Exponential asymptotics with a small exponent
Analytic and numerical solutions are considered to a simple model problem which contains a surprisingly complicated solution structure. Asymptotic solutions are sought when a parameter that appears as an exponent in the independent variable is small, the solution then exhibiting a sudden change in slope over a region that is exponentially thin. A straightforward approach using matched asymptotic expansions immediately reveals inadequacies of this method due to the requirement of an outer solution that needs to be evaluated beyond all orders in order to match to a suitable inner solution. This behaviour is elucidated by studying first the asymptotic structure of the solution using an exact integral, which explicitly reveals the need for the inclusion of exponentially small terms in the expansions. It is then shown how a direct asymptotic solution of the differential equation can be obtained by using Borel summation to evaluate the outer solution to exponential accuracy. Further, as a practical alternative, it is shown how these exponentially improved approximations can be made when an exact numerical solution is available and without recourse to the general term of the outer or inner expansions.