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About stochastic differential equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg19572] About stochastic differential equations
  • From: Dimitris Kugiumtzis <dimitris at mpipks-dresden.mpg.de>
  • Date: Wed, 1 Sep 1999 23:07:05 -0400
  • Sender: owner-wri-mathgroup at wolfram.com


Some time ago I posted a question for how to solve (using
NDSolve) differential equations with noise.

I got some response and I solved the problem (I believe) thanks to
David Park who sent me a notebook. Since some others showed
interest in this problem I attach the notebook (with his permission).
Blimbaum Jerry pointed an easier way to solve the problem (which
I did not test because I had already implemented the notebook
of David):
soln = NDSolve[{x'[t] == Random[] + Sin[t], x[0] == 1}, x[t], {t, 0,
5}] // First
Plot[x[t] /. soln, {t, 0, 5}, PlotRange -> All];

Thanks also to Jens-Peer Kuska and Robert L. Thrash for their
comments.

I should say that I was pleased to get all this response.

Dimitris


--
Dimitris Kugiumtzis
Max-Planck-Institute for Physics of Complex Systems
Noethnitzer Str. 38, 01187 Dresden, Germany
office tel: +49-351-8712211, fax: +49-351-8711199
home tel  : +49-351-4702378, mob: +49-171-1646864
official e-mail: dimitris at mpipks-dresden.mpg.de
lifetime e-mail: dimitris.kugiumtzis at iname.com
http://www.mpipks-dresden.mpg.de/~dimitris





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David Park
djmp at earthlink.net
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Perhaps you can you this as a model for working out your system.\
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Here the equations have been changed to evaluate Random dynamically. I had to \
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(***********************************************************************
End of Mathematica Notebook file.
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--------------8AAB8EAB10A822084B24C553


--------------AEFC719D6E6960B45DD6EE67

Some time ago I posted a question for how to solve (using
NDSolve) differential equations with noise.

I got some response and I solved the problem (I believe) thanks to
David Park who sent me a notebook. Since some others showed
interest in this problem I attach the notebook (with his permission).
Blimbaum Jerry pointed an easier way to solve the problem (which
I did not test because I had already implemented the notebook
of David):
soln = NDSolve[{x'[t] == Random[] + Sin[t], x[0] == 1}, x[t], {t, 0,
5}] // First
Plot[x[t] /. soln, {t, 0, 5}, PlotRange -> All];

Thanks also to Jens-Peer Kuska and Robert L. Thrash for their
comments.

I should say that I was pleased to get all this response.

Dimitris


--
Dimitris Kugiumtzis
Max-Planck-Institute for Physics of Complex Systems
Noethnitzer Str. 38, 01187 Dresden, Germany
office tel: +49-351-8712211, fax: +49-351-8711199
home tel  : +49-351-4702378, mob: +49-171-1646864
official e-mail: dimitris at mpipks-dresden.mpg.de
lifetime e-mail: dimitris.kugiumtzis at iname.com
http://www.mpipks-dresden.mpg.de/~dimitris



--------------AEFC719D6E6960B45DD6EE67

<HTML>
Some time ago I posted a question for how to solve (using
<BR>NDSolve)&nbsp;differential equations with noise.

<P>I&nbsp;got some response and I solved the problem (I&nbsp;believe) thanks
to
<BR>David Park who sent me a notebook. Since some others showed
<BR>interest in this problem I&nbsp;attach the notebook (with his permission).
<BR>Blimbaum Jerry pointed an easier way to solve the problem (which
<BR>I&nbsp;did not test because I had already implemented the notebook
<BR>of David):
<BR>soln = NDSolve[{x'[t] == Random[] + Sin[t], x[0] == 1}, x[t], {t, 0,
<BR>5}] // First
<BR>Plot[x[t] /. soln, {t, 0, 5}, PlotRange -> All];

<P>Thanks also to Jens-Peer Kuska and Robert L. Thrash for their
<BR>comments.

<P>I&nbsp;should say that I&nbsp;was pleased to get all this response.&nbsp;

<P>Dimitris
<BR>&nbsp;
<PRE>--&nbsp;
Dimitris Kugiumtzis
Max-Planck-Institute for Physics of Complex Systems
Noethnitzer Str. 38, 01187 Dresden, Germany
office tel: +49-351-8712211, fax: +49-351-8711199
home tel&nbsp; : +49-351-4702378, mob: +49-171-1646864
official e-mail: dimitris at mpipks-dresden.mpg.de
lifetime e-mail: dimitris.kugiumtzis at iname.com
<A HREF="http://www.mpipks-dresden.mpg.de/~dimitris";>http://www.mpipks-dresden.mpg.de/~dimitris</A></PRE>
&nbsp;</HTML>

--------------AEFC719D6E6960B45DD6EE67--

--------------8AAB8EAB10A822084B24C553

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Cell["Stochastic Differential Equations", "Title"],

Cell["\<\
David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/\
\>", "Subtitle"],

Cell[BoxData[
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Cell[TextData[{
  "This is an example of simulating two differential equations with random \
noise present. It is from ",
  StyleBox["Flux Induced Biochemical Differentiation, J. theor. Biol (1975) \
54, 363-379. ",
    FontSlant->"Italic"],
  StyleBox["The  random noise is generated in advance of each simulation. If \
the random noise is not going to be entirely random but depend on the \
variables of your equations, other than t, then I don't know if it can be \
done in ",
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  StyleBox[".",
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Cell["\<\
Perhaps you can you this as a model for working out your system.\
\>", "Text"],

Cell[CellGroupData[{

Cell["Simulating the Differentiation of Two Cells", "Section"],

Cell[CellGroupData[{

Cell["A Random Noise Generator", "Subsection"],

Cell[TextData[{
  "This generates a random noise function for ",
  Cell[BoxData[
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  "."
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Cell["\<\
These are the equations. (Two cells being driven by a flux g0, with enzymatic \
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equation, and also material leaving the cells with rate constant a. If the \
flux is high enough the cells are driven to different concentrations of \
material. Which cell obtains the higher concentration depends upon the \
noise.)\
\>", "Text"],

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Cell["\<\
This is a time dependent driving function for the cells (input flow to each \
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\>", "Text"],

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This runs a specific simulation. The noise is regenerated each time the \
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\>", "Text"],

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If you evaluate the last two cells a number of times you will see that \
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\>", "Text"]
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Cell["What I learned from your question", "Section"],

Cell["\<\
Even though Method is not shown as an option for NDSolve, it is an option. I \
think the options are Automatic (which is adaptive, I think), RungeKutta, \
Adams and Gear. RungeKutta seems to work OK. WRI doesn't tell us too much \
about this because maybe things will change in future releases.\
\>", "Text"],

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These are the equations. (Two cells being driven by a flux g0, with enzymatic \
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material. Which cell obtains the higher concentration depends upon the \
noise.)\
\>", "Text"],

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Here the equations have been changed to evaluate Random dynamically. I had to \
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\>", "Text"],

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This is a time dependent driving function for the cells (input flow to each \
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\>", "Text"],

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If you evaluate the last two cells a number of times you will see that \
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