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MathGroup Archive 1998

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Re: variation of constant (in ODE)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14632] Re: variation of constant (in ODE)
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Wed, 4 Nov 1998 13:47:01 -0500
  • Sender: owner-wri-mathgroup at wolfram.com

[Please contact Paul to obtain the notebook mentioned below - mod.]

Alexander D. Khilinsky wrote:

> Does anybody know, where I can find the Mathematica package,which
> implement the method  "variation of constant" (uses in solving ODE).
> Particularly, I need the implementation of methods of perturbation
> theory, for example,method of Bogolubov-Krylov-Mitropolsky.

The following package by Stephen Kaufmann, available from MathSource,
should do what you want:

Perturbation
       The aim of this package is to show a possible
       implementation of perturbation methods with Mathematica.
       It can be used to generate educational examples of
       perturbation exapansions. The methods of straightforward
       expansions, strained coordinates, and matched and composite
       solutions are implemented.

  http://www.mathsource.com/Content/Applications/Mathematics/0204-129

You also need the following package:

NonNegativeQ
       The Mathematica functions Positive, Negative, and
       NonNegative evaluate for numbers only. They can be used to
       define properties of symbols but for combinations of such
       symbols, the properties are not evaluated any further. The
       function NonNegativeQ tries to find out if the result cannot
       be negative. In such cases, it returns True, otherwise False.

  http://www.mathsource.com/Content/Enhancements/Algebraic/0204-062

I had already downloaded these Notebooks.  Unfortunately, it looks like
there were some conversion problems when they were updated from
V2.2->V3.0. In particular, the section on Matched and Composite
Expansions is slightly broken.  However, most functions work fine.  I
have attached (slightly) edited versions of these Notebooks.  I've also
Cc:d this to Stephan as he might want to fix the problems I've
encountered.


> For example, I have equation :
>
> u''(x) + u(x) + eps*u(x)^2 == 0
>
> and I know that u(x) = a*cos(t+b), where a=a(t),b=b(t). I want to know
> the OD equations for a(t), and b(t).

Using Stephan's package, I get the following solution:

Cos[t] + (eps (-3 + 2 Cos[t] + Cos[2 t])) / 6 +

                                     2
           2                    5 eps
       (eps  (-48 + 29 Cos[(1 - ------) t] +
                                  12
                          2
                     5 eps
       16 Cos[2 (1 - ------) t] +
                       12

                         2
                    5 eps
       3 Cos[3 (1 - ------) t])) / 144
                      12

Cheers,
	Paul

____________________________________________________________________
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA  6907                     mailto:paul at physics.uwa.edu.au
AUSTRALIA                        http://www.physics.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________



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