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MathGroup Archive 2006

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Solve problems

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64233] Solve problems
  • From: Joerg Schaber <schaber at molgen.mpg.de>
  • Date: Tue, 7 Feb 2006 03:36:06 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,

I have a system of polynomial equations where Solve cannot find the
right solutions. Any hints? By the way, actually I just want to find the
steady states of a differential equation system. If there is another
clever way, please let me know.
There are also coonstraints that all variable must be >=0, but including 
those constraints and using Reduce also does not yield a valid solution. 
There exisits a valid solution. I checked this solving the differential 
equation system with NDSolve and let it run into the steady state. But 
this is not very elegant and principally the steady states can be 
calculated directly.

eqns1={0 == 0. c1 + 0.001 c3 - 721.9 c1 c4 + 0.001 c2 c8,
0 == 0.001 c3 - 721.9 c1 c4 + 0.001 c5 - 346.09 c4 c6,
0 == 0.001 c5 - 346.09 c4 c6 - 989.77 c6 c7 + 0.001 c8,
0 == -989.77 c6 c7 + 0.001 c8, c1 + c2 + c3 == 5.7,
c3 + c4 + c5 == 19.3,
c5 + c6 + c8 == 4.,
c7 + c8 == 1.};

sol = Solve[eqns1, {c1, c2, c3, c4, c5, c6, c7, c8}, VerifySolutions ->
True];

\!\({{c2 -> 0.`, c3 ->
         5.69999921807596`, c5 -> 3.502051416006559`, c1 -> \
7.819240400365867`*^-7, c7 -> 0.5020524180816763`, c8 -> 
0.4979475819183236`, \
c6 -> 1.0020751177226038`*^-6, c4 -> 10.097949365917481`}, {c2 -> 0.`, 
c3 -> \
5.699999640315395`,
       c5 -> \(-8.352092810365317`\), c1 -> 3.5968460447115093`*^-7,
          c7 -> \(-11.352093909700997`\), c8 -> 12.352093909700997`, c6 -> \
\(-1.0993356796497231`*^-6\), c4 -> 21.95209317004992`}, {c2 -> 0.`,
       c3 -> 8.687577015258206`, c5 -> 10.612427012862872`,
       c1 -> \(-2.987577015258207`\), c7 -> \(-1.3272190975834564`*^-7\),
         c8 -> 1.0000001327219097`, c6 -> \(-7.612427145584781`\),
        c4 -> \(-4.028121078693263`*^-6\)}, {c2 -> 0.`, c3 -> \
19.300001049304143`,
         c5 -> 9.165049916617935`*^-7, c1 -> \(-13.600001049304144`\), 
c7 -> \
\(-3.0000004306092727`\), c8 -> 4.000000430609273`,
         c6 -> \(-1.3471142644128086`*^-6\), c4 -> \
\(-1.965809135229149`*^-6\)}, {c2 -> 0.`,
         c3 -> 24.324407481766347`, c5 -> \(-5.024405672582244`\), c1 -> \
\(-18.624407481766347`\), c7 -> 1.259078407457304`*^-7, c8 -> \
0.9999998740921593`, c6 -> 8.024405798490085`,
        c4 -> \(-1.8091841042296682`*^-6\)}}\)


Einsetzen ergibt:

eqns1 /. sol

{{True, False, False, False, True, True, True, True}, {True, True,
     False, False, True, True, True, True}, {True, False, False, True,
     True, True, True, True}, {True, False, True, True, True, True,
     True, True}, {False, False, False, True, True, True, True, True}}


best wishes,

joerg


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