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A numerical methodology for the Painlevé equations. (English) Zbl 1220.65092

Summary: 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 ordinary differential equation 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.

65L05Initial value problems for ODE (numerical methods)
34M55Painlevé and other special equations; classification, hierarchies
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
DLMF; Chebfun
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