First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants

Abstract

We construct a relation between the leading pre-factor function A(z) and the singulants u(0)(z), u(1)(z), and recurrence relation of the singulants at higher levels for the solution of singularly-perturbed first-order ordinary general differential equation with a small parameter via the method of multi-level asymptotics. The particular equation is chosen due to its appearance at every level of multi-level asymptotic approach for the first-order differential equations. By the relations derived by the asymptotic analysis from the equation, Stokes and anti-Stokes lines can be extracted more quickly and so which exponentials of the expansions are actually contributed in each sector of the complex plane can be deduced faster. Multilevel asymptotic analysis of the first-order singular equations and the Stokes phenomenon may be done straightaway from the higher levels of the analysis.