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The second Painlevé equation in the electrostatic-probe theory: Numerical solutions. (English) Zbl 0962.34016
The second Painleve equation y x x =2y 3 +xy-ν is investigated on the basis of the asymptotic analysis of a boundary value problem for singularly perturbed nonlinear ordinary differential equations describing the operation of an electrostatic probe in a collisional plasma. The behavior of certain properties of the equation and its regular solutions with the asymptotics yν/x as x+ is considered within the framework of electrostatic-probe theory. Conditions satisfied by these solutions at a certain point x 0 are indicated, which makes it possible to calculate the solutions numerically.
MSC:
34B16Singular nonlinear boundary value problems for ODE
78A30Electro- and magnetostatics
34E15Asymptotic singular perturbations, general theory (ODE)
35Q53KdV-like (Korteweg-de Vries) equations
65L10Boundary value problems for ODE (numerical methods)
34M55Painlevé and other special equations; classification, hierarchies