$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Newton's method; nonlinear equations at the end points
$$
\begin{align*}
F_i &= -\frac{1}{2\Delta x^2}
((\dfc(u_i)+\dfc(u_{i+1}))(u_{i+1}-u_i) -
(\dfc(u_{i-1})+\dfc(u_{i}))\times \\
&\qquad (u_{i}-u_{i-1})) + au_i - f(u_i) = 0
\end{align*}
$$
At \( i=0 \), replace \( u_{-1} \) by formula from Neumann condition.
- Exclude Dirichlet condition as separate equation:
replace \( u_i \), \( i=N_x \), by \( D \) in \( F_{i} \), \( i=N_x-1 \)
- Include Dirichlet condition as separate equation:
$$ F_{N_x}(u_0,\ldots,u_{N_x}) = u_{N_x} - D = 0\tp$$
Note: The size of the Jacobian depends on 1 or 2.
