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JOURNALS // Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki // Archive

Zh. Vychisl. Mat. Mat. Fiz., 2011 Volume 51, Number 12, Pages 2269–2278 (Mi zvmmf9592)

Mathieu functions and coulomb spheroidal functions in the electrostatic probe theory

A. V. Kashevarov

Zhukovsky Central Institute of Aerohydrodynamics, ul. Zhukovskogo 1, Zhukovskii, Moscow oblast, 140186 Russia

Abstract: A spherical probe placed in a slowly moving collisional plasma with a large Debye length $\lambda_{\mathrm D}\to\infty$ is considered. The partial differential equation describing the electron concentration around the probe is reduced to two ordinary differential equations, namely, to the equation for Coulomb spheroidal functions and Mathieu’s modified equation with the parameter $a$ of the latter related to the eigenvalue $\lambda$ of the former by the relation $a=\lambda+1/4$. It is shown that the solutions of Mathieu’s equation are Mathieu functions of half-integer order, which are expressed as series in terms of spherical Bessel functions and series of products of Bessel functions. These Mathieu functions are numerically constructed for Mathieu’s modified and usual equations.

Key words: Mathieu functions, Coulomb spheroidal functions, electrostatic probe.

UDC: 519.624.2

Received: 08.10.2010
Revised: 30.06.2011


 English version:
Computational Mathematics and Mathematical Physics, 2011, 51:12, 2137–2145

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