Selected Asymptotic Methods with Applications to Electromagnetics and Antennas

Fikioris, George; Tastsoglou, Ioannis; Bakas, Odysseas

doi:10.2200/s00525ed1v01y201307cem031

Cited by 4 publications

(2 citation statements)

References 52 publications

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“…We proceed to give the main result. By applying to Q ± (ξ ± ) the Mellin transform with respect to q [49,50], we obtain the low-order expansion (see Appendix D)…”

Section: Long-wavelength Asymptotic Expansion Under Zero Indexmentioning

confidence: 99%

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Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials

Margetis¹,

Maier²,

Stauber³

et al. 2020

J. Phys. A: Math. Theor.

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By an integral equation approach to the time-harmonic classical Maxwell equations, we describe the dispersion in the nonretarded frequency regime of the edge plasmon-polariton (EPP) on a semi-infinite flat sheet. The sheet has an arbitrary, physically admissible, tensor valued and spatially homogeneous conductivity, and serves as a model for a family of two-dimensional conducting materials. We formulate a system of integral equations for the electric field tangential to the sheet in a homogeneous and isotropic ambient medium. We show how this system is simplified via a length scale separation. This view entails the quasi-electrostatic approximation, by which the tangential electric field is replaced by the gradient of a scalar potential, ϕ. By the Wiener-Hopf method, we solve an integral equation for ϕ in some generality. The EPP dispersion relation comes from the elimination of a divergent limiting Fourier integral for ϕ at the edge. We connect the existence, or lack thereof, of the EPP dispersion relation to the index for Wiener-Hopf integral equations, an integer of topological character. We indicate that the values of this index may express an asymmetry due to the material anisotropy in the number of wave modes propagating on the sheet away from the edge with respect to the EPP direction of propagation. We discuss extensions such as the setting of two semi-infinite, coplanar sheets. Our theory forms a generalization of the treatment by Volkov and Mikhailov (1988 Sov. Phys.

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“…We proceed to give the main result. By applying to Q ± (ξ ± ) the Mellin transform with respect to q [49,50], we obtain the low-order expansion (see Appendix D)…”

Section: Long-wavelength Asymptotic Expansion Under Zero Indexmentioning

confidence: 99%

“…In this appendix, we outline a framework for the expansion of dispersion relation (20) when |q| is sufficiently small, and derive (23). To this end, we formally apply the Mellin transform technique, which is amenable to systematic calculations [49,50].…”

Section: Appendix D Long-wavelength Expansion Via Mellin Transformmentioning

confidence: 99%

Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials

Margetis¹,

Maier²,

Stauber³

et al. 2020

J. Phys. A: Math. Theor.

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On convergence and inherent oscillations within computational methods employing fictitious sources

Fikioris

Tsitsas

2015

Computers & Mathematics with Applications

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Asymptotic approximations elucidating the Gibbs phenomenon and Fejér averaging

Fikioris

Andrianesis

2017

Asymptotic Analysis

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This paper deals with two trigonometric sums that are pervasive in the literature dealing with the Gibbs phenomenon. In particular, the two sums often serve as test cases for methods – such as the method of Fejér averaging – that aim to overcome the Gibbs phenomenon. Each of the two is the partial sum of a convergent infinite series with a discontinuous limit function. Our starting points are some recently-published results, both exact and asymptotic, for the two sums. For the case of a large number of terms, we proceed from those results to develop simple and revealing asymptotic formulas for the two sums and, also, for their Fejér averages. These formulas break down as we approach the point of discontinuity, so we further develop similar formulas that are appropriate near the discontinuity point. We repeat all these tasks for a third sum whose corresponding infinite series exhibits a logarithmic singularity. Our asymptotic results, which view the three sums from a new perspective, illuminate many aspects of the Gibbs phenomenon and the Fejér averaging method. As an illustrative example of the applicability of our formulas, we exploit their properties to develop a convergence acceleration method. We then use this method to accelerate some (more complicated) sums that exhibit logarithmic singularities, including a sum that arises in several physics applications.

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Selected Asymptotic Methods with Applications to Electromagnetics and Antennas

Cited by 4 publications

References 52 publications

Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials

Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials

On convergence and inherent oscillations within computational methods employing fictitious sources

Asymptotic approximations elucidating the Gibbs phenomenon and Fejér averaging

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