*(English)*Zbl 0914.34056

An investigation is made of the nature of asymptotic solutions to linear differential equations of arbitrary order in the neighborhood of an irregular singularity of rank one with distinct characteristic values. The authors introduce a classification of the solutions into two types, explicit and implicit, and describe their properties.

They investigate the computation of solutions by numerical integration of the differential equation. It is shown that it is the exception rather than the rule for the investigation process to be stable, particularly so for implicit solutions. To overcome these instabilities, the authors develop boundary value methods, complete with error analysis. The paper contains numerical examples, which illustrate the computation of both, explicit and implicit solutions, and associated Stokes multipliers.

This paper is founded on the previous paper of the second author [Methods Appl. Anal. 4, No. 4, 375-403 (1997; Zbl 0896.34049)].

##### MSC:

34E05 | Asymptotic expansions (ODE) |

34M99 | Differential equations in the complex domain |

65L20 | Stability and convergence of numerical methods for ODE |