$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The linear system in Picard iteration
$$
F_i = \int_0^L (\dfc(u^{-})u^{\prime}\baspsi_i^{\prime} + au\baspsi_i)\dx -
\int_0^L f(u^{-})\baspsi_i\dx + C\baspsi_i(0)
$$
This is a linear problem \( a(u,v)=L(v) \) with bilinear and linear forms
$$ a(u,v) = \int_0^L (\dfc(u^{-})u^{\prime}v^{\prime} + auv)\dx,\quad
L(v) = \int_0^L f(u^{-})v\dx - Cv(0)$$
The linear system now is computed the standard way.
