Environmental Engineering Reference
In-Depth Information
7.2 Separation of Variables and Fourier Sin/Cos Transformation
The Fourier method of expansion is effective once the function base for expanding
the solution is available. It is not, however, straightforward to know such a base
for some boundary conditions. We must first apply separation of variables, in gen-
eral, to find eigenvalues and eigenfunctions which form a function base for solution
expansion. For some problems, it is more convenient to solve by a Fourier sin/cos
transformation in a finite region.
7.2.1 Separation of Variables
Example 1.
Solve the internal Dirichlet problem:
u
xx
+
x
2
y
2
a
2
u
yy
=
0
,
+
<
,
(7.11)
u
|
x
2
=
F
(
x
,
y
)
.
y
2
a
2
+
=
Solution.
Consider
x
=
r
cos
θ
,
y
=
r
sin
θ
. The PDS (7.11) becomes, in a polar
coordinate system
⎧
⎨
2
u
2
u
∂θ
∂
1
r
∂
u
r
2
∂
1
r
2
+
r
+
2
=
0
,
0
<
r
<
a
,
∂
∂
(7.12)
⎩
u
|
r
=
a
=
f
(
θ
)
,
f
(
θ
+
2
π
)=
f
(
θ
)
.
Let
u
=
R
(
r
)
Θ
(
θ
)
. Substituting it into the equation in PDS (7.12) yields
Θ
(
θ
)+
λΘ
(
θ
)=
0
,
(7.13)
Θ
(
θ
+
2
π
)=
Θ
(
θ
)
and
r
2
R
(
rR
(
r
)+
r
)
−
λ
R
(
r
)=
0
, |
R
(
0
)
| <
∞
,
(7.14)
where
is the separation constant.
By Eq. (7.13), we obtain the eigenvalues and the eigenfunctions
λ
n
2
λ
=
,
n
=
0
,
1
,
2
, ··· ,
Θ
n
(
θ
)=
C
n
cos
n
θ
+
D
n
sin
n
θ
,
where constants
C
n
and
D
n
are not all equal to zero.
Substituting
n
2
into Eq. (7.14) yields a homogeneous Euler equation
λ
=
r
2
R
(
rR
(
n
2
R
r
)+
r
)
−
(
r
)=
0
.
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