function [u,xgrid] = explicit_fd( payoff_fn, u_m_inf, u_p_inf, r, sigma, xLeft, xRight, Nx, tau_Max, M )
% 
% Solves the diffusion equation using an explicit finite-difference model
% 
% Input: 
%   tau_Max     = the maxium time to integrate the diffusion equation.
% 
% See the Epage 156 in Wilmotts book: The Mathematics of Financial Derivatives
% 
% Written by:
% -- 
% John L. Weatherwax                2008-06-11
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

xgrid = linspace( xLeft, xRight, Nx+1 ); 
dx    = (xRight-xLeft)/Nx; 
dt    = tau_Max/M; 

a = dt/(dx*dx); 
if( 0 && a>=0.5 )
  fprintf('a = %10.6f\n',a); 
  error('a to large .... this explicit method may not be stable ... increasing M may help ...\n'); 
end
k = r/(0.5*sigma^2); 

% get the initial conditions: 
tau  = 0.0; 
oldu = payoff_fn( xgrid, tau, k ); 

% bst   = tau/(0.5*sigma^2); 
% [C,P] = blackscholes(E,bst,E*exp(xgrid),r,sigma); %T-tau/(0.5*sigma^2)
% p_tmp = (P/E).*exp( 0.5*(k-1)*xgrid )*exp( 0.25*((k+1)^2)*tau );
% %figure; plot( xgrid, oldu, '-rx' ); hold on; plot( xgrid, p_tmp, '-go' ); pause; 

u      = zeros(M+1,Nx+1);    % <- space for our results ...
u(1,:) = oldu; 
newu   = zeros(1,Nx+1); 
for m=1:M,
  tau = m*dt; 
    
  % update the endpoints: 
  newu(1)   = u_m_inf( xgrid(1),   tau, k ); 
  newu(end) = u_p_inf( xgrid(end), tau, k ); 
  
  % update the interior of our grid: 
  newu( 2:(end-1) ) = oldu( 2:(end-1) ) + a * ( oldu( 1:(end-2) ) - 2*oldu( 2:(end-1) ) + oldu( 3:end ) ); 
  
  % prepare for the next pass: 
  oldu = newu;   
    
  % bst   = tau/(0.5*sigma^2); 
  %   [C,P] = blackscholes(E,bst,E*exp(xgrid),r,sigma); %T-tau/(0.5*sigma^2)
  %   p_tmp = (P/E).*exp( 0.5*(k-1)*xgrid )*exp( 0.25*((k+1)^2)*tau );
  %   %figure; plot( xgrid, oldu, '-rx' ); hold on; plot( xgrid, p_tmp, '-go' ); %pause; 
  
  % store 
  u(m+1,:) = newu; 
end