function example1s

%  Solves the BVP and then uses spline interpolation
%       y'' + p(x)y' + q(x)y= f(x)   for xL < x < xr  '
%  where
%      y(xl) = yL  and y(xR) = yR

%  p=0, q=-1, f=sin(2*pi*x)
%  xL=0, yL=0  and  xR=1, yR=0

% clear all previous variables and plots
clear *
clf

% set boundary conditions
	xL=0; yL=0;
	xR=1; yR=0;

% calculate and plot exact solution
xx=linspace(xL,xR,100);
exact=-sin(2*pi*xx)/(1+4*pi*pi);
plot(xx,exact,'k')
hold on

% define title and axes used in plot
xlabel('x-axis','FontSize',14,'FontWeight','bold')
ylabel('Solution','FontSize',14,'FontWeight','bold')
title('BVP: Example 1 (with splines)','FontSize',14,'FontWeight','bold')

% have MATLAB use certain plot options (all are optional)
box on
% Set the fontsize to 14 for the plot
set(gca,'FontSize',14); 

% loop used to increase N value
for in=1:3

	% set number of points along the x-axis
	if in==1
		n=2
	elseif in==2
		n=4
	elseif in==3
		n=8
	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);

	% calculate the coefficients of finite difference equation
	a=zeros(1,n); b=zeros(1,n); c=zeros(1,n); z=zeros(1,n);
	for i=1:n
		a(i)=-2+h*h*q(x(i+1));
		b(i)=1-0.5*h*p(x(i+1));
		c(i)=1+0.5*h*p(x(i+1));
		z(i)=h*h*f(x(i+1));
	end;
	z(1)=z(1)-yL*b(1);
	z(n)=z(n)-yR*c(n);

	% solve the tri-diagonal matrix problem
	y=tri(a,b,c,z);
	y=[yL, y, yR];
	
	%  use spline interpolation
	xs=linspace(xL,xR,200);
	ys=spline(x,y,xs);

	% plot the solution
	if in==1
		x1=x; y1=y;
		%plot(x1,y1,'sr','LineWidth',1,'MarkerSize',7)
		plot(xs,ys,'--r','LineWidth',1)
		legend(' Exact',' N = 2',2);
		set(findobj(gcf,'tag','legend'),'FontSize',14,'FontWeight','bold');
		pause
	elseif in==2
		x2=x; y2=y;
		%plot(x,y,'ob','LineWidth',1)
		plot(xs,ys,'--b','LineWidth',1)
		legend(' Exact',' N = 2',' N = 4',2);
		set(findobj(gcf,'tag','legend'),'FontSize',14,'FontWeight','bold');
		pause
	elseif in==3
		x3=x; y3=y;
		%plot(x,y,'*m','LineWidth',1)
		plot(xs,ys,'--m','LineWidth',1)
		legend(' Exact',' N = 2',' N = 4',' N = 8',2);
	end	
	% Set legend font to 14/bold    
	set(findobj(gcf,'tag','legend'),'FontSize',14,'FontWeight','bold'); 
end

plot(x1,y1,'sr','LineWidth',1,'MarkerSize',7)
plot(x2,y2,'ob','LineWidth',1,'MarkerSize',7)
plot(x3,y3,'*m','LineWidth',1,'MarkerSize',7)

hold off


function g=q(x)
g=-1;

function g=p(x)
g=0;

function g=f(x)
g=sin(2*pi*x);


% tridiagonal solver
function y = tri( 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