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%%
%% This is file `mfpic4ode.tex',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% mfpic4ode.dtx (with options: `tex')
%%
%% This is a generated file.
%%
%% Copyright (C) 2007 by Robert Marik <marik@mendelu.cz>
%%
%% This file may be distributed and/or modified under the conditions of
%% the LaTeX Project Public License, either version 1.2 of this license
%% or (at your option) any later version. The latest version of this
%% license is in:
%%
%% http://www.latex-project.org/lppl.txt
%%
%% and version 1.2 or later is part of all distributions of LaTeX version
%% 1999/12/01 or later.
%%
\catcode`\@=11
\newif\ifcolorODEarrow
%%%\colorODEarrowfalse
\colorODEarrowtrue
%%% The line from one point to another
\def\ODEline#1#2{\lines{#1,#2}}
%%% The variable ODErhs is used to store the function from the right
%%% hand side of ODE in the form y'=f(x,y). We use command
%%% ODEdefineequation to set up this variable.
\def\ODEdefineequation#1{\fdef{ODErhs}{x,y}{#1}}
%%% Integral curve using Euler method. The step of this method is
%%% ODEstep and the number of steps is ODEstepcount. The points are
%%% stored in metapost variables x1,y1.
\def\trajectory#1#2{
\begin{connect}
\mfsrc{x1:=#1;y1:=#2;
for i=1 upto ODEstepcount:
x2:=x1+ODEstep;
y2:=y1+ODEstep*ODErhs(x1,y1);}
\ODEline{z1}{z2}
\mfsrc{
if ((y2>yneg) and (y2<ypos)): x1:=x2; y1:=y2 fi;
endfor
}
\end{connect}
}
%%% Integral curve using Runge--Kutta method.
\def\trajectoryRK#1#2{
\begin{connect}
\mfsrc{x1:=#1;y1:=#2;
for i=1 upto ODEstepcount:
k1:=ODErhs(x1,y1);
x3:=x1+(ODEstep/2);
y3:=y1+k1*(ODEstep/2);
k2:=ODErhs(x3,y3);
x2:=x1+ODEstep;
y2:=y1+ODEstep*k2;}
\ODEline{z1}{z2}
\mfsrc{
if ((y2>yneg) and (y2<ypos)): x1:=x2; y1:=y2 fi;
endfor
}
\end{connect}
}
%%% Integral curve using fourth order Runge--Kutta method.
\def\trajectoryRKF#1#2{
\begin{connect}
\mfsrc{x1:=#1;y1:=#2;
for i=1 upto ODEstepcount:
k1:=ODErhs(x1,y1);
x3:=x1+(ODEstep/2);
y3:=y1+k1*(ODEstep/2);
k2:=ODErhs(x3,y3);
y4:=y1+k2*(ODEstep/2);
k3:=ODErhs(x3,y4);
y5:=y1+k3*(ODEstep/2);
k4:=ODErhs(x3,y5);
kk:=(k1+2*k2+2*k3+k4)/6;
x2:=x1+ODEstep;
y2:=y1+ODEstep*kk;}
\ODEline{z1}{z2}
\mfsrc{
if ((y2>yneg) and (y2<ypos)): x1:=x2; y1:=y2 fi;
endfor
}
\end{connect}
}
\def\ODEarrow#1#2{
\mfsrc{x1:=#1; y1:=#2;
x3:=x1+(ODEarrowlength)/((xscale)++(ODErhs(#1,#2)*yscale));
y3:=y1+(ODEarrowlength*ODErhs(#1,#2))/((xscale)++(ODErhs(#1,#2)*yscale));
if y3>y1:ODEcolorarrow:=blue else: ODEcolorarrow:=red fi;
}
\ifcolorODEarrow
\drawcolor{ODEcolorarrow} \headcolor{ODEcolorarrow}
\fi
\draw\arrow\lines{z1,z3}
}
\def\ODEarrows#1{\ODE@cycle@points#1;,;}
\def\trajectories#1{\ODE@cycle@IC#1;,;}
\def\ODE@last@point{}
\def\ODE@cycle@points#1,#2;{\def\temp{#1}\ifx\temp\ODE@last@point\let\next\relax
\else\ODEarrow{#1}{#2}\relax\let\next\ODE@cycle@points\fi\next}
