From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Help needed with difference equation
Date: 11 May 1999 15:31:08 GMT
Newsgroups: sci.math.num-analysis
Keywords: difference equations like differential equations?
Christian Som wrote:
>For a biology paper, I desperately need to analytically solve the
>following difference equation:
>
>u(t+1) = u(t)W[(u(t) + a) / (u(t)c + d)]
>
>with c = bV+ 1
>and d = b(aV + 1)
>
>Starting conditions: u(0) = uo,
>
>a, b, W and V are non-negative constants.
You can't expect to find a closed form for u(t) for a general
difference equation of the form u(t+1) = u(t)*(c1*u(t) + c2)/(c3*u(t)+c4),
since in particular this would mean you would be able to take, say,
c1=-c2, c3=0, c4=1 and get a closed form for the quadratic recurrence
related to the logistic map. This is well-known to exhibit chaotic behaviour
for large enough values of c2.
Your extra conditions amount to the statements that c1 >= 0, c2 >= 0,
c3 >= 1, and c4 c1 >= c2 (c3 -1), which precludes the specific case
I just mentioned, but certainly seems insufficient to expect a "closed
form" of any reasonable type.
On the other hand, if you can argue that the difference equation is
merely an approximation to the "true" situation, represented by the
corresponding differential equation
d u(t) (c1 u(t) + c2)
-- u(t) = u(t) + ---------------------
dt c3 u(t) + c4
then you're in luck because this is an autonomous differential equation
whose only integral is of a rational function of low degree; while you
can't get u = u(t) explicitly, you can get an implicit equation
F(u,t)=0 which ought to be enough for anybody :-)
dave
==============================================================================
From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Help!--Analogies between differential equations and recurrence relations.
Date: 25 May 1999 09:04:44 GMT
Newsgroups: sci.math.research
In article <373C1C0C.465C82FF@hanara.kmaritime.ac.kr>,
jgbae writes:
snip
|> Were there any fine study on the analogies between differential equation
|>
|> and recurrence relation? I'm interested in the nonlinear cases.
|> If someone has any information about this, please give me a line.
|> Thanks in advance.
|>
yes, there is a complete analogy, coming from the theory of the jordan
canonical form of a matrix. for an introduction
(with some pointers to nonlinear problems) see
Elaydi, Saber N.
An introduction to difference equations / Saber N. Elaydi
New York [u.a.] : Springer , 1996 - XIII, 389 S. : Ill.
- Undergraduate texts in mathematics
ISBN: 0-387-94582-2 - 3-540-94582-2
Lakshmikantham, V.
Theory of difference equations / numerical methods and applications / V. Lakshmikantham ; D. Trigiante
Boston [u.a.] : Academic Press , 1988 - X, 242 S.
Literaturverz. S. 231-237 - Mathematics in Science and Engineering
ISBN: 0-12-434100-4
hope this helps
peter