MATH 220: SEPARATION OF VARIABLES
Separation of variables is a method to solve certain PDEs which have a ‘warped
product’ structure.
The general idea is the following: suppose we have a linear
PDE
Lu
= 0 on a space
M
x
×
N
y
. We look for solutions
u
(
x,y
) =
X
(
x
)
Y
(
y
). In
general, there are no nontrivial solutions (the identically 0 function being trivial),
but in special cases we might be able to find some. We cannot expect even then
that all solutions of the PDE are of this form. However, if we have a family
u
n
(
x,y
) =
X
n
(
x
)
Y
n
(
y
)
, n
∈ I
,
of separated solutions, where
I
is some index set (e.g. the positive integers), then,
this being a linear PDE,
u
(
x,y
) =
summationdisplay
n
∈I
c
n
u
n
(
x,y
) =
summationdisplay
n
∈I
c
n
X
n
(
x
)
Y
n
(
y
)
solves the PDE as well in a distributional sense for any constants
c
n
∈
C
,
n
∈ I
,
provided the sum converges in distributions, and we may be able to choose the
constants so that this in fact gives an arbitrary solution of the PDE.
We emphasize that our endeavor, in general, is very unreasonable. Thus, we may
make assumptions as we find it fit – we need to justify our results
after
we derive
them.
As an example, consider the wave equation
u
tt
−
c
2
Δ
x
u
= 0
on
M
x
×
R
t
, where
M
is the space – for instance,
M
is
R
n
, or a cube [
a,b
]
n
or a ball
B
n
=
{
x
∈
R
n
:

x
 ≤
1
}
.
A separated solution is one of the form
u
(
x,t
) =
X
(
x
)
T
(
t
). Substituting into the PDE yields
X
(
x
)
T
′′
(
t
)
−
c
2
T
(
t
)(Δ
x
X
)(
x
) = 0
.
Rearranging, and assuming
T
and
X
do not vanish,
T
′′
(
t
)
c
2
T
(
t
)
=
Δ
x
X
(
x
)
X
(
x
)
.
Now, the left hand side is a function independent of
x
, the right hand side is a
function independent of
t
, so they are both equal to a constant,
−
λ
, namely pick
your favorite value of
x
0
and
t
0
, and then for any
x
and
t
,
RHS(
x
) = LHS(
t
0
) = RHS(
x
0
) = LHS(
t
)
,
so the constant in question is LHS(
t
0
). Thus, we get two ODEs:
T
′′
(
t
) =
−
λc
2
T
(
t
)
,
(Δ
x
X
)(
x
) =
−
λX
(
x
)
.
Now typically one has additional conditions.
For instance, one has boundary
conditions at
∂M
:
u

∂M
×
R
= 0 (DBC) or
∂u
∂n

∂M
×
R
= 0 (NBC)
.
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