function  A = nma_FDM_matrix_laplace_1D_Neumann_scheme_1(N )
% builds finite difference A matrix for 1-D laplace

%-----------------------------------------------------------------
% nma_FDM_matrix_laplace_1D_Neumann_scheme_1(N )
%
% returns the A matrix, which is the system finite difference matrix for
% numerical solution of 1-D laplace equation Uxx = f, with nuemman
% boundary conditions on both sides of the element. Based on 3 point
% scheme    U'' =  U(j-1)-2*U(j)+U(j+1)  for middle points and based on
% scheme  ((-2*U(0)+2*U(1))/(h^2))=f(0)+((2*a)/h) for the left most point, 
% where a = U'(0), and h is the spacing. and for for the right most point
% ((2*U(N)-2*U(N+1))/(h^2)) = f(N+1)-((2*b)/h) where b=U'(1)
%
% notice that spacing h between the nodes is not used. The caller must
% divide by h^2 this A matrix abone return.
%
% INPUT:  N, the number of nodes
% OUTPUT: A, the matrix , see below
%
% EXAMPLE:
% A = nma_FDM_matrix_laplace_1D_Neumann_scheme_1(5)
%     -2     2     0     0     0
%      1    -2     1     0     0
%      0     1    -2     1     0
%      0     0     1    -2     1
%      0     0     0     2    -2
%
%
% A = A/h^2;  % h is space between points
% U = A\f;  % solve for U, where Uxx=f, need to set f as function before.
%
% copyright: Nasser M. Abbasi
% 10/18/2010

A = zeros(N);
for i=1:N
    for j=1:N
        if(i==j)
            A(i,j) = -2;
        else
            if( i==j+1 || i==j-1 )
                A(i,j) = 1;
            end
        end
    end
end

%fix A above for Von Neumann B.C. only 1st and last rows need fixed
A(1,2)       = 2;
A(end,end-1) = 2;
end