\documentclass[11pt]{article}
\usepackage{color}
%\usepackage{af}

%%%
\textheight 240mm
\textwidth 160mm 
\columnsep 10pt 
\columnseprule 0pt 

\oddsidemargin 0pt 
\evensidemargin 0pt 
\marginparwidth 90pt 
\marginparsep 11pt 

\topmargin 0pt 
\headheight 0pt 
\headsep 0pt 
\footskip 30pt 

\parskip 0pt plus 1pt 
\parindent 1.5em 
\partopsep 3pt plus 2pt minus 2pt 
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\makeatletter
\def\eqnarray{\stepcounter{equation}\let\@currentlabel=\theequation
\global\@eqnswtrue
\global\@eqcnt\z@\tabskip\@centering\let\\=\@eqncr
$$\halign to \displaywidth\bgroup\@eqnsel\hskip\@centering
%%  $\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne 
    $\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne 
%%  \hskip 2\arraycolsep \hfil${##}$\hfil
                         \hfil$\displaystyle{{}##{}}$\hfil
%%  &\global\@eqcnt\tw@ \hskip 2\arraycolsep $\displaystyle\tabskip\z@{##}$\hfil 
    &\global\@eqcnt\tw@                      $\displaystyle\tabskip\z@{##}$\hfil 
   \tabskip\@centering&\llap{##}\tabskip\z@\cr}
\makeatother

\renewcommand{\labelenumi}{(\theenumi)}
\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\theenumii}{(\arabic{enumi})}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}

\def\baselinestretch{1.20}
\def\r#1{{\rm#1}}
\def\ddt{\frac{\partial}{\partial t}}
\def\ddr{\frac{\partial}{\partial r}}
\def\dds{\frac{\partial}{\partial s}}
\def\dd#1{\frac{\partial #1}{\partial s}}
\def\ave#1{\left\langle#1\right\rangle}
\def\me{m_\r{e}}
\def\mi{m_\r{i}}
\def\mb{m_\r{b}}
\def\mue{\mu_\r{e}}
\def\mui{\mu_\r{i}}
\def\De{D_\r{e}}
\def\Di{D_\r{i}}
\def\chie{\chi_\r{e}}
\def\chii{\chi_\r{i}}
\def\nee{n_\r{e}}
\def\ni{n_\r{i}}
\def\nb{n_\r{b}}
\def\uer{u_{\r{e}r}}
\def\uir{u_{\r{i}r}}
\def\ueth{u_{\r{e}\theta}}
\def\uith{u_{\r{i}\theta}}
\def\ubth{u_{\r{b}\theta}}
\def\ueph{u_{\r{e}\phi}}
\def\uiph{u_{\r{i}\phi}}
\def\ubph{u_{\r{b}\phi}}
\def\Er{E_r}
\def\Eth{E_\theta}
\def\Eph{E_\phi}
\def\Bth{B_\theta}
\def\Bph{B_\phi}
\def\Athd{\dot{A}_\theta}
\def\Aphd{\dot{A}_\phi}
\def\Ath{A_\theta}
\def\Aph{A_\phi}
\def\Te{T_\r{e}}
\def\Ti{T_\r{i}}
\def\nna{n_{01}}
\def\nnb{n_{02}}
\def\nnc{n_{03}}
\def\Zi{Z_\r{i}}
\def\Zb{Z_\r{b}}
\def\Pb{P_\r{b}}
\def\Eb{E_\r{b}}
\def\PRFe{P_\r{RFe}}
\def\PRFi{P_\r{RFi}}
\def\PBr{P_\r{Br}}
\def\EH{E_\r{H}}
\def\Tna{T_{01}}
\def\Tnb{T_{02}}
\def\Tnc{T_{03}}
\def\fei{f_\r{ei}}
\def\nbrp{n_\r{b}^\r{rp}}
\def\Pbpara{P_{\r{b}\parallel}}
\def\Pbperp{P_{\r{b}\perp}}
\def\nueff{\nu_\r{eff}}
\def\ubrp{u_\r{b}^\r{rp}}
\def\Dbrp{D_\r{b}^\r{rp}}
\def\Db{D_\r{b}}
\def\fCX{f_\r{CX}}
\def\Palpe{P_{\alpha\r{e}}}
\def\Palpi{P_{\alpha\r{i}}}

\def\nuNCe{\nu_\r{NCe}}
\def\nuNCi{\nu_\r{NCi}}
\def\nube{\nu_\r{be}}
\def\nubi{\nu_\r{bi}}
\def\nune{\nu_\r{0e}}
\def\nuni{\nu_\r{0i}}
\def\nunb{\nu_\r{0b}}
\def\nuL{\nu_\r{L}}
\def\nuCX{\nu_\r{CX}}
\def\nuion{\nu_\r{ionize}}
\def\nub{\nu_\r{b}}
\def\nuTei{\nu_\r{Tei}}
\def\vb{v_\r{b}}
\def\nediv{n_\r{ediv}}
\def\Tediv{T_\r{ediv}}
\def\Tidiv{T_\r{idiv}}
\def\nuLTe{\nu_{\r{L}T_\r{e}}}
\def\nuLTi{\nu_{\r{L}T_\r{i}}}
\def\nuOL{\nu_\r{OL}}
\def\nubL{\nu_\r{bL}}
\def\vbph{v_{\r{b}\phi}}
\def\vbth{v_{\r{b}\theta}}

\newcommand{\Frac}[2]{%
  {\displaystyle {\displaystyle #1\over \displaystyle #2}}%
}
\newcommand{\rdprd}[2]{\Frac{\partial #1}{\partial #2}}
\newcommand{\aveaveB}[1]{\left< \!\! \left< #1 \right> \!\! \right>}
\newcommand{\hatomega}{\widehat{\omega}}

