I was watching really nice video by Steve Brunton on applications of the Fourier Transform to solving partial differential equations. The approach is actually quite interesting for solving the nonlinear Burgers' equation:
$$ u_t + uu_x = \nu u_{xx} $$
In previous examples, Brunton used the Fourier transform to convert the PDE equation to a system of ODEs in the frequency domain, solve the equations numerically in the frequency domain, and then transform back to the original time domain. However, in the Burgers' equation problem, he showed the Fourier transformation of spatial variable $u(x, t)$, the computation of the derivatives in the spatial domain, and then the inverse transformation of $u_x, u_{xx}$ back to the original domain for computation in some standard ODE scheme like RK.
I can follow the method just fine, but it was not clear what the criteria is for transforming a PDE into the spatial versus frequency domains? How would I choose which domain for the transformation, or do I get the same result independent of my choice of spatial or frequency domain? In this Burgers' equation case, it seems like transforming the spatial variable lets me apply the transform to both the first and second derivatives--instead of a single time derivative. But I was looking for some more general principle if anything like that exists. In other words, do I get different results or accuracy when I Fourier transform this problem to the frequency domain instead the spatial domain?