Fourier-Laplace transform of mixed PDE?

[ThatsRightJack]

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 -(k^2)*F(u) where F(u) is again the Fourier transform of u. My question is, how do handle the time derivative part with a Laplace transform? What would the Fourier-Laplace transform of the given PDE look like?

 

[HallsofIvy]

A point about your notation: (du/dx)(du/dx)(du/dt) is NOT a third derivative, it is a product of three first derivatives.

Now, a Laplace-Fourier transform has to be taken with respect to a specific variable. If you are taking the transform of \partial^3f/\partial x^2\partial t with respect to x, it the same as transform of the second derivative. If with respect to t, it is the same as the transform of the first derivative.

 

[ThatsRightJack]

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 to t with the transform variable "w". I'm still a little unclear as to what the final transform function looks like?

If this is the Fourier transform of d^2(u)/(dx^2), with respect to x using the tansform variable "k":
-(k^2)*F(u)
where F(u) is the Fourier transform of u, what would the Laplace transform of that be with respect to t using the transform variable "w" ?