Separation of Variables
The idea is just as the name suggests: start with an equation involving
more than one variable and perform manipulations that result in getting all
appeara nces of the variable t on one side of the equation and all
appearances of the other variables are on the other side.
Our equation is:
where Du represents the (spatial)
Laplacian operator. Since we want to separate the time dependence
from the spatial variables, we look for special solutions of the form
In this case, the left-hand side becomes
and the right-hand side becomes f(x,y)
Setting these equal and gathering the terms involving t
on one side and the ones involving x and y on the other, we
Now pick some fixed time t0 and allow x and
y to vary. Clearly, the right-hand side cannot change, because
v depends only on t and we have fixed that at
t0. Therefore, the right-hand side is a constant. We
call this constant -L, because we know
(in hindsight) that it has to be negative for our applications.
We have now separated the variables. We are looking for eigenfunctions of the Laplacian of f, each with
its own temporal companion v(t), and they turn out to be
either sine or cosine functions. We assemble these special solutions to
make the general one .
This is the essence of the argument showing how to break down the general
solution to our PDE problem into a superposition of particular solutions.
Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998,
The Mathematical Association of America.
All rights reserved.
Created: 12 Aug 1998 ---
Last modified: Sep 30, 2003 9:37:16 AM
Comments to: CVM@maa.org