A numerical methodology for the Painlevé equations

Bengt Fornberg, J.A.C. Weideman

Research output: Contribution to journalArticlepeer-review

42 Scopus citations

Abstract

The six Painlevé transcendents PI-PVI have both applications and analytic properties that make them stand out from most other classes of special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation for being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as 'numerical mine fields'. In the present work, we note that the Painlevé property in fact provides the opportunity for very fast and accurate numerical solutions throughout such fields. When combining a Taylor/Padé-based ODE initial value solver for the pole fields with a boundary value solver for smooth regions, numerical solutions become available across the full complex plane. We focus here on the numerical methodology, and illustrate it for the PI equation. In later studies, we will concentrate on mathematical aspects of both the PI and the higher Painlevé transcendents. © 2011 Elsevier Inc.
Original languageEnglish (US)
Pages (from-to)5957-5973
Number of pages17
JournalJournal of Computational Physics
Volume230
Issue number15
DOIs
StatePublished - Jul 2011
Externally publishedYes

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