\def\red#1{\textcolor{red}{#1}}

\begin{document}

\begin{center}
TASK/TX V4.56 [09/09/28]
\end{center}

\section{Equations in $r$ coordinates}
\vspace{-5mm}

\begin{eqnarray}
%=======%
% \phi  %
%=======%
  0 &=&
%
    \frac{1}{r} \ddr r \ddr \phi
%
  + \frac{1}{\epsilon_0}
%
    \left(-e \nee + \Zi e \ni + \Zb e \nb + \Zb e g \nbrp \right)
\\[5mm]
%==========%
% r \Athd  %
%==========%
  \frac{1}{c^2} \ddt \Athd &=&
%
    \ddr \frac{1}{r} \ddr r \Ath
%
  + \mu_0 (-e \nee \ueth + \Zi e \ni \uith + \Zb e \nb \ubth)
\\[5mm]
%========%
% \Aphd  %
%========%
  \frac{1}{c^2} \ddt \Aphd &=&
%
  \frac{1}{r} \ddr r \ddr \Aph
%
  + \mu_0 (-e \nee \ueph + \Zi e \ni \uiph + \Zb e \nb \ubph)
%
%  + \mu_0 j_\r{virtual}
\\[5mm]
%=======%
% \Aph  %
%=======%
  \ddt \Aph &=&
%
  \Aphd
\\[5mm]
%=======%
% \Ath  %
%=======%
  \ddt \Ath &=&
%
  \Athd
\\[5mm]
%=======%
% \nee  %
%=======%
  \ddt\nee &=&
%
  - \frac{1}{r} \ddr r \nee \uer 
%
  + \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, (\nna + \nnb + \nnc)
%
  - \nuL (\nee - \nediv)
\nonumber\\[3mm]&&{}
%
  + (1 - \fCX) \frac{\Pb}{\Eb}
\\[5mm]
%============%
% \nee \uer  %
%============%
  \ddt \nee \uer &=&
%
  - \frac{1}{r} \ddr r \uer \nee \uer
%
  + \frac{1}{r} \ueth \nee \ueth
%
  - \ddr \frac{\nee \Te}{\me}
\nonumber\\[3mm]&&{}
%
  - \frac{e}{\me} \nee \Er
%
  - \frac{e}{\me} \Bph \nee \ueth
%
  + \frac{e}{\me} \Bth \nee \ueph
\\[5mm]
%=============%
% \nee \ueth  %
%=============%
  \ddt \nee \ueth &=&
%
  - \frac{1}{r^2} \ddr r^2 \uer \nee \ueth
%
  + \frac{1}{r^2} \ddr r^3 \nee \mue \ddr \frac{\ueth}{r}
%
  - \frac{e}{\me} \nee \Eth
%
  + \frac{e}{\me} \Bph \nee \uer
\nonumber\\[2.5mm]&&{}
%
  - \nuNCe \nee \ueth
%
  - \nu_\r{ei1} \nee (\ueth - \uith)
%
  - \nu_\r{ei2} \nee (\ueph - \uiph)
\nonumber\\[2.5mm]&&{}
%
  - \frac{\mb}{\me} \nu_\r{be1} \nb (\ueth - \ubth)
%
  - \frac{\mb}{\me} \nu_\r{be2} \nb (\ueph - \ubph)
\nonumber\\[2.5mm]&&{}
%
  - \frac{e^2 \Bph^2 \De}{\me\Te}\nee
    \left[(\ueth - \uith) - \frac{\Bth}{\Bph}(\ueph - \uiph)
     + \frac{B}{\Bph}\ave{\frac{\omega}{m}}_\r{e} r
     + \frac{1-\alpha_\r{e}}{e\Bph}\ddr \Te + \frac{\Er}{\Bph}\right]
%
%   + \frac{\Zi^2 e^2 \Bph^2 \Di}{\me\Ti}\ni
%    \left(\uith - \frac{\Bth}{\Bph}\uiph - \ave{\frac{\omega}{m}}_\r{i}
%     r \right)
\nonumber\\[2.5mm]&&{}
%
%%  + (-e) \Bph
%%    \left\{
%%     \nee \De
%%      \left(
%%        - \rdprd{}{r} \ln \nee
%%        + \frac{(-e) E_r}{\Te}
%%        + \aveaveB{\frac{\omega}{m}} r
%%          \frac{(-e) \Bph}{\Te}
%%      \right)
%%      - \nee \left( \mue - \frac{\De}{2} \right) \rdprd{}{r} \ln \Te
%%    \right\}
%%\nonumber\\[2.5mm]&&{}
%%  - \Zi e \Bph
%%   \left\{
%%     \ni \Di
%%      \left(
%%        - \rdprd{}{r} \ln \ni
%%        + \frac{\Zi e E_r}{\Ti}
%%        + \aveaveB{\frac{\omega}{m}} r
%%          \frac{\Zi e \Bph}{\Ti}
%%      \right)
%%      - \ni \left( \mui - \frac{\Di}{2} \right) \rdprd{}{r} \ln \Ti
%%    \right\}
%%\nonumber\\[2.5mm]&&{}
%
  - 2 \nuL \nee \ueth
%
  - \nune \nee \ueth
%
%  - \nu_\r{eHL} (1 - u_\r{H0}^2) \nee \ueth
%
%  + \nu_\r{eHL} u_{\r{H}\phi} u_{\r{H}\theta} \nee \ueph
%
  + \mu_{e2} \frac{\Bph}{\Bth^2} \frac{1}{\me e} \ddr \Te
\\[5mm]
%=============%
% \nee \ueph  %
%=============%
  \ddt \nee \ueph &=&
%
  - \frac{1}{r} \ddr r \uer \nee \ueph
%
  + \frac{1}{r} \ddr r \nee \mue \ddr \ueph
%
  - \frac{e}{\me} \nee \Eph
%
  - \frac{e}{\me} \Bth \nee \uer 
\nonumber\\[2.5mm]&&{}
%
  - \nu_\r{ei3} \nee (\ueph - \uiph)
%
  - \nu_\r{ei2} \nee (\ueth - \uith)
%
  - \frac{\mb}{\me} \nu_\r{be2} \nb (\ueth - \ubth)
%
  - \frac{\mb}{\me} \nu_\r{be3} \nb (\ueph - \ubph)
\nonumber\\[2.5mm]&&{}
%
  + \frac{e^2 \Bph^2 \De}{\me\Te}\frac{\Bth}{\Bph}\nee
    \left[(\ueth - \uith) - \frac{\Bth}{\Bph}(\ueph - \uiph)
