function leap

%  logistic eq: leapfrog
%  y' = ry(1-y)  with   y(0) = y0 '

% clear all previous variables and plots
clear *
clf

% set parameters for problem
M=10;
r=10;
t0=0; y0=0.1;
tmax=1;

% calculate and plot exact solution
tt=linspace(t0,tmax,100);
a0=(1-y0)/y0;
for it=1:100
	exact(it)=1/(1+a0*exp(-r*tt(it)));
end;
plot(tt,exact,'--k')
hold on

% define legend and axes used in plot
xlabel('t-axis','FontSize',14,'FontWeight','bold')
ylabel('Solution','FontSize',14,'FontWeight','bold')
title(['Logistic Equation:  M = ', int2str(M)],'Color','k','FontSize',14,'FontWeight','bold')

% have MATLAB use certain plot options (all are optional)
box on
axis([0 tmax -2 2])	
% Set the fontsize to 14 for the plot
set(gca,'FontSize',14); 

% calculate leapfrog solution
t=linspace(t0,tmax,M+1);
h=t(2)-t(1);
y=y0;
yy=1/(1+a0*exp(-r*h));
leap=[y, yy];
ts=[0 h];
for i=3:M+1
	yyy=y+2*r*h*yy*(1-yy);
	leap=[leap, yyy];
	ts=[ts t(i)];
	y=yy;
	yy=yyy;
	plot(ts,leap,'-or','LineWidth',1)
	if i==3
		legend(' Exact',' Leap',2)
		% Set legend font to 14/bold 
		set(findobj(gcf,'tag','legend'),'FontSize',14,'FontWeight','bold');
	end;

	pause(1);
	
end;        

hold off