How to handle fft of initial condition?

[John B. Thoo]

Hi.  I have a question about solving a differential equation using  
fft, specifically, how to handle the initial condition.

Suppose the equation is

  u_t = f(x, t),  u(x,0) = u0(x).

Applying the F.T., I think I get the system of ODEs

  d [u^]_k
  --------  =  [f^]_k(t),  [u^]_k(0) = [u0^]_k,
     dt

for  k = -infinity..infinity,  where  ^  denotes the F.T.  Then I  
think the solution  u  would be

  u(x,t) = \sum_{k=-infinity..infinity} [u^]_k(t) * exp(ikx).

My question is this: If I let  [u0^] = fft (u0),  how do I match a  
negative index  k  when Octave begins with  k = 1?  E.g., if  k = -3,  
then I think I can handle  [u^]_{-3}  using complex conjugates:

  d [u^]_{-3}     d [u']_3
  -----------  =  --------.
       dt            dt

But then what would be  [u0^]_{-3}?  Or am I approaching this wrongly  
altogether?

Please let me know if my question is not clear and I will try to  
restate it.  Thanks.

---John.