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Re: Solving a system of equations without having to define

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112586] Re: Solving a system of equations without having to define
  • From: "Niels R. Walet" <niels.walet at manchester.ac.uk>
  • Date: Tue, 21 Sep 2010 02:05:03 -0400 (EDT)

I don't know of any way doing that, but I can see that you can find the 
outer limits of your variables:
Define

pol[n]:= Sum[  Binomial[n + 1, i] k^(n + 1 - i)    Product[j t + mu, {j, 
0, i - 1}], {i, 0, n + 1}]

m[imax]=k^imax lambda /pol[imax]; n[imax-1]=k^(imax-1) lambda (k+imax 
t+mu)/pol[imax],....
ms[imax]=(k^imax lambda mu)/(imax t)/pol[imax];ms[imax-1]=k^(imax-1) 
lambda  mu (2 k + imax t + mu))/((imax-1) t),....

This is enough to fully specify the solution, e.g., by recursive 
solution....

Niels
Michael A. Gilchrist wrote:
> 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] - k m[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
>
>
>   


-- 
Prof. Niels R. Walet                   Phone:  +44(0)1613063693
School of Physics and Astronomy        Fax:    +44(0)1613064303
The University of Manchester           Mobile: +44(0)7905438934
Manchester, M13 9PL,  UK               room 7.7, Schuster Building
email: Niels.Walet at manchester.ac.uk 
web:   http://www.theory.physics.manchester.ac.uk/~mccsnrw



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