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Suppose I have a function $f(x,y,z)$. I need to know when one can write it as $$f(x,y,z)=a(x)\cdot b(y) \cdot c(z)$$ where $a, b, c$ are functions. I don't want to know what they are, but just whether it's possible to separate.

Motivation of the question was that many times, to solve a p.d.e we use separation of variables, I just want to know the justification.

mike65535
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avz2611
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  • I think this question is quite hard to give a full answer to. Even simple functions like $f(x,y,z)=x+y$ can't be written that way –  Feb 04 '16 at 11:40
  • The answer is: whenever it makes sense. If you can find $a$, $b$ and $c$ simpler than $f$ then you can separate the variables. At most you shold see the problem the other way round: What can I do to separate the variables?. The answer is that, in some case, you cna make a change in the variables that lets you find a separable form. – N74 Feb 04 '16 at 11:59

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From what I've experienced, we use separation of variables when we face linear PDEs and we can safely use them when we're faced with problems that have well-defined coordinates.

For example, in the heat, wave and Laplace's equation, we have the first one depending on space, the second one depending on space and time, and the last one again on space. We can use separation of variables because we make the initial - although implicit - assumption that theses variables are independent in relation to each other. Again, we know how the solutions are supposed to look, so this is a method that works.

There is not a completely settled way of generalizing when you'll be able to use separation of variables. There is a better topic where someone was able to do a good ammount of background check on that matter, which might answer your question: Why separation of variables works in PDEs?

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