Your simulation tool is using Discrete Fourier transforms, which are a way of breaking a continuous function (like an audio waveform) into its frequency spectrum. Given a function f defined over a finite period of time (say from 0 to 2*pi) then the n-th Fourier coefficient
is the value of
int(f(t)*exp(-i*n*t), t=0..2*pi) / (2*pi)
The Discrete Fourier Transform is an algorithm for taking sampled data and determining the Fourier coefficients of the function that was sampled for small values of n, and using those coefficients to express the function as a sum of sines and cosines, which is handy for all sorts of reasons.
This is similar but not the same as the Fourier transform used to help solve differential equations. That Fourier transform turns the function f(t) into the function F(w) defined by
F(w) = int(f(t)*exp(-i*w*t), t=-infinity..infinity) / sqrt(2*pi)
The differences are the domain of integration (everything, instead of some finite interval) and the fact that w doesn’t have to be an integer like n has to be. Oh, and that square root, which is there for annoying technical reasons.
But just to confuse the issue…I think the acronym “FFT” should stand for “Fast Fourier Transform”, which is a particular implementation of the Discrete Fourier Transform algorithm. In other words, FFT and DFT should refer to the same thing, but neither is what’s used to solve differential equations. Unless there’s another meaning of the acronym “FFT” that I’m not aware of, which is entirely possible (this isn’t my specialty, I only know this because I happen to be studying for my general examination at the moment).