An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system
This paper deals with the solution of a first-order linear system of differential equations with variable coefficients around a singularity of their coefficients. More precisely, the linear systems under consideration have the form \(x{\mathbf y}' =A(x) {\mathbf y}\), where \(A(x)= x^{-q} \sum_{j\geq 0} A_jx^j\) with an integer \(q\geq 0\) and \(A_j\) constant matrices, and the aim of the paper is to compute the exponential part of a fundamental matrix solution around the singularity \(x=0\). It is known [cf. \textit{W. Wasow}: Asymptotic expansions for ordinary differential equations (1965; Zbl 0133.35301)] that, after a meromorphic transformation, there is an integer \(s\geq 1\) -- the so-called ramification index -- such that a fundamental matrix of these systems can be written as \(Y(x)= \Phi(u)x^R \exp (Q(u))\), where \(u= x^{(-1/s)}\), \(\Phi(u)\) is a meromorphic series in \(u\), \(R\) a constant matrix and \(Q(u)= \text{diag} (q_1(u), \dots, q_n(u))\). In this context the author proposes an efficient algorithm to compute \(Q(u)\). Furthermore he gives two versions of this algorithm that can be implemented in the Maple computer algebra system and presents some examples to show the efficiency of the proposed algorithm.
scientific article; zbMATH DE number 978144