$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Computing the Jacobian
$$
\begin{align*}
J_{i,j} = \frac{\partial F_i}{\partial c_j}
& = \int_0^L \frac{\partial}{\partial c_j}
(\dfc(u)u^{\prime}\baspsi_i^{\prime} + au\baspsi_i -
f(u)\baspsi_i)\dx\\
&=
\int_0^L
((\dfc^{\prime}(u)\frac{\partial u}{\partial c_j}u^{\prime} +
\dfc(u)\frac{\partial u^{\prime}}{\partial c_j})\baspsi_i^{\prime}
+ a\frac{\partial u}{\partial c_j}\baspsi_i -
f^{\prime}(u)\frac{\partial u}{\partial c_j}\baspsi_i)\dx\\
&=
\int_0^L
((\dfc^{\prime}(u)\baspsi_ju^{\prime} +
\dfc(u)\baspsi_j^{\prime}\baspsi_i^{\prime}
+ a\baspsi_j\baspsi_i -
f^{\prime}(u)\baspsi_j\baspsi_i)\dx\\
&=
\int_0^L
(\dfc^{\prime}(u)u^{\prime}\baspsi_i^{\prime}\baspsi_j +
\dfc(u)\baspsi_i^{\prime}\baspsi_j^{\prime}
+ (a - f(u))\baspsi_i\baspsi_j)\dx
\end{align*}
$$
Use \( \dfc^{\prime}(u^{-}) \), \( \dfc(u^{-}) \), \( f^\prime (u^{-}) \), \( f(u^{-}) \)
and integrate expressions numerically (only known functions)
