Different methods to solving PDE's

[captain]

I just want to know when would you use a Fourier transform method to solve a PDE vs. separation of variables or Laplace transform? My guess is that a Fourier transform is for a problem in which the solution exists from negative infinity to positive infinity, whereas a Laplace transform would for a solution which is bounded from 0 to infinity, but I still don't know when you would use separation of variables even if you are in the domain in which you could use a Fourier or laplace transform. All this is somewhat loosely stated. Thanks to everyone in advance who can clear this up for me and confirm if what I said was correct above.

 

[gato_]

The answer is you can use (multiplicative) separation of variables if it works! That usualy means you can propose a solution of the form, say, u(x,t)=U(x)T(t), and when you insert that into the equation you are able to put it in the form L[U(x)]=M[T(t)], L and M being differential operators. If you get there, it turns out that one side depends only on x, and the other on t, so the only solution available is both sides being equal to a constant. Something similar happens to additive separabiliy.

 

[gato_]

And there is such a thing as a one-sided Fourier transform. It is the result os substituting s=iw in the laplace transform. This one-sided Fourier transform is related to the usual one via hilbert transform

 

[matematikawan]

I just want to know when would you use a Fourier transform method to solve a PDE vs. separation of variables or Laplace transform? My guess is that a Fourier transform is for a problem in which the solution exists from negative infinity to positive infinity, whereas a Laplace transform would for a solution which is bounded from 0 to infinity, but I still don't know when you would use separation of variables even if you are in the domain in which you could use a Fourier or laplace transform. All this is somewhat loosely stated. Thanks to everyone in advance who can clear this up for me and confirm if what I said was correct above.

I think I read some where (probably in this forum), we can only try separation of variables when the boundary conditions are homogeneous. If I'm not mistaken.