Quaternionic fundamental solutions for electromagnetic scattering problems and application
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.
scientific article; zbMATH DE number 1895118
scientific article