Geodesic
Numerical solution of the Cauchy problem for the Painlevé; I and II equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 3, pp. 379-387

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A numerical method for solving the Cauchy problem for the first and second Painlevé; differential equations is proposed. The presence of movable poles of the solution is allowed. The positions of the poles are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to an auxiliary system of differential equations in a neighborhood of a pole. The equations in this system and its solution have no singularities in either the pole or its neighborhood. Numerical results confirming the efficiency of this method are presented. 
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     author = {A. A. Abramov and L. F. Yukhno},
     title = {Numerical solution of the {Cauchy} problem for the {Painlev\'e;} {I} and {II} equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {379--387},
     publisher = {mathdoc},
     volume = {52},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_3_a2/}
}
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A. A. Abramov; L. F. Yukhno. Numerical solution of the Cauchy problem for the Painlevé; I and II equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 3, pp. 379-387. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_3_a2/