Geodesic
On series solutions of the magnetostatic integral equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 39 (1999) no. 4, pp. 630-637 
J. F. Ahner; V. V. Dyakin; V. Ya. Raevskii; R. Ritter. On series solutions of the magnetostatic integral equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 39 (1999) no. 4, pp. 630-637. http://geodesic.mathdoc.fr/item/ZVMMF_1999_39_4_a10/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the fundamental magnetostatic integral equation for homogeneous and isotropic solids. The construction of the orthogonal basis consisting of the eigenvectors of the magnetostatic operator is based on the coupling of the spectral characteristics of this operator and classic potential operators. Estimations of the error of the resulting field in terms of cut-off errors to the external magnetic field are given. Additionally we obtain explicit expressions for both the eigenvectors and eigenfunqtions of the magnetostatic operator for the case that the underlying body is a triaxial ellipsoid. These quantities are given in terms of Lame functions and ellipsoidal harmonics.

[1] Friedman M. J., “Mathematical study of nonlinear singular integral magnetic field equation”, SIAM J. Appl. Math., 39 (1980), 14–20 | DOI | MR | Zbl

[2] Raevskii V. Ya., “Some properties of the potential theory operators and their applications to investigation of the basic electro-magnetostatic equation”, Theor. and Math. Phys., 100:3 (1994), 323–331 | DOI | MR

[3] Dyakin V. V., Raevskii V. Ya., “Investigation of an equation of electrophysics”, USSR Comput. Math. Math. Phys., 30:1 (1990), 213–218 | DOI | MR

[4] Ahner J. F., Dyakin V. V., Raevskii V. Ya., “Eigenvectors of the magnetostatic operator for a spheroid”, Comput. Math. Math. Phys., 36:5 (1996), 643–647 | MR | Zbl

[5] Tselnik D. S., “A bound for the remainder of the Hilbert–Schmidt series and other results on representation of solutions to the functional equation of the second kind with a self-adjoint compact operator as an infinite series”, Comput. Math. Appl., 29:10 (1995), 61–68 | DOI | MR | Zbl

[6] Gohberg I., Krein M., Introduction to the linear nonself-adjoint operator theory in Hilbert spaces, Nauka, M., 1965 | MR

[7] Reed M., Simon B., Methods of modern mathematical physics. Analysis of operators, v. 4, Acad. Press, New York, 1978 | MR | Zbl

[8] Ritter S., “The spectrum of the electrostatic integral operator for an ellipsoid”, Inverse Scattering and Potential Problems in Math. Phys., Methoden und Verfahren der math. Phys., 40, Peter Lang, Frankfurt a.M./Bern, 1994, 157–167 | MR

[9] Hobson E. W., The theory of spherical and ellipsoidal harmonics, Chelsea, New York, 1955 | MR

[10] Walter H. G., “Lamé functions of the first kind generated by computer”, Celestial Mech., 4 (1971), 15–30 | DOI | MR | Zbl