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.

Formal computation and manipulation of formal series

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).

Numerical computation and visualization

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.

References
• [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.

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,