ODE15s solver

%ODE15s function function F = oden(t,ys) % F is same as dudt, U = ys(1:N,1);

        % Boundary
        U(1)=0;
        U(N)=1;

        F(1,1)=beta*((U(2)-2*U(1))/dx/dx)+alpha*(U(2)/(2*dx));
        F(2:N-1,1) = beta*((U(3:N)-2*U(2:N-1)+U(1:N-2))/dx/dx)+alpha*((U(3:N)-U(1:N-2))/(2*dx));
        F(N,1)=beta*((1-2*U(N)+U(N-1))/dx/dx)+alpha*((1-U(N-1))/(2*dx));

    end

end

This is my ODE15s solver for a finite difference for diffusion and convection, could someone tell what is wrong with this formula, because I keep not getting what I hoped for, which is a striaght line. The ghost point for 0 and N+1 is the same as 1 and N. Thank you.