% First example of fft:
% grid
L = 20;
n = 2^7;
x2 = linspace(-L/2,L/2,n+1); 
x = x2(1:n);
% function
u = exp(-x.^2);
ut = fft(u);
utshift = fftshift(ut);
%
%--------------------------
% derivative example:
clear all; close all;
% grid
L = 20;
n = 2^7;
x2 = linspace(-L/2,L/2,n);
x = x2(1:n);
k = 2*pi/L*[0:(n/2-1) (-n/2):-1];
%
u = sech(x);
ut = fft(u);
ut1 = -k.^2.*ut;
uprime_num = ifft(ut1);
uprime_exact = -sech(x).*tanh(x);
%
plot(x,uprime_num,'r--',x,uprime_exact,'b--')