**The DESIR Package
**

Differential Equations Singularities Irregular and Regular

The DESIR package contains commands in Maple that help you solve
linear ordinary differential equations in the neighborhood of singularities in the complex plane.

The first part consists in a new version of the DESIR-II package,
which was written in Maple V [1, 5]. Both versions are improved
forms of the DESIR-I package written in Reduce [2]. The purpose of
this part is the computation of formal solutions of homogenous
linear ordinary differential equations. In order to have access to
all information the formal solutions contain, the internal data
structure can be assigned to an optionally passed argument. The
Gevrey caracteristics (type and order) of the divergent series are
also stored in this data structure. Specific information can then
be extracted by using the primitive functions (irregpart,
regpart,param).

The second part collects some functions for the numerical
computation and the graphical visualization of the solutions in
the complex plane.

The principle of the representation is the following [6]: it
consists in plotting the image under the considered function f of
a circle or a circular arc around the singularity, in general 0
(or infinity). The color is used to associate a point in the
domain and its image: each point f(x) is plotted with a color
corresponding to the argument of x.

As the studied functions are in general multi-valued, we consider
them in the neighborhood of 0 as functions on the Riemann surface
of the logarithm and points in the domain are represented by their
Euler coordinates, whereas the image points are computed in
cartesian coordinates.

** Computation
of Stokes matrices **

The third part is new. It deals with the Stokes phenomenom, in the
neighborhood of an irregular singular point. The main function is
StokesMatrices, which describes this phenomenom as a list of
objects [arg, M_arg]: arg is the argument of a Stokes ray and
M_arg is the corresponding Stokes matrix. Indeed, the Stokes
constants can be numerically computed by the analysis of the
singularities of the Borel transform of the divergent series, and
this for a large class of differential equations of single rank k
>1: the function works under the hypothesis that the Borel
transforms don't have aligned singularities in the Borel plane,
and it allows there any polar, ramified or logarithmic
singularities [4]. The function monodromy computes the matrix of
formal monodromy. Then it is possible to use the knowledge of
these matrices to define a function (in Maple) which calculates
automatically, for a particular solution f, a suitable linear
combination, depending on the sector. This is the aim of the
function combli_variable.

- [1] Rational Newton
algorithm for computing formal solutions of linear
differential equations

M.A. Barkatou, ISSAC'88, Italy.

- [2] An algorithm to obtain formal solutions of a linear
homogeneous differential equation at an irregular singular
point

J. Della Dora, C. Di Crescenzo and E. Tournier

In EUROSAM 82, ed. J. Calmet, volume 144 of Lecture Notes in Computer Science page 273.

Springer-Verlag, Berlin and Heidelberg (1982).

- [3] Algorithms for
the splitting of formal series; applications to alien
differential calculus (file.pdf)

F. Fauvet, F. Richard-Jung, J. Thomann

in the proceedings of Transgressive Computing 2006, a Conference in honor of Jean della Dora, Granada, Spain, april 2006.

- [4] Automatic
computation of Stokes matrices

F. Fauvet, F. Richard-Jung, J. Thomann

Numerical Algorithms V 50, n°2, Feb. 2009.

- [5] On the latest
version of DESIR-II

E. Pfl\"ugel

Theor. Comput. Sci. 187, (1-2), 81-- 86, 1997.

- [6] An implicit
differential equation with MAPLE and IRONDEL (file.pdf)

F. Richard-Jung

in the proceedings of Computer Algebra in Scientific Computing 2004, 383--397, Saint Petersburg, Russia, july 2004.

** Code**

The directory desir_2016
contains two maple files,

maple.mla and maple.help,

that you have to download in your own directory "mydirectory".

In a maple session, you have first to add this directory to the
global variable libname:

libname:="mydirectory",libname;

and then to load the package: with(Desir);

The command ?Desir will give you an overview of the package.
All the functions that are accessible to the user are
documented.

To run the package, you need: Maple at least 10.0

Françoise JUNG Laboratoire Jean Kuntzmann 38401 Domaine universitaire de Saint Martin d'Hères, UGA, France. Francoise.Jung@imag.fr http://www-ljk.imag.fr/membres/Francoise.Jung |