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Integral equations in the wave scattering problem at an irregular interface of domains

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Abstract

Modeling the scattering of electromagnetic waves at an interface of media with different characteristics, one encounters the conjugation problem. Using the method of boundary integral equations and the theory of generalized potentials, we prove the classical resolvability of this problem. The boundary is assumed to be irregular. This means that the plane is divided into two domains by a curve which coincides with a straight line, except for a finite part, producing the irregularity. We propose algorithms for the approximate solution of the conjugation problem based on the spline methods for the solution of integral equations. We theoretically substantiate the computational scheme, namely, we prove the convergence and estimate the convergence rate.

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References

  1. E. K. Lipachev, “Boundary Value Problems for the Helmholtz Equation in Domains with an “Irregular” Boundary,” in N. I. Lobachevskii Math. Center (Kazansk. Matem. Ob-vo, Kazan, 2002), Vol. 17, pp. 79–89.

    Google Scholar 

  2. E. K. Lipachev, “On Boundary Integral Equation Method in Scattering Problem for Unbounded Domains,” SIAM Proc. in Appl. Math. 102, 509–512 (2000).

    MathSciNet  Google Scholar 

  3. V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  4. V. G. Maz’ya, “Boundary Integral Equations,” Itogi Nauki i Tekhniki, Sovr. Problemy Matem. Fundam. Napr. (VINITI, Moscow) 27, 131–228 (1988).

    Google Scholar 

  5. T. N. Galishnikova and A. S. Il’inskii, Numerical Methods in Diffraction Problems (Mosk. Gos. Univ., Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  6. N. N. Fedorov, Foundations of Electrodynamics (Vyssh. Shkola, Moscow, 1980) [in Russian].

    Google Scholar 

  7. R. Petit (Ed.), Electromagnetic Theory of Gratings (Berlin-Heidelberg-New York, 1980).

  8. R. Kress and G. F. Roach, “Transmission Problems for the Helmholtz Equation,” J. Math. Phys. 19, 1433–1437 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. G. Ramm, “Scattering by a Penetrable Body,” Journal Math. Phys. 25(3), 469–471 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  10. R. E. Kleinman and P. A. Martin, “On Single Integral Equations for the Transmission Problem of Acoustics,” SIAM J. Appl. Math. 48, 307–325 (1988).

    Article  MathSciNet  Google Scholar 

  11. N. B. Pleshchinskii, “The Fourier Transformation Method in Conjugation Problems for Electromagnetic Fields,” in N. I. Lobachevskii Math. Center (Kazansk. Matem. Ob-vo, Kazan, 2000), Vol. 6, pp. 153–185.

    Google Scholar 

  12. A. S. Il’inskii and Yu. V. Shestopalov, Application of the Spectral Theory Methods in Wave Propagation Problems (Mosk. Gos. Univ., Moscow, 1989) [in Russian].

    Google Scholar 

  13. Ye. K. Lipachyov, “Approximation Solution of the Boundary Value Problem of Wave Diffraction on Domains with Infinite Boundary,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 69–72 (2001) [Russian Mathematics (Iz. VUZ) 45, No. 4, 67–70].

  14. Yu. V. Shestopalov, “Application of the Method of Generalized Potentials for the Solution of Certain Problems of the Theory of Diffraction and Wave Propagation,” Zhurn. Vychisl. Matem. i Matem. Fiz. 30(7), 1081–1092 (1990).

    MATH  MathSciNet  Google Scholar 

  15. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983; Mir, Moscow, 1987).

    MATH  Google Scholar 

  16. B. G. Gabdulkhaev, Optimal Approximations of Solutions to Linear Problems (Kazansk. Gos. Univ., Kazan, 1980) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Kazan State University, ul. Kremlyovskaya 18, Kazan, 420008, Russia

    E. K. Lipachev

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  1. E. K. Lipachev
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Correspondence to E. K. Lipachev.

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Original Russian Text © E.K. Lipachev, 2007, published in Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 2007, No. 8, pp. 35–47.

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Lipachev, E.K. Integral equations in the wave scattering problem at an irregular interface of domains. Russ Math. 51, 33–44 (2007). https://doi.org/10.3103/S1066369X0708004X

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  • DOI: https://doi.org/10.3103/S1066369X0708004X

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