     + \frac{B}{\Bph}\ave{\frac{\omega}{m}}_\r{e}r
     + \frac{1-\alpha_\r{e}}{e\Bph}\ddr \Te + \frac{\Er}{\Bph}\right]
%
%  - \frac{\Zi^2 e^2 \Bph^2 \Di}{\me\Ti}\frac{\Bth}{\Bph}\ni
%    \left(\uith - \frac{\Bth}{\Bph}\uiph - \ave{\frac{\omega}{m}}_\r{i}
%     r \right)
\nonumber\\[2.5mm]&&{}
%
  - 2 \nuL \nee \ueph
%
  - \nune \nee \ueph
%
%  + \nu_\r{eHL} u_{\r{H}\phi} u_{\r{H}\theta} \nee \ueth
%
%  - \nu_\r{eHL} (1 - u_\r{H0}^2) \nee \ueph
\\[5mm]
%===========%
% \nee \Te  %
%===========%
  \ddt \, \frac{3}{2} \nee \Te &=&
%
 - \frac{1}{r} \ddr r
   \left( \frac{5}{2} \uer \nee \Te - \nee \chie \ddr \Te \right)
%
  +  \uer \ddr \nee \Te
%
  -  e \Eth \nee \ueth
%
  -  e \Eph \nee \ueph 
\nonumber\\[2.5mm]&&{}
%
  - \frac{3}{2} \nuTei \nee (\Te - \Ti)
%
%  - \frac{1}{2} \mb \vb \nube \nb (\ueph - \ubph)
%
  - \nuL \Te (\nee - \nediv)
%
  - \frac{3}{2} \nuLTe \nee (\Te - \Tediv)
\nonumber\\[2.5mm]&&{}
%
  - \EH \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, (\nna + \nnb + \nnc)
%
  + \fei \Pb
%
  + \Palpe
%
  + \PRFe
%
  - \PBr
\\[5mm]
%\end{eqnarray}
%\neewpage
%\begin{eqnarray}
%
%======%
% \ni  %
%======%
  \ddt\ni &=&
%
  - \frac{1}{r} \ddr r \ni \uir
%
  + \frac{\nuion}{\Zi}\, \frac{\nee}{\nna +\ nnb + \nnc}\, (\nna + \nnb
  + \nnc)
%
  - \frac{\nuL}{\Zi} (\nee - \nediv) 
\nonumber\\[2.5mm]&&{}
%
  + \nub \nb
%
  + \nub g \nbrp
%
  - \fCX \frac{\Pb}{\Eb}
%
  + \nubL \nb
%
  + S_\r{LC}
%
  - \nuOL \ni
%
\\[5mm]
%===========%
% \ni \uir  %
%===========%
  \ddt \ni \uir &=&
%
  - \frac{1}{r} \ddr r \uir \ni \uir
%
  + \frac{1}{r} \uith \ni \uith
%
  - \ddr \frac{\ni \Ti}{\mi}
\nonumber\\[3mm]&&{}
%
  + \Zi \frac{e}{\mi}\ni \Er
%
  + \Zi \frac{e}{\mi} \Bph \ni \uith
%
  - \Zi \frac{e}{\mi} \Bth \ni \uiph
\\[5mm]
%============%
% \ni \uith  %
%============%
  \ddt \ni \uith &=&
%
  - \frac{1}{r^2} \ddr r^2 \uir \ni \uith
%
  + \frac{1}{r^2} \ddr r^3 \ni \mui \ddr \frac{\uith}{r}
%
  + \Zi \frac{e}{\mi} \ni \Eth
%
  - \Zi \frac{e}{\mi} \Bph \ni \uir
\nonumber\\[2.5mm]&&{}
%
  - \nuNCi \ni \uith
%
  - \frac{\me}{\mi} \nu_\r{ei1} \nee (\uith - \ueth)
%
  - \frac{\me}{\mi} \nu_\r{ei2} \nee (\uiph - \ueph)
%
  - \frac{\mb}{\mi} \nubi \nb (\uith - \ubth)
\nonumber\\[2.5mm]&&{}
%
  + \frac{e^2 \Bph^2 \De}{\mi\Te}\nee
    \left[(\ueth - \uith) - \frac{\Bth}{\Bph}(\ueph - \uiph)
     + \frac{B}{\Bph}\ave{\frac{\omega}{m}}_\r{e} r
     + \frac{1-\alpha_\r{e}}{e\Bph}\ddr \Te + \frac{\Er}{\Bph}\right]
%
%  - \frac{\Zi^2 e^2 \Bph^2 \Di}{\mi\Ti}\ni
%    \left(\uith - \frac{\Bth}{\Bph}\uiph - \ave{\frac{\omega}{m}}_\r{i}
%     r \right)
%
%%  - (-e) \Bph
%%    \left\{
%%      \nee \De
%%      \left(
%%        - \rdprd{}{r} \ln \nee
%%        + \frac{(-e) E_r}{\Te}
%%        + \aveaveB{\frac{\omega}{m}} r
%%          \frac{(-e) \Bph}{\Te}
%%      \right)
%%      - \nee \left( \mue - \frac{\De}{2} \right) \rdprd{}{r} \ln \Te
%%    \right\}
%%\nonumber\\[2.5mm]&&{}
%%  + \Zi e \Bph
%%    \left\{
%%      \ni \Di
%%      \left(
%%        - \rdprd{}{r} \ln \ni
%%        + \frac{\Zi e E_r}{\Ti}
%%        + \aveaveB{\frac{\omega}{m}} r
%%          \frac{\Zi e \Bph}{\Ti}
%%      \right)
%%      - \ni \left( \mui - \frac{\Di}{2} \right) \rdprd{}{r} \ln \Ti
%%    \right\}
\nonumber\\[2.5mm]&&{}
%
  - 2 \nuL \ni \uith
%
  - \nuni \ni \uith
%
  - \nuCX \ni \uith
%
  + S_\r{LC\theta}
%
  - \nuOL \ni \uith
%\nonumber\\[2.5mm]&&{}
%
%  - \nu_\r{iHL} (1 - u_{\r{H}\theta}^2) \ni \uith
%
%  + \nu_\r{iHL} u_{\r{H}\phi} u_{\r{H}\theta} \ni \uiph
%
  - \mu_{i2} \frac{\Bph}{\Bth^2} \frac{1}{\mi \Zi e} \ddr \Ti
%
  + T_{\theta,\r{i}}
\\[5mm]
%============%
% \ni \uiph  %
%============%
  \ddt \ni \uiph &=&
%
  - \frac{1}{r} \ddr r \uir \ni \uiph
%
  + \frac{1}{r} \ddr r \ni \mui \ddr \uiph
%\nonumber\\[2.5mm]&&{}
%
  + \Zi \frac{e}{\mi} \ni \Eph
%
  + \Zi \frac{e}{\mi} \Bth \ni \uir 
