Yes, you're right. That was not the correct notation. Sorry!
As far as the Laplace-Fourier transform is concerned, the Fourier transform of the spatial derivatives is taken with respect to x with the transform variable "k" and the Laplace transform of the time derivative is taken with respect...
I have a third order derivative of a variable, say U, which is a function of both space and time.
du/dx * du/dx * du/dt or (d^3(U)/(dt*dx^2))
The Fourier transform of du/dx is simply ik*F(u) where F(u) is the Fourier transform of u. The Fourier transform of d^2(u)/(dx^2) is simply...