Optimal successive complementary expansion for singular differential equations

Abstract

In this article, we consider a singular ordinary differential equation of a two-point boundary value problem in an asymptotic sense. We discuss the method of successive complementary expansion (SCEM) with asymptotics beyond all orders approach thinking and also scrutinise the solutions that already exist in the literature along with their advantages and drawbacks. In particular, we present an approach using techniques in exponential asymptotics that extends the SCEM for singular differential equations well beyond the leading-order terms and consider the interaction of small parameter in the solutions. We optimally truncate the divergent outer solution and do not eliminate the growing exponentials. Through this analysis, we show that the extended method exhibits more information, such as the effects of exponential smallness and Stokes lines, than the regular SCEM that cannot reveal the information from such singular differential equations.