$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Using Newton's method on the logistic ODE with typical math notation
Set iteration start as \( u^{n,0}= u^{n-1} \) and iterate
with explicit indices for time (\( n \)) and
Newton iteration (\( k \)):
$$
u^{n,k+1} = u^{n,k} +
\frac{\Delta t (u^{n,k})^2 + (1-\Delta t)u^{n,k} - u^{n-1}}
{2\Delta t u^{n,k} + 1 - \Delta t}
$$
Compare notation with
$$
u = u^{-} +
\frac{\Delta t (u^{-})^2 + (1-\Delta t)u^{-} - u^{(1)}}
{2\Delta t u^{-} + 1 - \Delta t}
$$
