       Re: Solving a system of equations without having to define

• To: mathgroup at smc.vnet.net
• Subject: [mg112587] Re: Solving a system of equations without having to define
• From: "Michael A. Gilchrist" <mikeg at utk.edu>
• Date: Tue, 21 Sep 2010 02:05:14 -0400 (EDT)

```Hi Niels,

Thanks for the suggestion.  I've run into systems of equations like this
before, but was unable to solve them generally (I'm more biologist than
mathematician).  I'll give them a second look to see if I can find a recursive
solution.

With resepect to the original post.  I think I can now refine my question as,
"Is there a way to set up Mathematica to solve a set of a recursive equations
with two boundary conditions?"

Would appreciate any help.

Mike

-----------------------------------------------------
Department of Ecology & Evolutionary Biology
569 Dabney Hall
University of Tennessee
Knoxville, TN 37996-1610

phone:(865) 974-6453
fax:  (865) 974-6042

web: http://eeb.bio.utk.edu/gilchrist.asp
-----------------------------------------------------

On Mon, 20 Sep 2010, Niels R. Walet wrote:

> 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 - (k + mu) m},
>>     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 + t ms - delta ms},
>>     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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