Olver's error bound methods applied to linear ordinary differential equations having a simple turning point

Abstract

Uniform asymptotic solutions of linear ordinary differential equations having a large parameter and a simple turning point are well known. Classical expansions involve Airy functions and their derivatives, and one of Frank Olver's major achievements was obtaining explicit and realistic error bounds. Here alternative expansions are considered, which involve the Airy function alone (and not its derivative). This is based on the early work of Cherry, and using Olver's techniques explicit error bounds are derived. The derivative of asymptotic solutions of turning point problems is also considered, and again using Olver's techniques, sharper error bounds are derived via the differential equation satisfied by such solutions.