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On the Solution of the Electron Boltzmann Equation in Higher Order Legendre Polynomial Expansion
Author(s) -
Winkler R.,
Wilhelm J.,
Hess A.
Publication year - 1983
Publication title -
beiträge aus der plasmaphysik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.531
H-Index - 47
eISSN - 1521-3986
pISSN - 0005-8025
DOI - 10.1002/ctpp.19830230505
Subject(s) - legendre polynomials , boltzmann equation , associated legendre polynomials , legendre function , polynomial , mathematics , hierarchy , mathematical analysis , distribution function , legendre's equation , function (biology) , boltzmann constant , differential equation , physics , quantum mechanics , orthogonal polynomials , evolutionary biology , economics , market economy , biology , classical orthogonal polynomials , gegenbauer polynomials
The paper deals with the mathematical structure of the general solution and the selection of the physical relevant solution of the hierarchy resulting from the electron Boltzmann equation by the Legendre polynomial expansion for a weakly ionized plasma with elastic and exciting collisions and under the action of an electric field. At first the properties of the general solution for the two and four term approximation of the distribution function are analyzed and then the study is extended to arbitrarily even order of the hierarchy using the theory of weakly and strongly singular differential equation systems for the investigation of the solution behaviour of the truncated hierarchy at small and at large energies, respectively. In this way, especially the free parameters of the non‐singular part of each general solution, which is obtained for small and large energies and is of interest for the construction of the desired solution, could be found, and the procedure for their final determination is explained. A first illustrative example is given of the application of these studies for the determination of the velocity distribution function of the electrons in four term approximation for a model plasma.