% Script for solving u'' = 1/(1-0.6cos(2*pi*x)) - 5/4, 1-periodic BCs
% using Fourier spectral method.
% Solution is plotted
% for N = 2, 4  8, 16

  clf
  subplot(2,2,1)
  color = 'rgbk'

  L = 1;
  pmax = 4;
  pintp = 5;
  for p = 1:pmax
    N = 2^p;
    h = L/N;
    k = [0:(N/2-1) (-N/2):(-1)]*2*pi/L;
    k(1) = 1e-6;          % Redefine k(1) nonzero to prevent 0/0 division later
    x = (0:(N-1))*h;    % These are only the grid points covering [0 1]
     
    %    Define right hand side over extended domain 
    
    f = 1./(1-0.6*cos(2*pi*x)) - 5/4;
       
    %    DFT, solve, and inverse DFT

    fhat = fft(f);
    fhat(1) = 0;
    u = real(ifft(-fhat./k.^2));
    uintp = interpft(u,2^pintp);
    plot(x,u,['x' color(p)],(0:2^pintp - 1)/2^pintp,uintp,['-' color(p)])
    if (p==1)
      xlabel('x')
      ylabel('u')
      title('Solution')
      hold on
    end
  end
  legend('N=2',' ','N=4',' ','N=8',' ','N=16',' ')
  hold off