$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Newton's method: details of the Jacobian
$$
\begin{align*}
J_{i,j,i-1,j} &= \frac{\partial F_{i,j}}{\partial u_{i-1,j}}\\
&= \frac{\Delta t}{h^2}(\dfc^{\prime}(u_{i-1,j})(u_{i,j}-u_{i-1,j})
+ \dfc(u_{i-1,j})(-1)),\\
J_{i,j,i+1,j} &= \frac{\partial F_{i,j}}{\partial u_{i+1,j}}\\
&= \frac{\Delta t}{h^2}(-\dfc^{\prime}(u_{i+1,j})(u_{i+1,j}-u_{i,j})
- \dfc(u_{i-1,j})),\\
J_{i,j,i,j-1} &= \frac{\partial F_{i,j}}{\partial u_{i,j-1}}\\
&= \frac{\Delta t}{h^2}(\dfc^{\prime}(u_{i,j-1})(u_{i,j}-u_{i,j-1})
+ \dfc(u_{i,j-1})(-1)),\\
J_{i,j,i,j+1} &= \frac{\partial F_{i,j}}{\partial u_{i,j+1}}\\
&= \frac{\Delta t}{h^2}(-\dfc^{\prime}(u_{i,j+1})(u_{i,j+1}-u_{i,j})
- \dfc(u_{i,j-1}))\tp
\end{align*}
$$
