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I am wondering if there are some conclusions as to when a series solution using the separate variable method to a PDE exists; i.e. for what requirements on the PDE, what requirements on the initial and boundary conditions so that one can assume the solution as products of terms involving only a single dependent variable, and then proceed to solve the equation in Fourier's way.

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    since you are taking of PDE ,then it is impossible to assume their solution as a function of 1 independant variable2011-07-18
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    @muhammad: [separation of variables](http://en.wikipedia.org/wiki/Separation_of_variables) means the ansatz that the solution is a *product* of function, each of only one variable, but each is a function of a different variable. In other words, you write the solution $U(x,y,z) = X(x) Y(y) Z(z)$.2011-07-18
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    A good reference on this is _Methods of Theoretical Physics_ by Philip McCord Morse and Herman Feshbach.2012-12-25

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