\displaystyle u_x = u_t ; 0 < x < 1, t > 0
Bondaries
\displaystyle u(0,t) = u(1,t) ; t > 0
\displaystyle u(x,0) = u(x) ; 0 < x < 1

I'd like to preform a von Neumann analysis on approimation.

\displaystyle  \frac{u^{n+1}_j - u^n_j}{\Delta t} = \frac{u^{n+1}_{j+1}- u^{n+1}_{j-1}}{2 \Delta x} ; j = 0,1,... N-1 , n = 0,1...

\displaystyle  u^{n}_j  = u^{n}_{N+1} ; all j

\displaystyle  u^{o}_j  = f(x_j) ;j = 0,1,... N-1

I did an analysis unsing Taylor expantion and got first order in space and first order in time. If I do a von Neumann I don't really know what how to start.

Should I do a Fourier transformation on the approximation or on the equation it self?