function [ a, f ] = adjust_boundary ( node_num, node_xy, node_boundary, ib, ...
  time, a, f )

%% ADJUST_BOUNDARY modifies the linear system for boundary conditions.
%
%  Modified:
%
%    06 April 2006
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer NODE_NUM, the number of nodes.
%
%    Input, real NODE_XY(2,NODE_NUM), the coordinates of nodes.
%
%    Input, integer NODE_BOUNDARY(NODE_NUM), is
%    0, if a node is an interior node;
%    1, if a node is a Dirichlet boundary node.
%
%    Input, integer IB, the half-bandwidth of the matrix.
%
%    Input, real TIME, the current time.
%
%    Input, real A(3*IB+1,NODE_NUM), the NODE_NUM by NODE_NUM
%    coefficient matrix, stored in a compressed format.
%
%    Input, real F(NODE_NUM), the right hand side.
%
%    Output, real A(3*IB+1,NODE_NUM), has been adjusted for boundary conditions.
%
%    Output, real F(NODE_NUM), has been adjusted for boundary conditions.
%

%
%  Get the exact solution at every node.
%
  u_exact = exact_u ( node_num, node_xy, time );

  for node = 1 : node_num

    if ( node_boundary(node) ~= 0 )

      jlo = max ( node - ib, 1 );
      jhi = min ( node + ib, node_num );

      for j = jlo : jhi
        a(node-j+2*ib+1,j) = 0.0;
      end

      a(node-node+2*ib+1,node) = 1.0;
    
      f(node) = u_exact(node);

    end

  end