On the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations

Abstract

An investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank. We introduce a classification of the solutions into two types, explicit and implicit. For the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached. No such sector exists for implicit solutions. In consequence, the two types of solution have different uniqueness properties. Another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions. We also investigate the computation of the solutions by numerical integration of the differential equation. It is shown that it is the exception rather than the rule for the integration process to be stable, particularly so for implicit solutions. To overcome these instabilities we develop boundary-value methods, complete with error analysis. Numerical examples illustrate the computation of both explicit and implicit solutions, and also the associated Stokes multipliers.