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MathGroup Archive 2006

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Re: Problem solving a difference equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68332] Re: Problem solving a difference equation
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 1 Aug 2006 07:00:04 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <eaer8e$5a4$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <eaer8e$5a4$1 at smc.vnet.net>,
 "aTn" <ayottes at dms.umontreal.ca> wrote:

> I am trying to solve the following difference equation:
> 
> s[i] = s[i-1] ( 1 + d ( a/ (b + c s[i-1]) - 1 ))
> 
> , where a,b,c and d are fixed real numbers.
> 
> Here is what happens when I try RSolve:
> 
> In[1] :=  RSolve[s[i] == s[i-1] ( 1+ d ( a/ (b + c s[i-1]) - 1)),
> s[i],i]
> Out[1] :=  RSolve[s[i] == s[i-1] (1+d ( a/ (b + c s[i-1]) - 1)),
> s[i],i]
> 
> ,that is, nothing much happens :).
> 
> I have a few questions:
> 
> 1) Why are the input and output of RSolve the same ?

Because RSolve cannot solve the difference equation, at least as posed.

> 2) Can one solve the difference equation using RSolve ?

I do not know if RSolve can solve rational difference equations. 

Note that as i -> Infinity, the limiting behaviour is

  Solve[s == s ((a/(b + c s) - 1) d + 1), s]

  {{s -> 0}, {s -> (a - b)/c}}

> 3) If the answer to question (2) is no, then do you have any
> suggestions on how to solve the equation.

It may be interesting to know how the recurrence arises.

Have a look at the literature on (first order) rational difference 
equations. For example, Yanagihara [1] shows that for any rational 
function R(y), the difference equation 

  y[z + 1] == R(y[z]) 

has a non-trivial meromorphic solution. 

Also, perhaps "generatingfunctionology" by Herbert Wilf, at

  http://www.math.upenn.edu/~wilf/DownldGF.html

will help.

[1] Yanagihara N 1980 Meromorphic solutions of some difference equations 
Funkcial. Ekvac. 23 309­26

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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