Exponential asymptotics with a small exponent

Analytic and numerical solutions are considered to a simple model problem w hich contains a surprisingly complicated solution structure. Asymptotic sol utions are sought when a parameter that appears as an exponent in the indep endent variable is small, the solution then exhibiting a sudden change in s lope over a region that is exponentially thin. A straightforward approach u sing matched asymptotic expansions immediately reveals inadequacies of this method due to the requirement of an outer solution that needs to be evalua ted beyond all orders in order to match to a suitable inner solution. This Behaviour is elucidated by studying first the asymptotic structure of the s olution 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 obtain ed by using Borel summation to evaluate the outer solution to exponential a ccuracy. Further, as a practical alternative, it is shown how these exponen tially improved approximations can be made when an exact numerical solution is available and without recourse to the general term of the outer or inne r expansions.