function explicit_step

%  Solves the heat equation for various M values
%       diff(u,x,x) = diff(u,t) + f(x,t)   for xL < x < xr, 0 < t < tmax
%  where
%      u = 0  at x=xL,xR  and  u = step at a & b at t = 0

% clear all previous variables and plots
clear *
clf

% set parameters
N=26;
tmax=0.1;
xL=0;
xR=1;
a=0.25;
b=0.75;

fprintf('\n    Solution Computed with N = %3.0f\n\n',N)

% calculate exact solution at t=tmax
xx=linspace(xL,xR,100);
for i=1:100
	s=0;
	for ii=1:50
		an=2*(cos(a*pi*ii)-cos(b*pi*ii))/(pi*ii);
		s=s+an*exp(-pi*pi*ii*ii*tmax)*sin(pi*ii*xx(i));
	end;
	u(i)=s;
end;

% generate the points along the x-axis, x(1)=xL and x(N+2)=xR
x=linspace(xL,xR,N+2);
h=x(2)-x(1);

% get(gcf)
%set(gcf,'Position', [1155 497 575 470]);
plotsize(560,725)

% calculate the explicit solution using different M values
for im=1:3
	M=143+im;

	% generate the points along the t-axis, t(1)=0 and t(M)=tmax
	t=linspace(0,tmax,M+1);
	k=t(2)-t(1);
	lamda=k/h^2;
	fprintf('\n    Lamda = %5.2e\n\n',lamda)
	
	ue=explicit2(x,t,N+2,M+1,h,k,lamda);

	% plot results
	subplot(3,1,im)
	plot(x,ue(:,M+1),'r')
	hold on

	% define axes used in plot
	xlabel('x-axis','FontSize',14,'FontWeight','bold')
	ylabel('Solution','FontSize',14,'FontWeight','bold')
	
	% have MATLAB use certain plot options (all are optional)
	set(gca,'FontSize',14);
	axis([0 1 0 0.4]);
	set(gca,'ytick',[0 0.2 0.4]);
	plot(xx,u,'--k')
	box on
	legend([' \lambda = ',num2str(lamda,'%3.3f')],' Exact','Location','South');
	set(findobj(gcf,'tag','legend'),'FontSize',12,'FontWeight','bold');
	if im==1
		say=['Heat Equation: exact vs explicit method at t = ',num2str(tmax,'%3.1f')];
		title(say,'FontSize',14,'FontWeight','bold')
	end;
	hold off
end;





% explicit method
function UE=explicit2(x,t,N,M,h,k,lamda)
UE=zeros(N,M);
for i=1:N
	UE(i,1)=g(x(i));
end;
for j=2:M
	for i=2:N-1
	  UE(i,j)=lamda*UE(i+1,j-1)+(1-2*lamda)*UE(i,j-1)+lamda*UE(i-1,j-1)-k*f(x(i),t(j-1));
	end;
end;

% subfunction f(x,t)
function q=f(x,t)
q=0;

% subfunction g(x)
function q=g(x)
q=0.5*(sign(x-0.25)-sign(x-0.75));

% tridiagonal solver
function y = tridiag( a, b, c, f )
N = length(f);
v = zeros(1,N);   
y = v;
w = a(1);
y(1) = f(1)/w;
for i=2:N
    v(i-1) = c(i-1)/w;
    w = a(i) - b(i)*v(i-1);
    y(i) = ( f(i) - b(i)*y(i-1) )/w;
end;
for j=N-1:-1:1
   y(j) = y(j) - v(j)*y(j+1);
end;

% subfunction plotsize
function plotsize(width,height)
siz=get(0,'ScreenSize');
bottom=max(siz(4)-height-95,1);
set(gcf,'Position', [2 bottom width height]);