\nonumber\\[2.5mm]&&{}
%
  - \frac{\me}{\mi} \nu_\r{ei3} \nee (\uiph - \ueph)
%
  - \frac{\me}{\mi} \nu_\r{ei2} \nee (\uith - \ueth)
%
  - \frac{\mb}{\mi} \nubi \nb (\uiph - \ubph)
\nonumber\\[2.5mm]&&{}
%
  - \frac{e^2 \Bph^2 \De}{\mi\Te}\frac{\Bth}{\Bph}\nee
    \left[(\ueth - \uith) - \frac{\Bth}{\Bph}(\ueph - \uiph)
     + \frac{B}{\Bph}\ave{\frac{\omega}{m}}_\r{e} r
     + \frac{1-\alpha_\r{e}}{e\Bph}\ddr \Te + \frac{\Er}{\Bph}\right]
%
%  + \frac{\Zi^2 e^2 \Bph^2 \Di}{\mi\Ti}\frac{\Bth}{\Bph}\ni
%    \left(\uith - \frac{\Bth}{\Bph}\uiph - \ave{\frac{\omega}{m}}_\r{i}
%     r \right)
\nonumber\\[2.5mm]&&{}
%
  - 2 \nuL \ni \uiph
%
  - \nuni \ni \uiph
%
  - \nuCX \ni \uiph
%
  + S_\r{LC\phi}
%
  - \nuOL \ni \uiph
%
  + T_{\phi,\r{i}}
%\nonumber\\[2.5mm]&&{}
%
%  - \nu_\r{iHL} u_{\r{H}\phi} u_{\r{H}\theta} \ni \uith
%
%  + \nu_\r{iHL} (1 - u_{\r{H}\theta}^2) \ni \uiph
\\[5mm]
%==========%
% \ni \Ti  %
%==========%
  \ddt \, \frac{3}{2} \ni \Ti &=&
%
 - \frac{1}{r} \ddr r
   \left( \frac{5}{2} \uir \ni \Ti - \ni \chii \ddr \Ti \right)
%
  + \uir \ddr \ni \Ti
%
  + \Zi e \Eth \ni \uith
%
  + \Zi e \Eph \ni \uiph
\nonumber\\[2.5mm]&&{}
%
  - \frac{3}{2} \nuTei \nee (\Ti - \Te)
%
%  - \frac{1}{2} \mb \vb \nubi \nb (\uiph - \ubph)
  + \mb \vb \frac{\Bth \uith + \Bph \uiph}{B} \frac{\Pbpara}{\Eb}
%
  - \frac{\nuL}{\Zi} \Ti (\nee - \nediv)
\nonumber\\[2.5mm]&&{}
%
  - \frac{3}{2} \nuLTi \ni (\Ti - \Tidiv)
%
  + \frac{3}{2} \frac{\nuion}{\Zi} \frac{\nee}{\nna + \nnb + \nnc}\, (\nna \Tna +
  \nnb \Tnb + \nnc \Tnc)
\nonumber\\[2.5mm]&&{}
%
  - \frac{3}{2} \nuCX \ni (\Ti - \Tna)
%
  + (1 - \fei) \Pb
%
  + \Palpi
%
  + \PRFi
\\[5mm]
%======%
% \nb  %
%======%
  \ddt \nb &=&
%
    \frac{\Pb}{\Eb}
%
  - g \frac{\Pbperp}{\Eb}
%
  - \nub \nb
%
  - \nubL \nb
%
  - \nueff g (\sqrt{\delta} \nb - \nbrp)
%
  + \frac{1}{r} \ddr r f_t \Db \ddr \nb
\\[5mm]
%============%
% \nb \ubth  %
%============%
  \ddt \nb \ubth &=&
%
    \Zb \frac{e}{\mb} \nb \Eth
%
%  - \nuNCi \nb \ubth
%
  - \nu_\r{be1} \nb (\ubth - \ueth)
%
  - \nu_\r{be2} \nb (\ubph - \ueph)
%
  - \nubi \nb (\ubth - \uith)
\nonumber\\[2.5mm]&&{}
%
  - \nunb \nb \ubth
%
  - \nuCX \nb \ubth
%
  + \vbth \frac{\Pbpara}{\Eb}
%
  - \nubL \nb \ubth
%
  - \nueff g \sqrt{\delta} \nb \ubth
\nonumber\\[2.5mm]&&{}
%
  + \frac{1}{r} \ddr r \ubth f_t \Db \ddr \nb
\\[5mm]
%=============%
% \nb  \ubph  %
%=============%
  \ddt \nb \ubph &=&
%
  \Zb \frac{e}{\mb} \nb \Eph
%
  - \nu_\r{be3} \nb (\ubph - \ueph)
%
  - \nu_\r{be2} \nb (\ubth - \ueth)
%
  - \nubi \nb (\ubph - \uiph)
\nonumber\\[2.5mm]&&{}
%
  - \nunb \nb \ubph
%
  - \nuCX \nb \ubph
%
  + \vbph \frac{\Pbpara}{\Eb}
%
  - \nubL \nb \ubph
%
  - \nueff g \sqrt{\delta} \nb \ubph
\nonumber\\[2.5mm]&&{}
%
  + \frac{1}{r} \ddr r \ubph f_t \Db \ddr \nb
\\[5mm]
%=======%
% \nna  %
%=======%
  \ddt \nna &=& 
%
    \frac{1}{r} \ddr r D_{01} \ddr \nna
%
  - \frac{1}{\Zi}\, \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, \nna
%
  - \nuCX\, \frac{\ni}{\nna + \nnb}\, \nna
\nonumber\\[2.5mm]&&{}
%
  + \gamma_0 \frac{\nuL}{\Zi} \, (\nee - \nediv)
\\[5mm]
%=======%
% \nnb  %
%=======%
  \ddt \nnb &=& 
%
    \frac{1}{r} \ddr r D_{02} \ddr \nnb
%
  - \frac{1}{\Zi}\, \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, \nnb
%
  + \nuCX\, \frac{\ni}{\nna + \nnb}\, \nna
%
%  + \fCX \frac{\Pb}{\Eb}
\\[5mm]
%=======%
% \nnc  %
%=======%
  \ddt \nnc &=& 
%
    \frac{1}{r} \ddr r D_{03} \ddr \nnc
%
  - \frac{1}{\Zi}\, \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, \nnc
%
  + \fCX \frac{\Pb}{\Eb}
\\[5mm]
%========%
% \nbrp  %
%========%
  \ddt \nbrp &=&
%
    \frac{\Pbperp}{\Eb}
%
  + \nueff (\sqrt{\delta} \nb - \nbrp)
%
  - \nub \nbrp
%
  - \frac{1}{r} \ddr \left( r \ubrp \nbrp - r \Dbrp \ddr \nbrp \right)
%
\end{eqnarray}

