Lecturer: Moulay Barkatou (Université de Limoges, France)

Title: Symbolic methods for solving systems of linear ordinary differential equations

Abstract: A large part of the known algorithms for solving linear ordinary equations which are implemented in the common computer algebra systems is dedicated to scalar differential equations. This fact is due to two main reasons : 1) most algorithmic problems that occur in the theory of linear differential equations are easier to solve for a single equation than for a system of equations, 2) there exists elementary algorithms that convert a given system into a single (or several uncoupled) scalar equations. Despite the fact that this approach may be very costly, especially for systems of large dimension, it still very popular and this probably because the non experts users know little about other methods. However the study of linear differential systems by direct methods (i. e. without first reducing them to a scalar equation) has a long history and has become very active in the last twenty years. The main purpose of this tutorial is to present and explain symbolic methodsfor studying systems of linear ordinary differential equations with emphasis on direct methods and their implementation in computer algebra systems.

Whether one is interested in global problems (finding closed form solutions, testing reducibility, computing properties of the differential Galois group) or in local problems (computing formal local invariants or local formal solutions) of linear differential scalar equations or systems, one has to develop and use appropriate tools for local analysis the purpose of which is to describe the behavior of the solutions near a point without knowing these solutions in advance.

After introducing the basic tools of local analysis we present the state of the art of existing algorithms and programs for solving the main local problems such as determining the type of a given singularity, computing the rank of a singularity, computing the Newton polygon and Newton polynomials at a given singularity, finding the formal solution, etc. Next we explain how by piecing together the local information around the different singularities one can solve some global problems such as finding rational solutions, exponential solutions, factoring a given differential system, etc.