function ex6bvp
%EX6BVP  Example 6 of the BVP tutorial.
%   This is Example 2 from M. Kubicek et alia, Test examples for comparison 
%   of codes for nonlinear boundary value problems in ordinary differential 
%   equations, B. Childs et al., eds., Codes for Boundary-Value Problems in 
%   Ordinary Differential Equations, Lecture Notes in Computer Science #76, 
%   Springer, New York, 1979.  This example shows how to deal with a singular
%   coefficient arising from reduction of a partial differential equation to
%   an ODE by symmetry.  Also, for the physical parameters considered here,
%   the problem has three solutions.


  %  Define the physical parameters for this problem.
  %  These parameters will be visible in nested functions.
  f = 0.6;
  g = 40;
  b = 0.2;
  
  guess  = [1; 0.5; 0];
  others = [0.9070; 0.3639; 0.0001];      % Values reported by Kubicek et alia.
  
  fprintf('y(0):  bvp4c    Kubicek et al.\n')
  figure
  
  for index = 1:3
    solinit = bvpinit(linspace(0,1,5),[guess(index) 0]);
    sol = bvp4c(@ex6ode,@ex6bc,solinit);
    
    fprintf('      %6.4f       %6.4f\n',sol.y(1,1),others(index))
    plot(sol.x,sol.y(1,:))
    if index == 1
      axis([0 1 -0.1 1.1])
      title('Multiple solutions to spherical catalyst problem.')
      xlabel('x')
      ylabel('y')
      hold on
    end
    drawnow
  end
  hold off
  
  % --------------------------------------------------------------------------
  % Nested functions
  %
  
  function dydx = ex6ode(x,y)
  %EX6ODE  ODE function for Example 6 of the BVP tutorial.
    dydx = [y(2); 0];
    temp = f^2 * y(1) * exp(g*b*(1-y(1))/(1+b*(1-y(1))));
    if x == 0
      dydx(2) = (1/3)*temp;
    else
      dydx(2) = -(2/x)*y(2) + temp;
    end
  end % ex6ode

  % --------------------------------------------------------------------------

  function res = ex6bc(ya,yb)
  %EX6BC  Boundary conditions for Example 6 of the BVP tutorial.
    res = [ ya(2)
            yb(1) - 1];      
  end % ex6bc

  % --------------------------------------------------------------------------

end  % ex6bvp