\bigskip

\section{Relations Between Fields and Potentials}
\setcounter{equation}{0}

\begin{eqnarray}
%======%
% \Er  %
%======%
  \Er &=&
%
  - \ddr \phi
\\[5mm]
%=======%
% \Eth  %
%=======%
  \Eth &=&
%
  - \ddt \Ath = - \Athd
\\[5mm]
%=======%
% \Eph  %
%=======%
  \Eph &=&
%
  - \ddt \Aph = - \Aphd
\\[5mm]
%=======%
% \Bth  %
%=======%
  \Bth &=&
%
  - \ddr \Aph
\\[5mm]
%=======%
% \Bph  %
%=======%
  \Bph &=&
%
  \frac{1}{r} \ddr r \Ath
\end{eqnarray}

\bigskip

\section{Equations in $s(=r^2)$ coordinates}
\setcounter{equation}{0}

\begin{eqnarray}
%=======%
% \phi  %
%=======%
  0 &=&
%
    4 \dds s \dds \red{\phi}
%
  + \frac{1}{\epsilon_0}
%
    \left(-e \red{\nee} + \Zi e \red{\ni} + \Zb e \red{\nb} + \Zb e g \red{\nbrp} \right)
\\[5mm]
%==========%
% r \Athd  %
%==========%
  \frac{1}{c^2} \ddt \red{r \Athd} &=&
%
    \left(   4 \dds s \dds \red{r \Ath}
%
           - 4 \dds \red{r \Ath} \right)
%
  + \mu_0 (-e \red{r \nee \ueth} + \Zi e \red{r \ni \uith} + \Zb e \red{r \nb \ubth})
\\[5mm]
%========%
% \Aphd  %
%========%
  \frac{1}{c^2} \ddt \red{\Aphd} &=&
%
    4 \dds s \dds \red{\Aph}
%
  + \mu_0 (-e \red{\nee \ueph} + \Zi e \red{\ni \uiph} + \Zb e \red{\nb \ubph})
%
%  + \mu_0 j_\r{virtual}
\\[5mm]
%=======%
% \Aph  %
%=======%
  \ddt \red{\Aph} &=&
%
  \red{\Aphd}
\\[5mm]
%=========%
% r \Ath  %
%=========%
  \ddt \red{r \Ath} &=&
%
  \red{r \Athd}
\\[5mm]
%=======%
% \nee  %
%=======%
  \ddt \red{\nee} &=&
%
  - 2 \dds \red{r \nee \uer} 
%
  + \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, (\red{\nna} + \red{\nnb}
  + \red{\nnc})
%
  - \nuL (\red{\nee} - \nediv)
\nonumber\\[3mm]&&{}
%
  + (1 - \fCX) \frac{\Pb}{\Eb}
\\[5mm]
%==============%
% r \nee \uer  %
%==============%
  \ddt \red{r \nee \uer} &=&
%
  - 2 r \dds \uer \red{r \nee \uer}
%
  + \frac{\ueth}{r} \red{r \nee \ueth}
%
  - 2 s \dds \frac{\red{\nee \Te}}{\me}
\nonumber\\[3mm]&&{}
%
  + 2 \frac{e}{\me} s \nee \dds \red{\phi}
%
  - \frac{e}{\me} \Bph \red{r \nee \ueth}
%
  - 2 \frac{e}{\me} s \dd{\Aph} \red{\nee \ueph}
\\[5mm]
%===============%
% r \nee \ueth  %
%===============%
  \ddt \red{r \nee \ueth} &=&
%
  - 2 \dds r \uer \red{r \nee \ueth}
%
  + \left(    4 \dds s \mue \dds \red{r \nee \ueth}
%
            - 4 \dds s \mue r \ueth \dd{\red{\nee}}
%
            - 4 \dds \mue \red{r \nee \ueth} \right)
\nonumber\\[2.5mm]&&{}
%
  + \frac{e}{\me} \nee \red{r \Athd}
%
  + \frac{e}{\me} \Bph \red{r \nee \uer}
%
  - \nuNCe \red{r \nee \ueth}
%
  - \nu_\r{ei1} \red{r \nee} (\red{\ueth} - \red{\uith})
%
  + 2 s \dd{\Aph} \frac{\nu_\r{ei2}}{\Bth} \red{\nee} (\red{\ueph} - \red{\uiph})
\nonumber\\[2.5mm]&&{}
%
  - \frac{\mb}{\me} \nu_\r{be1} \red{r \nb} (\red{\ueth} - \red{\ubth})
%
  + 2 \frac{\mb}{\me} s \dd{\Aph} \frac{\nu_\r{be2}}{\Bth} \red{\nee}
  (\red{\ueph} - \red{\ubph})
\nonumber\\[2.5mm]&&{}
%
  - \frac{e^2 \Bph^2 \De}{\me\Te}\red{\nee}
    \left[(\red{r \ueth} - \red{r \uith})+
     2\dd{\Aph}\frac{s}{\Bph}(\red{\ueph} - \red{\uiph})
     + s \frac{B}{\Bph}\ave{\frac{\omega}{m}}_\r{e}\right.
\nonumber\\[2.5mm]&&{}
%
    \left.+ 2 s \frac{1-\alpha_\r{e}}{e\Bph} \dds \Te
     - 2 s \frac{1}{\Bph} \dds \phi\right]
%
%   + \frac{\Zi^2 e^2 \Bph^2 \Di}{\me\Ti}\ni
%    \left(r \uith + 2\dd{\Aph}\frac{s}{\Bph}\uiph - s
%     \ave{\frac{\omega}{m}}_\r{i} \right)
%
  - 2 \nuL \red{r \nee \ueth}
%
  - \nune \red{r \nee \ueth}
\nonumber\\[2.5mm]&&{}
%
%  - \nu_\r{eHL} (1 - u_\r{H0}^2) r \nee \ueth
%
%  + r \nu_\r{eHL} u_{\r{H}\phi} u_{\r{H}\theta} \nee \ueph
%
  + 2 \mu_{e2} \frac{\Bph}{\Bth^2} \frac{s}{\me e \nee} \dds \red{\nee\Te}
%
  - 2 \mu_{e2} \frac{\Bph}{\Bth^2} \frac{s}{\me e \nee^2} \dd{\nee}
  \red{\nee\Te}
\\[5mm]
%=============%
% \nee \ueph  %
%=============%
  \ddt \red{\nee \ueph} &=&
%
  - 2 \dds r \uer \red{\nee \ueph}
%
  + \left(   4 \dds s \mue \dds \red{\nee \ueph}
%
           - 4 \dds s \mue \ueph \dd{\red{\nee}} \right)
\nonumber\\[2.5mm]&&{}
%
  + \frac{e}{\me} \nee \red{\Aphd}
%
  + 2 \frac{e}{\me} \dd{\Aph} \red{r \nee \uer} 
%
  - \nu_\r{ei3} \red{\nee} (\red{\ueph} - \red{\uiph})
%
  + 2 \dd{\Aph} \frac{\nu_\r{ei2}}{\Bth} \red{r \nee} (\red{\ueth} - \red{\uith})
\nonumber\\[2.5mm]&&{}
%
  + 2 \frac{\mb}{\me} \dd{\Aph} \frac{\nu_\r{be2}}{\Bth} \red{r \nee} (\red{\ueth} - \red{\uith})
%
  - \frac{\mb}{\me} \nu_\r{be3} \red{\nb} (\red{\ueph} - \red{\ubph})
\nonumber\\[2.5mm]&&{}
%
  - 2\frac{e^2 \Bph \De}{\me\Te}\dd{\Aph}\red{\nee}
    \left[(\red{r \ueth} - \red{r \uith})
