# Asymptotic behaviour of a solution to a nonlinear equation modelling capillary rise

## TL;DR

This paper analyzes the asymptotic behavior of solutions to a nonlinear ODE modeling capillary rise, proving convergence and developing approximation methods despite non-Lipschitz nonlinearities.

## Contribution

It introduces a novel analysis of a singular nonlinear ODE for capillary rise, including convergence proofs and asymptotic approximation techniques for non-Lipschitz nonlinearities.

## Key findings

- Proves convergence of solutions as a parameter approaches zero
- Develops an accurate asymptotic approximation method for large times
- Addresses nonlinear ODEs with non-Lipschitz components

## Abstract

We are concerned with the asymptotics and perturbation analysis of a singular second-order nonlinear ODE that models capillary rise of a fluid inside a narrow vertical tube. We prove the convergence of the exact solution to a unperturbed solution when a nondimensional parameter decrease to zero. Furthermore, we provide an accurate method of approximating the asymptotic solution for large times. Due to the a fact that the nonlinear component in the main equation does not satisfy the Lipschitz continuity condition the methods used to prove the main theorems are nonstandard, require careful analysis, and can be useful in dealing with similar nonlinear ODEs.