$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The algorithm of Picard iteration with classical math notation
- \( u^k \): computed approximation in iteration \( k \)
- \( u^{k+1} \) is the next approximation (unknown)
$$ au^k u^{k+1} + bu^{k+1} + c = 0\quad\Rightarrow\quad u^{k+1}
= -\frac{c}{au^k + b},\quad k=0,1,\ldots$$
Or with a time level \( n \) too:
$$ au^{n,k} u^{n,k+1} + bu^{n,k+1} - u^{n-1} = 0\quad\Rightarrow\quad u^{n,k+1}
= \frac{u^{n-1}}{au^{n,k} + b},\quad k=0,1,\ldots$$
