Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-20T05:37:54.374Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral analysis of integro-differential operators applied in linear antenna modelling

Part of: Ordinary differential operators Eigenvalue problems

Published online by Cambridge University Press:  12 April 2012

Dave J. Bekers  and
Stephanus J. L. van Eijndhoven
Dave J. Bekers
Affiliation:
TNO Defence, Security and Safety, Business Unit Observation Systems, 2597 AK Den Haag, The Netherlands (dave.bekers@tno.nl)
Stephanus J. L. van Eijndhoven
Affiliation:
Faculty of Mathematics and Computer Science, Technische Universiteit Eindhoven, 5600 MB, Eindhoven, The Netherlands (s.j.l.v.eijndhoven@tue.nl)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The current on a linear strip or wire solves an equation governed by a linear integro-differential operator that is the composition of the Helmholtz operator and an integral operator with a logarithmically singular displacement kernel. Investigating the spectral behaviour of this classical operator, we first consider the composition of the second-order differentiation operator and the integral operator with logarithmic displacement kernel. Employing methods of an earlier work by J. B. Reade, in particular the Weyl–Courant minimax principle and properties of the Chebyshev polynomials of the first and second kind, we derive index-dependent bounds for the ordered sequence of eigenvalues of this operator and specify their ranges of validity. Additionally, we derive bounds for the eigenvalues of the integral operator with logarithmic kernel. With slight modification our result extends to kernels that are the sum of the logarithmic displacement kernel and a real displacement kernel whose second derivative is square integrable. Employing this extension, we derive bounds for the eigenvalues of the integro-differential operator of a linear strip with the complex kernel replaced by its real part. Finally, for specific geometry and frequency settings, we present numerical results for the eigenvalues of the considered operators using Ritz's methods with respect to finite bases.

Keywords

eigenvalue problemsintegro-differential operatorslogarithmic kernellinear antennas

