.39)
l
p
n
+
h
2
2
(
)
a
n
=
f
(
x
)
cos
p
n
x
d
x
.
(
p
n
+
h
2
)+
l
h
0
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