Narrow Results

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.

Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of Olver.