# Fourier-Laplace transform of mixed PDE?

1. Jun 3, 2014

### 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?

2. Jun 4, 2014

### HallsofIvy

Staff Emeritus
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.

3. Jun 4, 2014

### 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" ?

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