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Zbl 1220.65092
Fornberg, Bengt; Weideman, J.A.C.
A numerical methodology for the Painlevé equations. (English)
[J] J. Comput. Phys. 230, No. 15, 5957-5973 (2011). ISSN 0021-9991

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.

MSC 2000:: *65L05 Initial value problems for ODE (numerical methods)
34M55 Painlevé and other special equations
65L60 Finite numerical methods for ODE

Keywords: Painlevé transcendents; $P_{I}$ equation; Taylor series method; Padé approximation; Chebyshev collocation method; numerical examples; initial value solver; boundary value solver

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