Talk:Separation of variables

Why can solutions to PDEs be written as sums of products?
In the article it is stated that solutions to PDEs can be expressed as a sum in which each term is a product of several functions which each depend on only one variable. For example, given the equation

$$(\frac{\partial}{\partial x}^2 + \frac{\partial}{\partial y}^2)\Psi(x,y)=0$$

we are told to assume that the solution can be written

$$\Psi (x,y) = \sum_{m,n=-\infty}^{\infty} c_{mn}~X_m(x)Y_n(y).$$

This should be explained. The explanation is really quite simple. Since any function

$$\Psi (x,y)$$ can be expressed as a Fourier series (assuming a compact domain where x and y run from zero to one) we have

$$\Psi (x,y) = \sum_{m,n=-\infty}^{\infty} c_{mn}~e^{2 \pi i (mx+ny)} = \sum_{m,n=-\infty}^{\infty} c_{mn}~e^{2 \pi i mx}e^{2 \pi i ny} $$

which is a sum of products of functions depending on x and y independently.

This explanation should be given in the article.

128.36.90.229 03:15, 24 February 2007 (UTC)

Separation of variables using sums
I have never seen separation of variables using sums before, but this should of course not be a problem. However (and this connects to a remark below) the space of functions of the form $$X(x) +Y(y)+Z(z)$$ is not dense in the space of functions of three variables $$x,y,z$$. In particular in example (I) solutions of the form $$F(x,y,z) = f(x-y,x-z)$$ are missed, for non-linear functions $$f$$. Therefore this method clearly does not find all solutions and at least a warning should be issued there. Tasar (talk) 14:23, 17 April 2008 (UTC)

Redundant examples
The two examples are redundant! It would make more sense to have either one example or two different examples. I am going to erase one ... Bvds (talk) 02:52, 1 May 2014 (UTC)

When is the method valid?
The German language version of this article includes a theorem (and proof) about when the method is actually applicable. Would it be a good idea to translate this and add it to the English version?

Eigenermann (talk) 00:01, 31 January 2019 (UTC)
 * I think it would be a good idea. I'm going to add a section on the applicability of separation of variables which will include the translated material and also explain the link to the spectral theorem for PDEs (which subsumes the Fourier series case). I think an example where separation of variables is inapplicable would be good too, but I can't think of one at the moment. Ducksforever (talk) 21:56, 8 January 2021 (UTC)

Software
Some CAS-software can do separation of variables: Xcas among others.

Xcas does separation of variables with this command:

split((x+1)*(y-2),[x,y]) = [x+1,y-2]

source:

http://www-fourier.ujf-grenoble.fr/~parisse/giac/cascmd_en.pdf

MacApps (talk) 11:41, 24 August 2021 (UTC)MacApps