Asymptotic Methods of ODEs: Exploring Singularities of the Second Kind

TL;DR

This paper introduces symbolic asymptotic methods for solving linear ODEs, focusing on singularities of the second kind, and enhances numerical stability through mollification techniques.

Contribution

It develops new symbolic asymptotic approximation methods for linear ODEs and stabilizes numerical calculations by recasting equations in mollified form.

Findings

Effective asymptotic approximations for second-kind singularities

Enhanced numerical stability through mollification

Applicable to both first-order systems and higher-order equations

Abstract

We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use to them stabilize numerical calculations. Our method follows classical analysis for first-order systems and higher-order scalar equations where growth behavior is expressed in terms of elementary functions. We then recast our equations in mollified form - thereby obtaining stability.