A new algorithm for computing the asymptotic solutions of a class of linear differential systems

TL;DR

The paper introduces a symbolic-algebraic algorithm to compute asymptotic solutions of linear differential systems of the form $Z'(x)=\rho(x)\{D+R(x)\}Z(x)$ with $R(x)\to0$, allowing periodic coefficients. It employs a sequence of near-diagonal transformations $Z_m=(I+P_m)Z_{m+1}$ to iteratively suppress perturbations, yielding explicit sub-dominant terms and a back-transformation factor $I+P_0(x)$ that delivers prescribed accuracy. The method builds a rigorous order-of-magnitude framework (via $\theta_j$, $\sigma$ and $K$), and is implemented in Mathematica through three algorithms that generate recurrences, evaluate matrices with derivatives, and bound the error $\|E_m\|$. Demonstrated on examples with purely algebraic, periodic, and periodic-ODE coefficients, the approach accommodates periodicity and non-commutative matrix algebra, extending prior work (e.g., BEEM95) and providing practical, error-controlled asymptotics for complex coefficient structures.

Abstract

This paper reports on a new algorithm to compute the asymptotic solutions of a linear differential system. A feature of the algorithm is the ability to accommodate periodic coefficients.