# Separation of Variables vs Fourier Transform (for PDE)

I would like to know how can I know if I have to solve a PDE (Heat Equation, Laplace Equation, Wave Equation, etc.) using Separation of Variables or Fourier Transform.

Which boundary conditions do I have to see? What's the difference?

Thanks!!

• math.stackexchange.com/questions/221137/… – mattos Feb 10 '15 at 3:27
• @Mattos thanks for the answer. I already read that thread, but that's not my question. I know Fourier Series are just for periodic functions, but, expanding a function, I can put it always as a periodic function (for example, if it's given between 0 and a, I can put that the same happens from a to 2a and it's periodic). So, how can I know if I have to use one or another, looking at the BOUNDARY CONDITIONS? – Unnamed Feb 10 '15 at 6:32
• Yes, look domain. Infinite domain $\implies$ Fourier Transform. – mattos Feb 10 '15 at 6:53
• @Mattos I have some exercises where $0<x<2$ and $t>0$ and I have to use Fourier Series (it's infinite in t) and I have to solve a Wave Equation. On the other hand, I have some exercises where $0<x<2$ and $y>0$ and I have to use Fourier Cosine Transform to solve a Laplace Equation. Then, I have another with $y>0$ (x can be any value) and I have to use Fourier Transform to solve Laplace Equation. Any more help? :S – Unnamed Feb 10 '15 at 15:32
• I'll write a post. – mattos Feb 10 '15 at 15:43

Remember, $t$ is a parameter in these situations. Being infinite in $t$ was not what I meant when I said infinite domain $\implies$ Fourier Transform..
However, $x$ and $y$ are dimensions, these are what I meant when I said infinite domain $\implies$ Fourier Transform. Notice in your last two examples, your PDEs depended on $x$ and $y$ but not $t$, and you needed to Fourier Transform both of them. This is because the domain of $x$ or $y$ (or both) in those examples is infinite.
1. If you see $t \gt 0$, that does not mean you have an infinite domain.
2. If you see $x, y \in (-\infty, \infty)$ or some variation of this, use Fourier Transform. If the domain is discrete (i.e. $x, y \in (a, b), \ a, b \in \mathbb{R}$) then use Fourier Series.