function U=molexample(J,tf)
% MOLEXAMPLE  Simple demonstration of method of lines on nonlinear 
% heat equation
%     u_t = 0.1 u_xx - 2 u^2,   (x,t) in (0,1) x (0,tf),
% with boundary conditions
%     u_x(0,t)=0,  u(1,t)=0
% and initial condition
%     u(x,0) = sin(20 x^2).
% Uses  ode23s,  with default error tolerance,  to solve semidiscretized 
% problem.
% 
%      U = molexample(J,tf)
% where:
%      U = approximate solution at time tf
%      J = number of subintervals
%     tf = final time
%
% Example:
%    >> J=100;   x=linspace(0,1,J+1);   U=molexample(J,1.0);
%    >> plot(x,U),  xlabel x,  ylabel U
% ELB 4/13/05
% See also: MOLMANUF

dx=1/J;   x=0:dx:1-dx;   U0=sin(20*x.^2);
[t,UU]=ode23s(@molderivs,[0 tf],U0);  % pass function *handle* in this case
U=UU(end,:);  % UU has soln at lots of times; pick last time
U=[U 0];  % add right boundary condition
          
function dUdt=molderivs(t,U);
% U,dUdt are a column vectors;  t not used
J=length(U);  dx=1/J;  nu = 0.1/(dx*dx);  dUdt=zeros(J,1);
dUdt(1)=-2*nu*U(1)+2*nu*U(2)-2*U(1)^2;
for k=2:J-1
   dUdt(k)=nu*U(k-1)-2*nu*U(k)+nu*U(k+1)-2*U(k)^2;   end
dUdt(J)=nu*U(J-1)-2*nu*U(J)-2*U(J)^2;