     + s \frac{B}{\Bph}\ave{\frac{\omega}{m}}_\r{e}
     + 2 s \frac{1-\alpha_\r{e}}{e\Bph} \dds \Te
     - 2 s \frac{1}{\Bph} \dds \phi \right]
\nonumber\\[2.5mm]&&{}
%
  - \frac{e^2 \Bph^2
  \De}{\me\Te}\left(\frac{\Bth}{\Bph}\right)^2\red{\nee}(\red{\ueph} - \red{\uiph})
%
%   + 2\frac{\Zi^2 e^2 \Bph \Di}{\me\Ti}\dd{\Aph}\ni
%    \left(r \uith - s \ave{\frac{\omega}{m}}_\r{i} \right)
%\nonumber\\[2.5mm]&&{}
%
%   + \frac{\Zi^2 e^2 \Bph^2
%   \Di}{\me\Ti}\left(\frac{\Bth}{\Bph}\right)^2\ni\uiph
%
  - 2 \nuL \red{\nee \ueph}
%
  - \nune \red{\nee \ueph}
%
%  + \nu_\r{eHL} u_{\r{H}\phi} u_{\r{H}\theta} \nee \ueth
%
%  - \nu_\r{eHL} (1 - u_\r{H0}^2) \nee \ueph
\\[5mm]
%===========%
% \nee \Te  %
%===========%
  \ddt \, \frac{3}{2} \red{\nee \Te} &=&
%
   - 5 \dds r \uer \red{\nee \Te}
%
   + \left(   4 \dds s \chie \dds \red{\nee \Te}
%
            - 4 \dds s \chie \Te \dd{\red{\nee}} \right)
%
  + 2 r \uer \dds \red{\nee \Te}
%
\nonumber\\[2.5mm]&&{}
%
  -  e \frac{\Eth}{r} \red{r \nee \ueth}
%
  -  e \Eph \red{\nee \ueph} 
%
  - \frac{3}{2} \nuTei \red{\nee} (\red{\Te} - \red{\Ti})
%
%  - \frac{1}{2} \mb \vb \nube \nb (\ueph - \ubph)
%
\nonumber\\[2.5mm]&&{}
%
  - \nuL \Te (\red{\nee} - \nediv)
%
  - \frac{3}{2} \nuLTe \red{\nee} (\red{\Te} - \Tediv)
\nonumber\\[2.5mm]&&{}
%
  - \EH \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, (\red{\nna} + \red{\nnb} + \red{\nnc})
%
  + \fei \Pb
%
  + \Palpe
%
  + \PRFe
%
  - \PBr
\\[5mm]
%======%
% \ni  %
%======%
  \ddt \red{\ni} &=&
%
  - 2 \dds \red{r \ni \uir}
%
  + \frac{\nuion}{\Zi}\, \frac{\nee}{\nna + \nnb + \nnc}\, (\red{\nna} +
  \red{\nnb} + \red{\nnc})
%
  - \frac{\nuL}{\Zi} (\red{\nee} - \nediv) 
\nonumber\\[2.5mm]&&{}
%
  + \nub \red{\nb}
%
  + \nub g \red{\nbrp}
%
  - \fCX \frac{\Pb}{\Eb}
%
  + \nubL \red{\nb}
%
  + S_\r{LC}
%
  - \nuOL \red{\ni}
\\[5mm]
%=============%
% r \ni \uir  %
%=============%
  \ddt \red{r \ni \uir} &=&
%
  - 2 r \dds \uir \red{r \ni \uir}
%
  + \frac{\ueth}{r} \red{r \ni \uith}
%
  - 2 s \dds \frac{\red{\ni \Ti}}{\mi}
\nonumber\\[3mm]&&{}
%
  - 2 \Zi \frac{e}{\mi} s \ni \dds \red{\phi}
%
  + \Zi \frac{e}{\mi} \Bph \red{r \ni \uith}
%
  + 2 \Zi \frac{e}{\mi} s \dd{\Aph} \red{\ni \uiph}
\\[5mm]
%==============%
% r \ni \uith  %
%==============%
  \ddt \red{r \ni \uith} &=&
%
  - 2 \dds r \uir \red{r \ni \uith}
%
  + \left(   4 \dds s \mui \dds \red{r \ni \uith}
%
           - 4 \dds s \mui r \uith \dd{\red{\ni}}
%
	   - 4 \dds \mui \red{r \ni \uith} \right)
\nonumber\\[2.5mm]&&{}
%
- \Zi \frac{e}{\mi} \ni \red{r \Athd}
%
  - \Zi \frac{e}{\mi} \Bph \red{r \ni \uir}
%
  - \nuNCi \red{r \ni \uith}
%
  - \frac{\me}{\mi} \nu_\r{ei1} \red{r \nee} (\red{\uith} - \red{\ueth})
\nonumber\\[2.5mm]&&{}
%
  + 2 \frac{\me}{\mi} s \dd{\Aph} \frac{\nu_\r{ei2}}{\Bth} \red{\nee} (\red{\uiph} - \red{\ueph})
%
  - \frac{\mb}{\mi} \nubi \red{r \nb} (\red{\uith} - \red{\ubth})
\nonumber\\[2.5mm]&&{}
%
  + \frac{e^2 \Bph^2 \De}{\mi\Te}\red{\nee}
    \left[(\red{r \ueth} - \red{r \uith}) +
     2\dd{\Aph}\frac{s}{\Bph}(\red{\ueph} - \red{\uiph})
     + s \frac{B}{\Bph}\ave{\frac{\omega}{m}}_\r{e}\right.
\nonumber\\[2.5mm]&&{}
%
    \left.+ 2 s \frac{1-\alpha_\r{e}}{e\Bph} \dds \Te
     - 2 s \frac{1}{\Bph} \dds \phi \right]
%
%  - \frac{\Zi^2 e^2 \Bph^2 \Di}{\mi\Ti}\ni
%    \left(r \uith + 2\dd{\Aph}\frac{s}{\Bph}\uiph - s
%     \ave{\frac{\omega}{m}}_\r{i} \right)
%\nonumber\\[2.5mm]&&{}
%
  - 2 \nuL \red{r \ni \uith}
%
  - \nuni \red{r \ni \uith}
%
  - \nuCX \red{r \ni \uith}
\nonumber\\[2.5mm]&&{}
%
  + r S_\r{LC\theta}
%
  - \nuOL \red{r \ni \uith}
%
%  - \nu_\r{iHL} (1 - u_{\r{H}\theta}^2) r \ni \uith
%
%  + r \nu_\r{iHL} u_{\r{H}\phi} u_{\r{H}\theta} \ni \uiph
%
  - 2 \mu_{i2} \frac{\Bph}{\Bth^2} \frac{s}{\mi \Zi e \ni} \dds \red{\ni\Ti}
%
  + 2 \mu_{i2} \frac{\Bph}{\Bth^2} \frac{s}{\mi \Zi e \ni^2} \dd{\ni} \red{\ni\Ti}
%
  + T_{\theta,\r{i}}
\\[5mm]
%============%
% \ni \uiph  %
%============%
  \ddt \red{\ni \uiph} &=&
%
  - 2 \dds r \uir \red{\ni \uiph}
%
  + \left(   4 \dds s \mui \dds \red{\ni \uiph}
%
           - 4 \dds s \mui \uiph \dd{\red{\ni}} \right)
%
  - \Zi \frac{e}{\mi} \ni \red{\Aphd}
%
  - 2 \Zi \frac{e}{\mi} \dd{\Aph} \red{r \ni \uir} 
\nonumber\\[2.5mm]&&{}
%
  - \frac{\me}{\mi} \nu_\r{ei3} \red{\nee} (\red{\uiph} - \red{\ueph})
%
  + 2 \frac{\me}{\mi} \dd{\Aph} \frac{\nu_\r{ei2}}{\Bth} \red{r \nee} (\red{\uith} - \red{\ueth})
%
  - \frac{\mb}{\mi} \nubi \red{\nb} (\red{\uiph} - \red{\ubph})
