% p35.m - Allen-Cahn eq. as in p34.m, but with boundary condition
%         imposed explicitly ("method (II)")

% Differentiation matrix and initial data:
  N = 20; [D,x] = cheb(N); D2 = D^2;    
  eps = 0.01; dt = min([.01,50*N^(-4)/eps]);
  t = 0; v = .53*x + .47*sin(-1.5*pi*x);

% Solve PDE by Euler formula and plot results:
  tmax = 100; tplot = 2; nplots = round(tmax/tplot);
  plotgap = round(tplot/dt); dt = tplot/plotgap;
  xx = -1:.025:1; vv = polyval(polyfit(x,v,N),xx);
  plotdata = [vv; zeros(nplots,length(xx))]; tdata = t;
  for i = 1:nplots
    for n = 1:plotgap
      t = t+dt; v = v + dt*(eps*D2*v + v - v.^3);        % Euler
      v(1) = 1 + sin(t/5)^2;  v(end) = -1;               % BC
    end
    vv = polyval(polyfit(x,v,N),xx);
    plotdata(i+1,:) = vv; tdata = [tdata; t];
  end
  clf, subplot('position',[.1 .4 .8 .5])
  mesh(xx,tdata,plotdata), grid on, axis([-1 1 0 tmax -1 2]),
  view(-60,55), colormap(1e-6*[1 1 1]); xlabel x, ylabel t, zlabel u