MSC classification

Secondary: 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 45C05: Eigenvalue problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 333 - 354
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Bekers, D. J., Finite antenna arrays: an eigencurrent approach, PhD Thesis, Technische Universiteit Eindhoven (2004; available at http://alexandria.tue.nl/extra2/200411410.pdf).Google Scholar
2.Bekers, D. J., van Eijndhoven, S. J. L. and Tijhuis, A. G., An eigencurrent approach for the analysis of finite antenna arrays, IEEE Trans. Antennas Propagat. 57 (2009), 3772–3782.CrossRefGoogle Scholar
3.Bekers, D. J., van Eijndhoven, S. J. L., van de Ven, A. A. F., Borsboom, P.-P. and Tijhuis, A. G., Eigencurrent analysis of resonant behavior in finite antenna arrays, IEEE Trans. Microwave Theory Tech. 54 (2006), 2821–2829.CrossRefGoogle Scholar
4.Bruno, O. P. and Haslam, M. C., Regularity theory and superalgebraic solvers for wire antenna problems, SIAM J. Sci. Comput. 29 (2007), 1375–1402.CrossRefGoogle Scholar
5.Claeys, X., On the theoretical justification of Pocklington's equation, Math. Models Meth. Appl. Sci. 19 (2009), 1325–1355.CrossRefGoogle Scholar
6.Davies, P. J., Duncan, D. B. and Funken, S. A., Accurate and efficient algorithms for frequency domain scattering from a thin wire, J. Computat. Phys. 168 (2001), 155–183.CrossRefGoogle Scholar
7.Dostanic, M. R., Spectral properties of the operator of Riesz potential type, Proc. Am. Math. Soc. 126 (1998), 2291–2297.CrossRefGoogle Scholar
8.Estrada, R. and Kanwal, R. P., Integral equations with logarithmic kernels, IMA J. Appl. Math. 43 (1989), 133–155.CrossRefGoogle Scholar
9.Fischer, B. E., Yagle, A. E. and Volakis, J. L., On the eigen-decomposition of electromagnetic systems and the frequency dependence of the associated eigenvalues, in Proc. Antennas and Propagation Int. Symp., Washington DC, 3–8 July 2005 (IEEE Press, 2005).Google Scholar
10.Gohberg, I. C. and Krein, M. G., Introduction to the theory of linear nonselfadjoint operators in Hilbert Space, Translations of Mathematical Monographs, Volume 18 (American Mathematical Society, Providence, RI, 1969).CrossRefGoogle Scholar
11.Harrington, R. F. and Mautz, J., Theory of characteristic modes for conducting bodies, IEEE Trans. Antennas Propagat. 19 (1971), 622–628.CrossRefGoogle Scholar
12.Jablonski, T. F. and Sowinski, M. J., Analysis of dielectric guiding structures by the iterative eigenfunction expansion method, IEEE Trans. Microwave Theory Tech. 37 (1989), 63–70.CrossRefGoogle Scholar
13.Jones, D. S., Note on the integral equation for a straight wire antenna, IEEE Proc. Microw. Antennas. Propag. 128 (1981), 114–116.Google Scholar
14.Kac, M., Distribution of eigenvalues of certain integral operators, Michigan Math. J. 3 (1956), 141–148.Google Scholar
15.Kartchevski, E. M., Nosich, A. I. and Hanson, G. W., Mathematical analysis of the generalized natural modes of an inhomogeneous optical fiber, SIAM J. Appl. Math. 65 (2005), 2033–2048.CrossRefGoogle Scholar
16.Kreyszig, E., Introductory functional analysis with applications (Wiley, 1978).Google Scholar
17.Lancellotti, V., de Ho, B. P. and Tijhuis, A. G., An eigencurrent approach to the analysis of electrically large 3-D structures using linear embedding via Green's operators, IEEE Trans. Antennas Propagat. 57 (2009), 3575–3585.CrossRefGoogle Scholar
18.Li, S. and Scharstein, R. W., Eigenmodes and scattering by an axial array of tape rings, IEEE Trans. Antennas Propagat. 54 (2006), 2096–2104.CrossRefGoogle Scholar
19.Magnus, W., Oberhettinger, F. and Soni, R. P., Formulas and theorems for the special functions of mathematical physics, Die Grundlehren der mathematischen Wissenschaften, Volume 52 (Springer, 1966).CrossRefGoogle Scholar
20.Marcuvitz, N., Waveguide handbook (McGraw-Hill, 1951).Google Scholar
21.Pearson, L. W., Separation of the logarithmic singularity in the exact kernel of the cylindrical antenna integral equation, IEEE Trans. Antennas Propagat. 23 (1975), 256–258.CrossRefGoogle Scholar
22.Ramm, A. G., Mathematical foundations of the singularity and eigenmode expansion methods (SEM and EEM), J. Math. Analysis Applic. 86 (1982), 562–591.CrossRefGoogle Scholar
23.Reade, J. B., Asymptotic behaviour of eigen-values of certain integral equations, Proc. Edinb. Math. Soc. 22 (1979), 137–144.CrossRefGoogle Scholar
24.Richter, G. R., On weakly singular Fredholm integral equations with displacement kernels, J. Math. Analysis Applic. 55 (1976), 32–42.CrossRefGoogle Scholar
25.Rynne, B. P., The well-posedness of the integral equations for thin wire antennas, IMA J. Appl. Math. 49 (1992), 35–44.CrossRefGoogle Scholar
26.Shestopalov, V. and Shestopalov, Y., Spectral theory and excitation of open structures (Peregrinus, Chicago, IL, 1996).CrossRefGoogle Scholar
27.Simíc, S., An estimation of the singular values of integral operator with logarithmic kernel, Facta Univ. Ser. Math. Inform. 21 (2006), 49–55.Google Scholar
28.Storer, J. E., Impedance of thin-wire loop antennas, AIEE Trans. 75 (1956), 606–619.Google Scholar
29.Tijhuis, A. G., Zhongqui, P. and Bretones, A. R., Transient excitation of a straight thin wire segment: a new look at an old problem, IEEE Trans. Antennas Propagat. 40 (1992), 1132–1146.CrossRefGoogle Scholar
30.van Beurden, M. C. and Tijhuis, A. G., Analysis and regularization of the thin-wire integral equation with reduced kernel, IEEE Trans. Antennas Propagat. 55 (2007), 120–129.CrossRefGoogle Scholar
31.Weidmann, J., Linear operators in Hilbert spaces (Springer, 1980).CrossRefGoogle Scholar