# Fourier Transforms: Solving PDE's

1. Oct 9, 2008

### Niles

1. The problem statement, all variables and given/known data
Hi all.

We have a function u(x,t), where x can go from (-infinity;infinity) and t>0. In my book it says:

"For fixed t, the function u(x,t) becomes a function of the spatial variable x, and as such, we can take its Fourier transform with respect to the x-variable. We denote this transform by $$\widehat{u}(x,t)$$:

$$\widehat{u}(\omega,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}{u(x,t)e^{-i\omega x}} dx$$

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Questions:

1: First of all, $$\widehat{u}(x,t)$$ is only a function of $$\omega$$, since we have fixed t, correct?

2: The author says that we have the following:

$$F(\frac{d}{dt}u(x,t))(\omega) = \frac{d}{dt}\widehat{u}(\omega,t),$$

where the large F denotes the Fourier transform of u(x,t) with respect to x. Since we have fixed t, then why are we differentiating with respect to t? Doesn't this give zero?