VW - CVM 1.1

# 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:

Du =
 ¶2u ¶x2
+
 ¶2u ¶y2
=
 ¶2u ¶t2

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

u(x,y,t) = f(x,y) v(t).

In this case, the left-hand side becomes

v(t)
 æ ç è
 ¶2f ¶x2
+
 ¶2f ¶y2
 ö ÷ ø
= v(t) Df(x,y)

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:

 Df(x,y) f(x,y)
=
 v''(t) v(t)

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.