\def\ODE@cycle@IC#1,#2;{\def\temp{#1}\ifx\temp\ODE@last@point\let\next\relax
\else
\trajectoryRKF{#1}{#2}\relax\let\next\ODE@cycle@IC\fi\next}
\mfsrc{path p,q;color ODEcolorarrow;}
%%% One-dimensional autonomous systems y'=f(y) where '=d/dx
\def\ODEharrow#1{
\mfsrc{x1:=#1;
if ODErhs(0,x1)>0: x3:=x1+ODEarrowlength else: x3:=x1-ODEarrowlength fi;
if ODErhs(0,x1)*ODErhs(0,x3)<0: x1:=-100;x3:=-100 fi;
if x3>x1:ODEcolorarrow:=blue else: ODEcolorarrow:=red fi;
}
\ifcolorODEarrow \drawcolor{ODEcolorarrow}
\headcolor{ODEcolorarrow} \fi
\pen{1.5pt}
\draw\arrow\lines{(x1,0),(x3,0)}
}
\def\ODEvarrow#1{
\mfsrc{x1:=#1;
if ODErhs(0,#1)>0:
x3:=x1+(ODEarrowlength/yscale) else: x3:=x1-(ODEarrowlength/yscale) fi;
if ODErhs(0,x1)*ODErhs(0,x3)<0: x1:=-100;x3:=-100 fi;
if x3>x1:ODEcolorarrow:=blue else: ODEcolorarrow:=red fi;
}
\ifcolorODEarrow \drawcolor{ODEcolorarrow}
\headcolor{ODEcolorarrow} \fi
\pen{1.5pt}
\draw\arrow\lines{(0,x1),(0,x3)}
}
%%% Two-dimensional autonomous systems x'=f(x,y), y'=g(x,y) where '=d/dt
\def\ASdefineequations#1#2{\fdef{ASf}{x,y}{#1}\fdef{ASg}{x,y}{#2}}
\def\AStrajectory#1#2{
\begin{connect}
\mfsrc{x1:=#1;y1:=#2;
for i=1 upto ODEstepcount:
x2:=x1+ODEstep*ASf(x1,y1);
y2:=y1+ODEstep*ASg(x1,y1);}
\ODEline{z1}{z2}
\mfsrc{
if ((y2>yneg) and (y2<ypos)): x1:=x2; y1:=y2 fi;
endfor
}
\end{connect}
}
\def\ASarrow#1#2{
\mfsrc{x1:=#1; y1:=#2;
x3:=x1+(ODEarrowlength*ASf(#1,#2))/((ASf(#1,#2)*xscale)++(ASg(#1,#2)*yscale ));
y3:=y1+(ODEarrowlength*ASg(#1,#2))/((ASf(#1,#2)*xscale)++(ASg(#1,#2)*yscale ));
if y3>y1:ODEcolorarrow:=blue else: ODEcolorarrow:=red fi;
}
\ifcolorODEarrow
\drawcolor{ODEcolorarrow} \headcolor{ODEcolorarrow}
\fi
\draw\arrow\lines{z1,z3}
}
\def\ASarrows#1{\AS@cycle@points#1;,;}
\def\AS@cycle@points#1,#2;{\def\temp{#1}\ifx\temp\ODE@last@point\let\next\relax
\else\ASarrow{#1}{#2}\relax\let\next\AS@cycle@points\fi\next}
\def\AStrajectories#1{\AS@cycle@IC#1;,;}
\def\AS@cycle@IC#1,#2;{\def\temp{#1}\ifx\temp\ODE@last@point\let\next\relax
\else
\AStrajectoryRKF{#1}{#2}\relax\let\next\AS@cycle@IC\fi\next}
\def\AStrajectoryRKF#1#2{
\begin{connect}
\mfsrc{x1:=#1;y1:=#2;
TIMEsteps:=abs(TIMEend/TIMEstep);
TIME:=0;
for i=1 upto TIMEsteps:
k1:=ASf(x1,y1);
l1:=ASg(x1,y1);
k2:=ASf(x1+(TIMEstep*k1/2),y1+(TIMEstep*l1/2));
l2:=ASg(x1+(TIMEstep*k1/2),y1+(TIMEstep*l1/2));
k3:=ASf(x1+(TIMEstep*k2/2),y1+(TIMEstep*l2/2));
l3:=ASg(x1+(TIMEstep*k2/2),y1+(TIMEstep*l2/2));
k4:=ASf(x1+(TIMEstep*k3),y1+(TIMEstep*l3));
l4:=ASg(x1+(TIMEstep*k3),y1+(TIMEstep*l3));
k5:=((k1)/6)+((k2)/3)+((k3)/3)+((k4)/6);
l5:=(l1/6)+(l2/3)+(l3/3)+(l4/6);
x2:=x1+(TIMEstep*k5);
y2:=y1+(TIMEstep*l5);}
\ODEline{z1}{z2}
\mfsrc{
if ((y2>yneg) and (y2<ypos) and (x2<xpos) and (x2>xneg)): x1:=x2; y1:=y2 fi;
endfor
}
\end{connect}
}
\catcode`\@12\relax
\endinput
%%
%% End of file `mfpic4ode.tex'.
|
