$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
The right-hand side
$$
\begin{equation}
b = \left(\begin{array}{c}
b_0\\
b_1\\
\vdots\\
b_i\\
\vdots\\
b_{N_x}
\end{array}\right)
\end{equation}
$$
with
$$
\begin{align}
b_0 &= 0\\
b_i &= u^{n-1}_i,\quad i=1,\ldots,N_x-1\\
b_{N_x} &= 0
\end{align}
$$
