function [ u, dudx, dudy ] = exact_u ( node_num, node_xy, time )

%% EXACT_U calculates the exact solution.
%
%  Discussion:
%
%    It is assumed that the user knows the exact solution and its
%    derivatives.  This, of course, is NOT true for a real computation.
%    But for this code, we are interested in studying the convergence
%    behavior of the approximations, and so we really need to assume
%    we know the correct solution.
%
%    As a convenience, this single routine is used for several purposes:
%
%    * it supplies the initial value function H(X,Y,T);
%    * it supplies the boundary value function G(X,Y,T);
%    * it is used by the COMPARE routine to make a node-wise comparison
%      of the exact and approximate solutions.
%    * it is used by the ERRORS routine to estimate the integrals of
%      the L2 and H1 errors of approximation.
%
%    DUDX and DUDY are only needed for the ERRORS calculation.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    09 April 2004
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer NODE_NUM, the number of nodes at which
%    a value is desired.
%
%    Input, real NODE_XY(2,NODE_NUM), the coordinates of 
%    the points where a value is desired.
%
%    Input, real TIME, the current time.
%
%    Output, real U(NODE_NUM), the exact solution.
%
%    Output, real DUDX(NODE_NUM), DUDY(NODE_NUM),
%    the X and Y derivatives of the exact solution.
%
  u(1:node_num) =         sin ( pi * node_xy(1,1:node_num)' ) ...
                       .* sin ( pi * node_xy(2,1:node_num)' ) ...
                        * exp ( -time );
 
  dudx(1:node_num) = pi * cos ( pi * node_xy(1,1:node_num)' ) ...
                       .* sin ( pi * node_xy(2,1:node_num)' ) ...
                        * exp ( -time );

  dudy(1:node_num) = pi * sin ( pi * node_xy(1,1:node_num)' ) ...
                       .* cos ( pi * node_xy(2,1:node_num)' ) ...
                        * exp ( -time );

  return
end