Geodesic
Numerical solution of the Painlevé V equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 1, pp. 58-71 Cet article a éte moissonné depuis la source Math-Net.Ru

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A numerical method for solving the Cauchy problem for the fifth Painlevé equation is proposed. The difficulty of the problem is that the unknown function can have movable singular points of the pole type; moreover, the equation has singularities at the points where the solution vanishes or takes the value 1. The positions of all of these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Numerical results illustrating the potentials of this method are presented. 
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A. A. Abramov; L. F. Yukhno. Numerical solution of the Painlevé V equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 1, pp. 58-71. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_1_a4/

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