Exponentially small expansions related to the parabolic cylinder function

TL;DR

This paper derives accurate exponentially small asymptotic expansions for solutions of a differential equation related to Weber's equation, correcting previous inaccuracies in standard expansions and validating results numerically.

Contribution

It introduces corrected exponentially small asymptotic expansions for solutions of Weber-related equations, improving upon standard methods.

Findings

New asymptotic expansions are more accurate than standard ones.

Numerical verification confirms the correctness of the derived expansions.

Standard asymptotics are shown to be incorrect for these solutions.

Abstract

The refined asymptotic expansion of the confluent hypergeometric function $M(a,b,z)$ on the Stokes line $\arg\,z=\pi$ given in {\it Appl. Math. Sci.} {\bf 7} (2013) 6601--6609 is employed to derive the correct exponentially small contribution to the asymptotic expansion for the even and odd solutions of a second-order differential equation related to Weber's equation. It is demonstrated that the standard asymptotics of the parabolic cylinder function $U(a,z)$ yield an incorrect exponentially small contribution to these solutions. Numerical results verifying the accuracy of the new expansions are given.