Factorial over a power approach is one of the fundamental techniques for deriving the late-order terms in the asymptotic approximation of integrals and differential equations. To our knowledge, although many differential equations depending on small or large parameters are addressed thoroughly and intensively by this approach in the literature to date, no explicit formula of the general representation of singularly-perturbed second order inhomogeneous ODEs in the form of this paper has yet been discussed generally in terms of their pre-factors. In this paper, we obtain a leading order asymptotic formula of the general asymptotic expansions suitable for the particular type of ODE by its pre-factors.

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.

Studying ordinary or partial differential equations or integrals using traditional asymptotic analysis, unfortunately, fails to extract the exponentially small terms and fails to derive some of their asymptotic features. In this paper, we discuss how to characterize an asymptotic behavior of a singular linear differential equation by the methods in exponential asymptotics. This paper is particularly concerned with the formulation of the series representation of a general second‐order differential equation. It provides a detailed explanation of the asymptotic behavior of the differential equation and its relation between the prefactor functions and the singulant of the expansion of the equation. Through having this relationship, one can directly uncover and investigate invisible exponentially small terms and Stokes phenomenon without doing more work for the particular type of equations. Here, we demonstrate how these terms and form of the expansion can be computed straight‐away, and, in a manner, this can be extended to the derivation of the potential Stokes and anti‐Stokes lines.

No abstract

In this article, we consider a singular ordinary differential equation of a two‐point boundary value problem in an asymptotic sense. We discuss the method of successive complementary expansion (SCEM) with asymptotics beyond all orders approach thinking and also scrutinise the solutions that already exist in the literature along with their advantages and drawbacks. In particular, we present an approach using techniques in exponential asymptotics that extends the SCEM for singular differential equations well beyond the leading‐order terms and consider the interaction of small parameter in the solutions. We optimally truncate the divergent outer solution and do not eliminate the growing exponentials. Through this analysis, we show that the extended method exhibits more information, such as the effects of exponential smallness and Stokes lines, than the regular SCEM that cannot reveal the information from such singular differential equations.