The Mathematical Institute, University of Oxford, Eprints Archive

A Numerical Methodology for the Painlevé Equations

Fornberg, B. and Weideman, J. A. C. (2011) A Numerical Methodology for the Painlevé Equations. Journal of Computational Physics . (Submitted)

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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.

Item Type:Article
Subjects:D - G > General
Research Groups:Oxford Centre for Collaborative Applied Mathematics
ID Code:1258
Deposited By:Peter Hudston
Deposited On:26 May 2011 07:43
Last Modified:09 Feb 2012 14:04

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