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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 123, Issue 1
  • January 1993, pp. 185-208

Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle

  • P. A. Martin (a1) and Petri Ola (a2)
  • DOI:
  • Published online: 14 November 2011

Time-harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle. The corresponding electromagnetic transmission problem is reduced to a single integral equation over S for a single unknown tangential vector field, where S is the interface between the obstacle and the surrounding medium. In fact, several different integral equations are derived and analysed, including two previously-known equations due to E. Marx and J. R. Mautz, and two new singular integral equations. Mautz's equation is shown to be uniquely solvable at all frequencies. A new uniquely solvable singular integral equation is also found. The paper also includes a review of methods using pairs of coupled integral equations over S. It is these methods that are usually used in practice, although single integral equations seem to offer some computational advantages.

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6R. F. Harrington . Boundary integral formulations for homogeneous material bodies. J. Electromagnetic Waves and Applications 3 (1989), 115.

8G. Jost . Integral equations with modified fundamental solution in time-harmonic electromagnetic scattering. IMA J. Appl. Math. 40 (1988), 129143.

9A. Kirsch . Surface gradients and continuity properties for some integral operators in classical scattering theory. Math. Meth. Appl. Sci. 11 (1989), 789804.

10R. E. Kleinman and P. A. Martin . On single integral equations for the transmission problem of acoustics. SIAM J. Appl. Math. 48 (1988), 307325.

11R. Kress . On boundary integral equation methods in stationary electromagnetic reflection. Lecture Notes in Mathematics 846 (1981), 210226.

13P. A. Martin . Identification of irregular frequencies in simple direct integral-equation methods for scattering by homogeneous inclusions. Wave Motion 13 (1991), 185192.

14E. Marx . Single integral equation for wave scattering. J. Math. Phys. 23 (1982), 10571065.

16J. R. Mautz . A stable integral equation for electromagnetic scattering from homogeneous dielectric bodies. IEEE Trans. Antennas and Propagation AP-37 (1989), 10701071.

18J. R. Mautz and R. F. Harrington , A combined-source solution for radiation and scattering from a perfectly conducting body. IEEE Trans. Antennas and Propagation AP-27 (1979), 445454.

19C. Müller . Foundations of the Mathematical Theory of Electromagnetic Waves (Berlin: Springer, 1969).

20G. Neave . A uniquely solvable integral eqùation for the exterior electromagnetic scattering problem. Quart. J. Mech. Appl. Math. 40 (1987), 5770.

21G. F. Roach . On the commutative properties of boundary integral operators. Proc. Amer. Math. Soc. 73 (1979), 219227.

22M. A. Shubin . Pseudodifferential Operators and Spectral Theory (Berlin: Springer, 1987).

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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