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MathGroup Archive 2010

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Solving a system of equations without having to define exact

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112528] Solving a system of equations without having to define exact
  • From: "Michael A. Gilchrist" <mikeg at utk.edu>
  • Date: Sat, 18 Sep 2010 07:27:44 -0400 (EDT)

Hi,

I'm working with a model that consists of a series of coupled ODEs and I am 
trying to study their equilibrium behavior.  Below is 
the code I use to define the equations and solve for the equilibrium state.

(*-----------------------------------------------------*)
(*define the variables *)
imax = 3;

valsI = Table[m[i], {i, 0, imax}];
valsII = Table[ms[i], {i, 0, imax}];

(*generate the equations *)
eqnsI = Join[{lambda + t m[1] - (k + mu) m[0]},
    Table[k m[i - 1] - km[i] + t (m[i + 1] (i + 1) -  m[i] i ) -
       mu m[i], {i, imax}] /. {m[imax + 1] -> 0}] ;

eqnsII = Join[{mu m[0] + t ms[1] - delta ms[0]},
    Table[ t (ms[i + 1] (i + 1) -  ms[i] i ) + mu m[i], {i,
       imax}] /. {ms[imax + 1] -> 0}] ;

(*solve the equation *)
sol = Solve[Map[0 == # &, Join[eqnsI, eqnsII]],
      Join[valsI, valsII]]// Simplify;

(*------------------------------------------------*)

If I set imax to a small integer value such as 2 to 8, Mathematica crunches 
out a solution quickly.  As imax gets bigger, Mathematica still comes up with 
a solution, but it gets ever more complex and difficult to calculate.

Based on this behavior, I surmise there is a general solution to these 
equations but it is sufficiently complex that I cannot intuit it from looking 
at the solutions with imax = 2, 3, 4, ....  I would love it if I could get 
Mathematica to give me a general solution such that the variable imax does not 
need to be explicitly defined.

Does anyone know of a way to pose such a problem (i.e. solve a set of 
equations where the exact number is unspecified) to Mathematica?  Or is this 
impossible? Any help would be greatly appreciated.

Thanks for your attention to this matter.

Mike



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