When can seperation of variables be applied?

[PhDorBust]

I have some confusion about when separation variables can be applied to a PDE. Can it be applied on any PDE that can separated for any domain? If so, is the use of more "powerful" techniques simply used to save effort? (As you might have to use superposition several times!)

 

[hosun]

To my knowledge, it can be use in a square domain only(i.e. x1 in [a,b] and x2 in [c,d]).
Non-square domain is usually transformed into square domain by the change of variable, in order to separate the variables.