# 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 D*u* 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

*u*(*x*,*y*,*t*) =
*f*(*x*,*y*) *v*(*t*).
In this case, the left-hand side becomes

and the right-hand side becomes *f*(*x*,*y*)
*v*''(*t*).
Setting these equal and gathering the terms involving *t*
on one side and the ones involving *x* and *y* on the other, we
find:

Now pick some fixed time *t*_{0} 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
*t*_{0}. 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

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