% example for finite difference approximation

clear
% example 1 
% f''=1, f(0)=1, f(1)=1
% exact solution: f = 1-x/2+x^2/2
% this can also be treated as the simplest case of a heat equation
% where source S(x)=1 (the right hand side of equation)

% set up matrix L
N = 50;

%L = zeros(N+1,N+1);

L = diag(-2*ones(1,N+1)) + diag(ones(1,N),1) + diag(ones(1,N),-1);

L(1,1)=1; L(1,2)=0;
L(N+1,N+1)=1; L(N+1,N)=0;

cond(L)
pause  % enter to proceed

dx = 1/N;
b = ones(N+1,1)*dx*dx;
b(1)=1;
b(N+1)=1;

F = L\b;

X = linspace(0,1,N+1);
plot(X,F,'x')
hold on
plot(X,ones(1,N+1)-X/2+X.^2/2,'r')