\nonumber\\[2.5mm]&&{}
%
  + 2\frac{e^2 \Bph \De}{\mi\Te}\dd{\Aph}\red{\nee}
    \left[(\red{r \ueth} - \red{r \uith})
     + s \frac{B}{\Bph}\ave{\frac{\omega}{m}}_\r{e}
     + 2 s \frac{1-\alpha_\r{e}}{e\Bph} \dds \Te
     - 2 s \frac{1}{\Bph} \dds \phi \right]
\nonumber\\[2.5mm]&&{}
%
  + \frac{e^2 \Bph^2
  \De}{\mi\Te}\left(\frac{\Bth}{\Bph}\right)^2\red{\nee}(\red{\ueph} - \red{\uiph})
%
%  - 2\frac{\Zi^2 e^2 \Bph \Di}{\mi\Ti}\dd{\Aph}\ni
%    \left(r \uith - s \ave{\frac{\omega}{m}}_\r{i} \right)
%\nonumber\\[2.5mm]&&{}
%
%  - \frac{\Zi^2 e^2 \Bph^2 \Di}{\mi\Ti}\left(\frac{\Bth}{\Bph}\right)^2\ni\uiph
%
  - 2 \nuL \red{\ni \uiph}
%
  - \nuni \red{\ni \uiph}
%
  - \nuCX \red{\ni \uiph}
\nonumber\\[2.5mm]&&{}
%
  + S_\r{LC\phi}
%
  - \nuOL \red{\ni \uiph}
%
  + T_{\phi,\r{i}}
%
%  - \nu_\r{iHL} u_{\r{H}\phi} u_{\r{H}\theta} \ni \uith
%
%  + \nu_\r{iHL} (1 - u_{\r{H}\theta}^2) \ni \uiph
\\[5mm]
%==========%
% \ni \Ti  %
%==========%
  \ddt \, \frac{3}{2} \red{\ni \Ti} &=&
%
  - 5 \dds r \uer \red{\nee \Te}
%
  + \left(   4 \dds s \chie \dds \red{\ni \Ti} 
%
           - 4 \dds s \chii \Ti \dd{\red{\ni}} \right)
%
  + 2 r \uir \dds \red{\ni \Ti}
%
\nonumber\\[2.5mm]&&{}
%
  + \Zi e \frac{\Eth}{r} \red{r \ni \uith}
%
  + \Zi e \Eph \red{\ni \uiph}
%
  - \frac{3}{2} \nuTei \red{\nee} (\red{\Ti} - \red{\Te})
%
%  - \frac{1}{2} \mb \vb \nubi \nb (\uiph - \ubph)
  + \mb \vb \frac{\Bth \uith + \Bph \uiph}{B} \frac{\Pbpara}{\Eb}  
\nonumber\\[2.5mm]&&{}
%
  - \frac{\nuL}{\Zi} \Ti (\red{\nee} - \nediv)
%
  - \frac{3}{2} \nuLTi \red{\ni} (\red{\Ti} - \Tidiv)
\nonumber\\[2.5mm]&&{}
%
  + \frac{3}{2} \frac{\nuion}{\Zi} \frac{\nee}{\nna + \nnb + \nnc}\, (\red{\nna} \Tna +
  \red{\nnb} \Tnb + \red{\nnc} \Tnc)
%
  - \frac{3}{2} \nuCX \red{\ni} (\red{\Ti} - \Tna)
\nonumber\\[2.5mm]&&{}
%
  + (1 - \fei) \Pb
%
  + \Palpi
%
  + \PRFi
\\[5mm]
%======%
% \nb  %
%======%
  \ddt \red{\nb} &=&
%
    \frac{\Pb}{\Eb}
%
  - g \frac{\Pbperp}{\Eb}
%
  - \nub \red{\nb}
%
  - \nubL \red{\nb}
%
  - \nueff g (\sqrt{\delta}\red{\nb} - \red{\nbrp})
%
  + 4 \dds s f_t \Db \dds \red{\nb}
\\[5mm]
%==============%
% r \nb \ubth  %
%==============%
  \ddt \red{r \nb \ubth} &=&
%
  - \Zb \frac{e}{\mb} \nb \red{r \Athd}
%
%  - \nuNCi r \nb \ubth
%
  - \nu_\r{be1} \red{r \nb} (\red{\ubth} - \red{\ueth})
%
  + 2 s \dd{\Aph} \frac{\nu_\r{be2}}{\Bth} \red{\nb} (\red{\ubph} - \red{\ueph})
%
  - \nubi \red{r \nb} (\red{\ubth} - \red{\uith})
\nonumber\\[2.5mm]&&{}
%
  - \nunb \red{r \nb \ubth}
%
  - \nuCX \red{r \nb \ubth}
%
  + r \vbth \frac{\Pbpara}{\Eb}
%
  - \nubL \red{r \nb \ubth}
%
  - \nueff g \sqrt{\delta} \red{r \nb \ubth}
\nonumber\\[2.5mm]&&{}
%
  + 4 \dds s f_t \Db \dds \red{r \nb \ubth}
%
  - 4 \dds \left( s f_t \Db \dd{r\ubth} \right) \red{\nb}
\\[5mm]
%============%
% \nb \ubph  %
%============%
  \ddt \red{\nb \ubph} &=&
%
  - \Zb \frac{e}{\mb} \nb \red{\Aphd}
%
  - \nu_\r{be3} \red{\nb} (\red{\ubph} - \red{\ueph})
%
  + 2 \dd{\Aph} \frac{\nu_\r{be2}}{\Bth} \red{\nb} (\red{\ubth} - \red{\ueth})
%
  - \nubi \red{\nb} (\red{\ubph} - \red{\uiph})
\nonumber\\[2.5mm]&&{}
%
  - \nunb \red{\nb \ubph}
%
  - \nuCX \red{\nb \ubph}
%
  + \vbph \frac{\Pbpara}{\Eb}
%
  - \nubL \red{\nb \ubph}
%
  - \nueff g \sqrt{\delta} \red{\nb \ubph}
\nonumber\\[2.5mm]&&{}
%
  + 4 \dds s f_t \Db \dds \red{\nb \ubph}
%
  - 4 \dds \left( s f_t \Db \dd{\ubph} \right) \red{\nb}
\\[5mm]
%=======%
% \nna  %
%=======%
  \ddt \red{\nna} &=& 
%
    4 \dds s D_{01} \dds \red{\nna}
%
  - \frac{1}{\Zi}\, \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, \red{\nna}
%
  - \nuCX\, \frac{\ni}{\nna + \nnb}\, \red{\nna}
\nonumber\\[2.5mm]&&{}
%
  + \gamma_0 \frac{\nuL}{\Zi} \, (\red{\nee} - \nediv)
\\[5mm]
%=======%
% \nnb  %
%=======%
  \ddt \red{\nnb} &=& 
%
    4 \dds s D_{02} \dds \red{\nnb}
%
  - \frac{1}{\Zi}\, \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, \red{\nnb}
%
  + \nuCX\, \frac{\ni}{\nna + \nnb}\, \red{\nna}
%
%  + \fCX \frac{\Pb}{\Eb}
\\[5mm]
%=======%
% \nnc  %
%=======%
  \ddt \red{\nnc} &=& 
%
    4 \dds s D_{03} \dds \red{\nnc}
%
  - \frac{1}{\Zi}\, \nuion\, \frac{\nee}{\nna + \nnb + \nnc}\, \red{\nnc}
%
  + \fCX \frac{\Pb}{\Eb}
\\[5mm]
%========%
% \nbrp  %
%========%
  \ddt \red{\nbrp} &=&
%
    \frac{\Pbperp}{\Eb}
%
  + \nueff (\sqrt{\delta} \red{\nb} - \red{\nbrp})
%
  - \nub \red{\nbrp}
%
  - 2 \dds r \ubrp \red{\nbrp} + 4 \dds s \Dbrp \dds \red{\nbrp}
%
\end{eqnarray}

