An asymptotic-numerical approach for examining global solutions to an ordinary differential equation

TL;DR

This paper introduces an asymptotic-numerical method that combines asymptotic analysis with numerical solutions to better identify global solutions of nonlinear ODEs on infinite intervals, overcoming limitations of purely numerical approaches.

Contribution

The paper presents a novel approach that integrates asymptotic information into numerical solvers to improve the detection of global solutions to nonlinear ODEs.

Findings

Enhanced accuracy in capturing asymptotic behavior of solutions

More comprehensive understanding of global solution structure

Improved numerical results through combined asymptotic-numerical method

Abstract

Purely numerical methods do not always provide an accurate way to find all the global solutions to nonlinear ODE on infinite intervals. For example, finite-difference methods fail to capture the asymptotic behavior of solutions, which might be critical for ensuring global existence. We first show, by way of a detailed example, how asymptotic information alone provides significant insight into the structure of global solutions to a nonlinear ODE. Then we propose a method for providing this missing asymptotic data to a numerical solver, and show how the combined approach provides more detailed results than either method alone.