Solutions of Painlev\'e II on real intervals: novel approximating sequences

TL;DR

This paper introduces novel approximating sequences for solutions of Painlevé II on finite intervals, demonstrating their convergence through numerical experiments even when traditional methods fail, and exploring their properties and construction.

Contribution

The paper constructs and analyzes extraordinary sequences of approximants for Painlevé II solutions, revealing their convergence and novel properties in boundary value problems.

Findings

Sequences strongly suggest convergence in many cases

Sequences are constructed from perturbation series of a related boundary problem

Intervals and constants in sequences converge to those of the Painlevé II solution

Abstract

Novel sequences of approximants to solutions of Painlev\'e II on finite intervals of the real line, with Neumann boundary conditions, are constructed. Numerical experiments strongly suggest convergence of these sequences in a surprisingly wide range of cases, even ones where ordinary perturbation series fail to converge. These sequences are here labeled extraordinary because of their unusual properties. Each element of such a sequence is defined on its own interval. As the sequence (apparently) converges to a solution of the corresponding boundary value problem for Painlev\'e II, these intervals themselves (apparently) converge to the defining interval for that problem, and an associated sequence of constants (apparently) converges to the constant term in the Painlev\'e II equation itself. Each extraordinary sequence is constructed in a nonlinear fashion from a perturbation series approximation to the solution of a supplementary boundary value problem, involving a generalization of Painlev\'e II that arises in studies of electrodiffusion.