Talk:Separation of variables

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[edit] 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)

[edit] Change

Before:

In mathematics, separation of variables is any of several methods of solving ordinary and partial differential equations, in which algebra allows to re-write an equation so that each of two variables occurs on only side of the equation and the other does not.

After:

In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on only one side of the equation.

I have a problem with this edit. Suppose the equation is

xy' + (y'')^2 = x^3.\,

This trivially becomes

Failed to parse (Cannot write to or create math output directory): xy' + (y'')^2 - x^3 = 0.\,


Then clearly each of the two variables occurs on only one side of the equation. But the variables have not been separated! Michael Hardy 02:43, 9 February 2006 (UTC)

{{sofixit}}? I edited the copy of that sentence, not the content. It seems to me that neither the before nor the after clearly defines separation of variables. Starwiz 17:02, 18 February 2006 (UTC)

[edit] 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
Ψ(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)