Quaternionic fundamental solutions for electromagnetic scattering problems and application
type: Item
instance of: scholarly article
title: Quaternionic fundamental solutions for electromagnetic scattering problems and application
zbMATH Open document ID: 1076.30052
DOI: 10.4171/ZAA/1136
author: Vladislav V. Kravchenko
author: Kira V. Khmelnytskaya
author: Vladimir Rabinovich
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2003-01-01T00:00:00Z
review text: V. D. Kupradse and M. A. Alexidse introduced the sixties of the last century a multipole-method based on fundamental solutions of differential operators, which was later applied by W. Freeden in geophysics. K. Gürlebeck and W. Sprößig found [\textit{K. Gürlebeck}, \textit{W. Sprößig} and \textit{M. Tasche}, ``Numerical realization of boundary collocation methods'', Multivariate approximation theory III, Proc. Conf., Oberwolfach/Ger. 1985, ISNM 75, 206--217 (1985; Zbl 0604.65088)] a corresponding collocation method in quaternion Sobolev spaces for a wide range of equations of mathematical physics. In the present paper the authors consider multipole methods for equations of Helmholtz type. For this reason the generalization to complex quaternions is necessary. A very interesting connection between quaternionic differential operators and Maxwell's system for chiral media is deduced. Using this relaion the completeness in Sobolev spaces with suitable multipliers is shown for Maxwell's equations in bounded and unbounded domains. In outer domains in case of the equation of Helmholtz type the well-known Sommerfeld condition is generalized and for Maxwell's equations the electric and magnetic fields the Silver-Müller radiation condition is derived. The reader can find numerical examples at the end of the paper.
Mathematics Subject Classification ID: 30G35
zbMATH DE Number: 1895118
zbMATH Keywords: scattering problems
zbMATH Keywords: Maxwell's equations
MaRDI profile type: Publication
cites work: Q3996854
cites work: On some properties of Beltrami fields in chiral media.
cites work: Q5552921
cites work: Approximation by Solutions of Partial Differential Equations
cites work: Q5518219
cites work: On the representation of inhomogeneous linear force-free fields
full work available at URL: https://doi.org/10.4171/zaa/1136
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description: scientific article; zbMATH DE number 1895118