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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