%
%  two simple plots
%

%  bessel coefficients

dt = 0.05;
t = dt:6*dt:1-dt;

nmax=50;

save = [];
clf;  subplot(2,1,1);  hold on
for ns = 1:nmax
  coeff = 2*(-1)^(ns-1)*besselj(ns,ns*t)./(ns*t);
  plot(log(ns), log(abs(coeff(end))),'rx')
  save = [save ; coeff];
end
axis([0 4 -7 0])
title('\bf bessel coefficients at t=0.95')
xlabel('\bf log(n)-axis')
ylabel('\bf log|a_n(0.95)|-axis')
text(0.2,-6,'(how close to the asymptotic slope does this give?)')

%  nonlinear wave solution

dx=pi/50;
x = -2*pi:dx:2*pi;

subplot(2,1,2);  hold on
for j=1:size(t,2)
  sol = 0;
  for ns =1:nmax
    sol = sol + save(ns,j)*sin(ns*x);
  end
  plot(x,sol,'b')
  axis([-2*pi 2*pi -1.3 1.1])
end
title(['\bf nonlinear wave: u(x,t), ' num2str(nmax) '-term series)'])
xlabel('\bf x-axis')
ylabel('\bf u-axis')
text(-3,0.5,'t = 0.05, 0.35, 0.65, 0.95')
text(0,-1.15,'(notice the weak resolution errors at t =0.95)')
