$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The structure of the equation system
Structure of nonlinear algebraic equations:
$$ A(u)u = b(u)$$
$$
\begin{align*}
A_{i,i} &= \frac{1}{2\Delta x^2}(-\dfc(u_{i-1}) + 2\dfc(u_{i})
-\dfc(u_{i+1})) + a\\
A_{i,i-1} &= -\frac{1}{2\Delta x^2}(\dfc(u_{i-1}) + \dfc(u_{i}))\\
A_{i,i+1} &= -\frac{1}{2\Delta x^2}(\dfc(u_{i}) + \dfc(u_{i+1}))\\
b_i &= f(u_i)
\end{align*}
$$
Note:
- \( A(u) \) is tridiagonal: \( A_{i,j}=0 \) for \( j > 1+1 \) and \( j < i-1 \).
- The \( i=0 \) and \( i=N_x \) equation must incorporate boundary conditions
