Handle URI:
http://hdl.handle.net/10754/597363
Title:
A numerical methodology for the Painlevé equations
Authors:
Fornberg, Bengt; Weideman, J.A.C.
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.
Citation:
Fornberg B, Weideman JAC (2011) A numerical methodology for the Painlevé equations. Journal of Computational Physics 230: 5957–5973. Available: http://dx.doi.org/10.1016/j.jcp.2011.04.007.
Publisher:
Elsevier BV
Journal:
Journal of Computational Physics
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
Jul-2011
DOI:
10.1016/j.jcp.2011.04.007
Type:
Article
ISSN:
0021-9991
Sponsors:
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.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorFornberg, Bengten
dc.contributor.authorWeideman, J.A.C.en
dc.date.accessioned2016-02-25T12:31:38Zen
dc.date.available2016-02-25T12:31:38Zen
dc.date.issued2011-07en
dc.identifier.citationFornberg B, Weideman JAC (2011) A numerical methodology for the Painlevé equations. Journal of Computational Physics 230: 5957–5973. Available: http://dx.doi.org/10.1016/j.jcp.2011.04.007.en
dc.identifier.issn0021-9991en
dc.identifier.doi10.1016/j.jcp.2011.04.007en
dc.identifier.urihttp://hdl.handle.net/10754/597363en
dc.description.abstractThe 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.en
dc.description.sponsorshipThe 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.en
dc.publisherElsevier BVen
dc.subjectChebyshev collocation methoden
dc.subjectPadé approximationen
dc.subjectPainlevé transcendentsen
dc.subjectPI equationen
dc.subjectTaylor series methoden
dc.titleA numerical methodology for the Painlevé equationsen
dc.typeArticleen
dc.identifier.journalJournal of Computational Physicsen
dc.contributor.institutionUniversity of Colorado at Boulder, Boulder, United Statesen
dc.contributor.institutionUniversiteit Stellenbosch, Stellenbosch, South Africaen
kaust.grant.numberKUK-C1-013-04en
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.