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

Say, Fatih

Comptes Rendus. Mathématique, Tome 359 (2021) no. 10, pp. 1267-1278.

We construct a relation between the leading pre-factor function $A\left(z\right)$ and the singulants $u_0\left(z\right)$, $u_1\left(z\right)$, 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. Multi-level asymptotic analysis of the first-order singular equations and the Stokes phenomenon may be done straightaway from the higher levels of the analysis.

Classification : 34M40, 34E20

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