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Zbl 0962.34016
Kashevarov, A.V.
The second Painlevé equation in the electrostatic-probe theory: Numerical solutions. (English. Russian original)
[J] Comput. Math. Math. Phys. 38, No.6, 950-958 (1998); translation from Zh. Vychisl. Mat. Mat. Fiz. 38, No.6, 992-1000 (1998). ISSN 0965-5425; ISSN 1555-6662/e

The second Painleve equation $y{{_x}{_x}}=2y^3+xy-\nu$ 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 $y\rightarrow\nu/x$ as $x\rightarrow+\infty$ 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.
[Alexey Tret'yakov (Siedlce)]

MSC 2000:: *34B16 Singular nonlinear boundary value problems
78A30 Electro- and magnetostatics
34E15 Asymptotic singular perturbations, general theory (ODE)
35Q53 KdV-like equations
65L10 Boundary value problems for ODE (numerical methods)
34M55 Painlevé and other special equations

Keywords: electrostatic-probe; boundary value problem; singularly perturbed; numerical solution

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