Math 124A – December 02, 2010
±
Viktor Grigoryan
18
Separation of variables: Neumann conditions
The same method of separation of variables that we discussed last time for boundary problems with
Dirichlet conditions can be applied to problems with Neumann, and more generally, Robin boundary
conditions. We illustrate this in the case of Neumann conditions for the wave and heat equations on the
ﬁnite interval.
Substituting the separated solution
u
(
x,t
) =
X
(
x
)
T
(
t
) into the wave Neumann problem
(
u
tt

c
2
u
xx
= 0
,
0
< x < l,
u
(
x,
0) =
φ
(
x
)
,
u
t
(
x,
0) =
ψ
(
x
)
,
u
x
(0
,t
) =
u
x
(
l,t
) = 0
,
(1)
gives the same equations for
X
and
T
as in the Dirichlet case,

X
00
=
λX,
and

T
00
=
cλ
2
T.
However, the boundary conditions now imply
X
0
(0)
T
(
t
) =
X
0
(
l
)
T
(
t
) = 0
,
∀
t
⇒
X
0
(0) =
X
0
(
l
) = 0
.
To ﬁnd all the separated solutions, we need to ﬁnd all the eigenvalues and eigenfunctions satisfying these
boundary conditions. To do this, we need to consider the cases
λ
= 0,
λ <
0 and
λ >
0 separately.
Assume
λ
= 0, then the equation for
X
is
X
00
= 0, which has the solution
X
(
x
) =
C
+
Dx
. The
derivative is then
X
0
(
x
) =
D
, and the boundary conditions imply that
D
= 0. So every constant
function,
X
(
x
) =
C
, is an eigenfunction for the eigenvalue
λ
0
= 0.
Next, we assume that
λ
=

γ
2
<
0, in which case the equation for
X
takes the form
X
00
=
γ
2
X.
The solution to this equation is
X
(
x
) =
Ce
γx
+
De
γx
, so
X
0
(
x
) =
Cγe
γx

Dγe

γx
. Checking the
boundary conditions gives
±
Cγ

Dγ
= 0
Cγe
γl

Dγe

γl
= 0
⇒
±
C
=
D
Cγ
(
e
2
γl

1) = 0
⇒
C
=
D
= 0
,
since
γ
6
= 0, and hence, also
e
2
γl

1
6
= 0. This leads to the identically zero solution