Airy series solution of Painlev\'e II in electrodiffusion: conjectured convergence

TL;DR

This paper develops an Airy function series solution for a nonlinear boundary-value problem in electrodiffusion, showing strong numerical agreement and conjecturing convergence for the Painlevé II equation describing the electric field.

Contribution

It introduces a novel Airy series approach to solve Painlevé II in electrodiffusion, extending the range of parameters where convergence is observed.

Findings

Series matches numerical solutions with high accuracy

Convergence is observed for a wide parameter range

Error measures decrease monotonically with more terms

Abstract

A perturbation series solution is constructed in terms of Airy functions for a nonlinear two-point boundary-value problem arising in an established model of steady electrodiffusion in one dimension, for two ionic species carrying equal and opposite charges. The solution includes a formal determination of the associated electric field, which is known to satisfy a form of the Painlev\'e II differential equation. Comparisons with the numerical solution of the boundary-value problem show excellent agreement following termination of the series after a sufficient number of terms, for a much wider range of values of the parameters in the model than suggested by previously presented analysis, or admitted by previously presented approximation schemes. These surprising results suggest that for a wide variety of cases, a convergent series expansion is obtained in terms of Airy functions for the Painlev\'e transcendent describing the electric field. A suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases. It is conjectured that the series does converge for this subset.