Graphics Reference
In-Depth Information
21.3
From Laplace to Fourier
Before defining Fourier series, the reader unfamiliar with the subject may find some
motivation helpful. To motivate some of the central ideas we start with the Laplace
equation.
One way to solve the Laplace equation is by expressing it in polar coordinates. If
we do this, equation (21.1) becomes
∂
∂
u
r
Ê
Ë
ˆ
¯
∂
r
2
1
∂
∂
u
+-
=
0
,
r
π
0
.
(21.5)
∂
r
r
2
q
For simplicity assume that
()
the function u r
,
is continuous for
0
££
r
1
.
(21.6)
Note that u(r,q+2p) = u(r,q). These conditions are not enough to define u. Assume
that u satisfies the boundary condition
()
=
()
u
1, qq
f
(21.7a)
for some periodic function f(q) with
(
)
=
()
f
qp q
+
2
f
.
(21.7b)
The problem of finding a function u(r,q) that satisfies (21.5)-(21.7) is called the
Dirich-
let problem
. One attempt to solve this problem involves using the “method of separa-
tion of variables” and to look for a solution of the form
()
=
() ()
ur
,
q
RrH
q
.
This reduces the problem to solving two
ordinary
differential equation that have solu-
tions of the form
()
=
u
r
,
q
aA
,
n
=
0
0
,
n
=
(
)
(
)
n
i
n
q
-
i
n
q
ar
Ae
+
Be
,
n
>
,
for some constants a, A, and B. Since any linear combination of solutions is also a
solution, one is led to suppose the more general solution
•
Â
()
=
nn
i
q
ur
,q
a r e
(21.8)
n
n
=-•
with boundary condition
