Recurrence relationship of successive level singulants

Abstract

We study the asymptotic behaviour of the solution to a singularly perturbed model differential equation. The analytical solution is asymptotically represented by a formal power series in the perturbation parameter. This paper provides an explanation of the relation between the pre-factor functions P(z) and Q(z) and the singulants of the expansion of the general second-order singularly perturbed differential equation in higher levels of the asymptotic analysis. Exponential asymptotics typically relies upon the derivation of the singulant function as it is crucial to determine the location of the Stokes lines. By doing the careful analysis, we connect the subsequent order singulants with the previous order singulants and determine the asymptotics of the late coefficients. By this relation, one can predict and investigate the Stokes phenomenon and exponential smoothing of the equations in the form of the model differential equation.