\bigskip

\section{Boundary Conditions}

\begin{tabular}{|c|c|c|c|c|}\hline
 num. & name & variable      & center & edge \\\hline
 1    & LQm1 & $\phi$        & N        & 0\\
 2    & LQm2 & $r \Athd$     & 0        & $2s_b B_\r{V}$\\
 3    & LQm3 & $\Aphd$       & N        & $-2bB_{\theta\r{b}}$\\
 4    & LQm4 & $\Aph$        & $\times$ & $\times$\\
 5    & LQm5 & $r \Ath$      & $\times$ & $\times$\\
 6    & LQe1 & $\nee$         & $\times$ & $\times$(N)\\
 7    & LQe2 & $r \nee \uer$  & 0        & 0($\times$)\\
 8    & LQe3 & $r \nee \ueth$ & 0        & 0\\
 9    & LQe4 & $\nee \ueph$   & N        & 0\\
 10   & LQe5 & $\nee \Te$     & N        & N\\
 11   & LQi1 & $\ni$         & $\times$ & $\times$(N)\\
 12   & LQi2 & $r \ni \uir$  & 0        & 0($\times$)\\
 13   & LQi3 & $r \ni \uith$ & 0        & 0\\
 14   & LQi4 & $\ni \uiph$   & N        & 0\\
 15   & LQi5 & $\ni \Ti$     & N        & N\\
 16   & LQb1 & $\nb$         & $\times$(N) & $\times$(N)\\
 17   & LQb3 & $r \nb \ubth$ & $\times$(0) & $\times$(0)\\
 18   & LQb4 & $\nb \ubph$   & $\times$(0) & $\times$(0)\\
 19   & LQn1 & $\nna$        & N        & $2bS_\r{gas}$\\
 20   & LQn2 & $\nnb$        & N        & 0\\
 20   & LQn3 & $\nnc$        & N        & 0\\
 21   & LQr1 & $\nbrp$       & N        & N\\\hline
 \multicolumn{5}{l}{0: Dirichlet condition $(u=0)$}\\
 \multicolumn{5}{l}{N: Neumann condition $(u'=0)$}\\
 \multicolumn{5}{l}{value: Neumann condition $(u'=\mbox{value})$}\\
 \multicolumn{5}{l}{$\times$: no condition}
\end{tabular}


\end{document}
