Fourier Transforms: Why Can't Homogeneous PDEs Be Solved?

[JohnSimpson]

Why can't homogeneous PDEs be solved by completely Fourier transforming both sides, down to an algebraic equation.

To clarify, consider the diffusion equation Del-Squared u = du/dt

If we Fourier transform both sides with respect to all 3 spatial variables and the time variable, we have something along the lines of

k^2 U = omega*U

which I can't seem to do anything useful with. What am I missing, or, if I'm not missing anything, is there a deeper reason behind why this approach won't work?

 

[Thaakisfox]

The right side doesn't depend on the spatial variables, and the left side on the time variable, hence if you just Fourier transform mindlessly, you will get dirac deltas everywhere

 

[JohnSimpson]

both sides depend on both space and time though, since on both sides we have u(x,y,z,t)

 

[Ben Niehoff]

You can use a four-dimensional Fourier transform, yes. Then solve the equation algebraically for U(k, w) and take the inverse transform.

The problem you will run into is how to integrate around the poles. There are several different possibilities, each of which ultimately depends on the boundary conditions.