The task is the following, how to extend Mathematica's FourierTransform command in order to be able to analytically deal with non-linear differential equations. To do so it should be able to deal with convolutions and derivatives
One way of dealing with complicated non-linear Partial Differential Equations is to transform them to Fourier space, where the products become convolutions. However the FourierTansform command does not transform products into convolutions. The command
Should result in something like
I tried to create an extension that (robustly) allows this but failed.
Furthermore, in order to deal with PDE's the extension should also be able to deal with derivatives, such that
I think such an extension would be extremely useful, but unfortunately FourierTransform is not very good at recognizing derivatives, and can simply not at all recognize convolutions...