function r = fisher_residual ( t, x )

%*****************************************************************************80
%
%% fisher_residual() computes the residual of the KPP Fisher equation.
%
%  Discussion:
%
%    ut = uxx + u * ( 1 - u )
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    03 May 2024
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Mark Ablowitz, Anthony Zeppetella,
%    Explicit solutions of Fisher's equation for a special wave speed,
%    Bulletin of Mathematical Biology,
%    Volume 41, pages 835-840, 1979.
%
%    Daniel Arrigo,
%    Analytical Techniques for Solving Nonlinear Partial Differential Equations,
%    Morgan and Clayfoot, 2019,
%    ISBN: 978 168 173 5351.
%
%  Input:
%
%    real T, X: the time and position where the solution is evaluated.
%
%  Output:
%
%    real R: the residual at that time and position.
%
  [ u, ut, ux, uxx ] = fisher_exact ( t, x );
  r = ut - uxx - u * ( 1.0 - u );

  return
end
