Speaker: Bengt Fornberg
Abstract. The six Painlevé transcendents P_I to 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 of being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as 'numerical mine fields'. We note in this present work that, on the contrary, the Painlevé property in fact provides excellent opportunities for very fast and accurate numerical solutions across the full complex plane. The numerical method will be described for the P_I equation, with solution illustrations given also for the P_II and P_IV equations.
The present work was carried out in collaborations with Prof. André Weideman (University of Stellenbosch) and Jonah Reeger (University of Colorado at Boulder).