A numerical methodology for the Painlevé equations

• Published in: Journal of computational physics Vol. 230; no. 15; pp. 5957 - 5973
• Main Authors: Fornberg, Bengt, Weideman, J.A.C.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 01.07.2011; Elsevier
• Subjects: Chebyshev collocation method; Computational techniques; Exact sciences and technology; Mathematical analysis; Mathematical methods in physics; Mathematical models; Methodology; Padé approximation; Painlevé transcendents; Physics; PI equation; Planes; Poles; Polyimide resins; Solvers; Stands; Taylor series method; Taylor series method; Padé approximation; Chebyshev collocation method; Painlevé transcendents; P I equation; Complex domain; equation; Taylor series; Calculation methods; P; Painlevé property; Numerical solution; Painlevé equation; Differential equations; Calculation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2011.04.007

The six Painlevé transcendents P I − P VI 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 P I equation. In later studies, we will concentrate on mathematical aspects of both the P I and the higher Painlevé transcendents.