A numerical methodology for the Painlevé equations

Bengt Fornberg, J.A.C. Weideman

Research output: Contribution to journalArticlepeer-review

54 Scopus citations


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
Issue number15
StatePublished - Jul 2011
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: The work of Bengt Fornberg was supported by the NSF Grants DMS-0611681 and DMS-0914647. Part of it was carried out in the fall of 2010 while he was an Oliver Smithies Lecturer at Balliol College, Oxford, and also a Visiting Fellow at OCCAM (Oxford Centre for Collaborative Applied Mathematics). The latter is supported by Award No. KUK-C1-013-04 to the University of Oxford, UK, by King Abdullah University of Science and Technology (KAUST). Andre Weideman was supported by the National Research Foundation of South Africa. He acknowledges the hospitality of the Numerical Analysis Group at the Mathematical Institute, Oxford University, during a visit in November 2010. We thank Jonah Reeger for creating Fig. 4.5. Discussions with Peter Clarkson, John Ockendon, Nick Trefethen, Ben Herbst, Bryce McLeod and Rodney Halburd are gratefully acknowledged. The present work was stimulated by the workshop 'Numerical solution of the Painleve Equations', held in May 2010 at the International Center for the Mathematical Sciences (ICMS), in Edinburgh.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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