Math 124A – November 30, 2010
Viktor Grigoryan
17
Separation of variables: Dirichlet conditions
Earlier in the course we solved the Dirichlet problem for the wave equation on the finite interval 0
< x < l
using the reflection method. This required separating the domain (
x, t
)
∈
(0
, l
)
×
(0
,
∞
) into different
regions according to the number of reflections that the backward characteristic originating in the regions
undergo before reaching the
x
axis. In each of these regions the solution was given by a different ex
pression, which is impractical in applications, and the method does not generalize to higher dimensions
or other equations. We now study a different method of solving the boundary value problems on the
finite interval, called
separation of variables
.
Let us start by considering the wave equation on the finite interval with homogeneous Dirichlet con
ditions.
(
u
tt

c
2
u
xx
= 0
,
0
< x < l,
u
(
x,
0) =
φ
(
x
)
,
u
t
(
x,
0) =
ψ
(
x
)
,
u
(0
, t
) =
u
(
l, t
) = 0
.
(1)
The idea of the separation of variables method is to find the solution of the boundary value problem
as a linear combination of simpler solutions (compare this to finding the simpler solution
S
(
x, t
) of the
heat equation, and then expressing any other solution in terms of the heat kernel). The building blocks
in this case will be the
separated solutions
, which are the solutions that can be written as a product of
two functions, one of which depends only on
x
, and the other only on
t
, i.e.
u
(
x, t
) =
X
(
x
)
T
(
t
)
.
(2)
Let us try to find all the separated solutions of the wave equation. Substituting (2) into the equation gives
X
(
x
)
T
00
(
t
) =
c
2
X
00
(
x
)
T
(
t
)
.
Dividing both sides of these identity by

c
2
X
(
x
)
T
(
t
), we get

X
00
(
x
)
X
(
x
)
=

T
00
(
t
)
c
2
T
(
t
)
=
λ.
(3)
Clearly
λ
is a constant, since it is independent of
x
from
λ
=

T
00
/
(
c
2
T
), and is independent of
t
from
λ
=

X
00
/X
. We will shortly see that the boundary conditions force
λ
to be positive, so let
λ
=
β
2
, for
some
β >
0. One can then rewrite (3) as a pair of separate ODEs for
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 Fall '08
 Ponce
 Math, Differential Equations, Equations, Partial Differential Equations, Boundary value problem, Partial differential equation, Joseph Fourier, separated solutions

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