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MathGroup Archive 2005

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Re: Mathematica 5.2: The 64-bit and multicore release

  • To: mathgroup at smc.vnet.net
  • Subject: [mg59004] Re: Mathematica 5.2: The 64-bit and multicore release
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 26 Jul 2005 04:03:08 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, England
  • References: <dbnj2n$hpn$1@smc.vnet.net> <dbou1g$snt$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Nasser Abbasi wrote:
> "Wolfram Research" <newsdesk at wolfram.com> wrote in message 
> news:dbnj2n$hpn$1 at smc.vnet.net...
> 
>>
> 
>>* New algorithms for symbolic differential equations
> 
> 
> hello;
> 
> How can one find more about the above?

Hi Nasser,

The following notebook _Solving Higher Order ODEs using Mathematica 
5.2_, posted today by Devendra Kapadia (Wolfram Research) on Mathsource 
may answer your question:

http://library.wolfram.com/infocenter/MathSource/5702/

"The Mathematica function DSolve has been equipped with several modern 
algorithms for solving higher order linear ordinary differential 
equations (ODEs) in Version 5.2. The aim of this notebook is to explain 
the motivation for these developments and to provide some information 
and examples which illustrate the new functionality.

In Mathematica 5.1, we had focussed on adding methods for solving first 
order and second order ODEs such as Abel equations, hypergeometric-type 
equations and equations with non-rational coefficients using DSolve. As 
explained in the Advanced Documentation for DSolve, the code structure 
for this function is hierarchical, so that the problem of solving ODEs 
of order greater than 2 is often reduced to that of solving a first 
order or second order ODE. Within the last few years, a deeper 
understanding of several aspects of higher order ODEs (such as 
factorization techniques) has emerged which makes it possible to carrry 
out this reduction in a systematic way. Also, higher order ODEs 
(particularly orders 3 and 4) are increasingly being seen in scientific 
models. Thus, we were interested in widening the application of the 
methods implemented in Version 5.1 to higher order ODEs.

In Section 2, we will review the methods for solving higher order ODEs 
which were already available in V 5.1.

In Section 3, we will discuss the implementation of the 
Bronstein-Mulders-Weil-van Hoeij algorithm for solving linear ODES of 
arbitrary order that are symmetric powers of second order ODEs.

Section 4 deals with a generalization of the notion of symmetric power 
in which we start with a pair of second order ODEs.

In Section 5, we deal with the important notion of factorization for a 
differential operator.

We end this introduction by noting that it will be convenient to switch 
back and forth between (homogeneous) differential equations and the 
corresponding differential operators since the algorithms really refer 
to the differential operators and the final step of integrating lower 
order ODEs to find the solutions is straightforward in all cases."

Best regards,
/J.M.


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