Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

On a uniquely solvable integral equation in a mixed Dirichlet-Neumann problem of acoustic scattering


Author: P. A. Krutitskii
Journal: Quart. Appl. Math. 59 (2001), 493-506
MSC: Primary 45B05; Secondary 31B20, 35J05, 35J25, 76Q05
DOI: https://doi.org/10.1090/qam/1848531
MathSciNet review: MR1848531
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The mixed Dirichlet--Neumann problem for the Helmholtz equation in the exterior of several bodies (obstacles) is studied in 2 and 3 dimensions. The problem is investigated by a special modification of the boundary integral equation method. This modification can be called the ``method of interior boundaries", because additional boundaries are introduced inside scattering bodies, where the Neumann boundary condition is given. The solution of the problem is obtained in the form of potentials on the whole boundary. The density in the potentials satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact, our method holds for any positive wave numbers. The Neumann and Dirichlet problems are particular cases of our problem.


References [Enhancements On Off] (What's this?)

  • [1] V. Ya. Arsenin, \cyr Metody matematicheskoĭ fiziki i spetsial′nye funktsii., Izdat. “Nauka”, Moscow, 1974 (Russian). MR 0363045
  • [2] David L. Colton and Rainer Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 700400
  • [3] J. Giroire and J.-C. Nédélec, Numerical solution of an exterior Neumann problem using a double layer potential, Math. Comp. 32 (1978), no. 144, 973–990. MR 0495015, https://doi.org/10.1090/S0025-5718-1978-0495015-8
  • [4] D. S. Jones, Integral equations for the exterior acoustic problem, Quart. J. Mech. Appl. Math. 27 (1974), 129–142. MR 0339669, https://doi.org/10.1093/qjmam/27.1.129
  • [5] D. S. Jones, Methods in electromagnetic wave propagation, 2nd ed., Oxford Engineering Science Series, vol. 40, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1369090
  • [6] P. A. Krutitskiĭ, The Dirichlet problem for the Helmholtz equation in the exterior of cuts in the plane, Zh. Vychisl. Mat. i Mat. Fiz. 34 (1994), no. 8-9, 1237–1258 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 34 (1994), no. 8-9, 1073–1090. MR 1300397
  • [7] P. A. Krutitskiĭ, The Neumann problem for the Helmholtz equation in the exterior of cuts in the plane, Zh. Vychisl. Mat. i Mat. Fiz. 34 (1994), no. 11, 1652–1665 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 34 (1994), no. 11, 1421–1431 (1995). MR 1307611
  • [8] P. A. Krutitskiĭ, A mixed problem for the Helmholtz equation in a multiply connected domain, Zh. Vychisl. Mat. i Mat. Fiz. 36 (1996), no. 8, 127–137 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 36 (1996), no. 8, 1087–1095 (1997). MR 1407732
  • [9] P. A. Krutitskii, The Neumann problem on wave propagation in a 2-D external domain with cuts, J. Math. Kyoto Univ. 38 (1998), no. 3, 439–451. MR 1661204
  • [10] P. A. Krutitskii, Wave propagation in a 2-D external domain bounded by closed and open curves, Nonlinear Anal. 32 (1998), no. 1, 135–144. MR 1491619, https://doi.org/10.1016/S0362-546X(97)00470-7
  • [11] R. Kussmaul, Ein numerisches Verfahren zur Lösung des Neumannschen Neumannschen Aussenraumproblems für die Helmholtzsche Schwingungsgleichung, Computing (Arch. Elektron. Rechnen) 4 (1969), 246–273 (German, with English summary). MR 0245219
  • [12] R. Leis, Vorlesungen über partielle Differentialgleichungen zweiter Ordnung, Bibliographisches Instit, Mannheim, 1967
  • [13] I. K. Lifanov, Singular integral equations and discrete vortices, VSP, Utrecht, 1996. MR 1451377
  • [14] W. L. Meyer, W. A. Bell, B. T. Zinn, and M. P. Stallybrass, Boundary integral solutions of three dimensional acoustic radiation problems, J. Sound and Vibration 59, 245-262 (1978)
  • [15] J.-C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in 𝑅³, Comput. Methods Appl. Mech. Engrg. 8 (1976), no. 1, 61–80. MR 0455503, https://doi.org/10.1016/0045-7825(76)90053-0
  • [16] Arnold F. Nikiforov and Vasilii B. Uvarov, Special functions of mathematical physics, Birkhäuser Verlag, Basel, 1988. A unified introduction with applications; Translated from the Russian and with a preface by Ralph P. Boas; With a foreword by A. A. Samarskiĭ. MR 922041
  • [17] O. I. Panich, To a problem on solvability of exterior boundary value problems for wave equation and Maxwell equations, Uspekhi Mathemat. Nauk. 20, 221-226 (1965)
  • [18] Siegfried Prössdorf and Bernd Silbermann, Numerical analysis for integral and related operator equations, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 84, Akademie-Verlag, Berlin, 1991 (English, with English and German summaries). MR 1206476
    Siegfried Prössdorf and Bernd Silbermann, Numerical analysis for integral and related operator equations, Operator Theory: Advances and Applications, vol. 52, Birkhäuser Verlag, Basel, 1991. MR 1193030
  • [19] S. Prössdorf and J. Saranen, A fully discrete approximation method for the exterior Neumann problem of the Helmholtz equation, Zeitschrift für Analysis und ihre Anwendungen 13, 683-695 (1999)
  • [20] Christoph Ruland, Ein Verfahren zur Lösung von (Δ+𝑘²)𝑢=0 in Aussengebieten mit Ecken, Applicable Anal. 7 (1977/78), no. 2, 69–79 (German, with English summary). MR 0474895, https://doi.org/10.1080/00036817808839177
  • [21] F. Ursell, On the exterior problems of acoustics, Proc. Cambridge Philos. Soc. 74 (1973), 117–125. MR 0315985
  • [22] F. Ursell, On the exterior problems of acoustics. II, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, 545–548. MR 503014, https://doi.org/10.1017/S0305004100055365
  • [23] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963
  • [24] V. S. Vladimirov, Equations of mathematical physics, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, vol. 3, Marcel Dekker, Inc., New York, 1971. MR 0268497
  • [25] Calvin H. Wilcox, Scattering theory for the d’Alembert equation in exterior domains, Lecture Notes in Mathematics, Vol. 442, Springer-Verlag, Berlin-New York, 1975. MR 0460927

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 45B05, 31B20, 35J05, 35J25, 76Q05

Retrieve articles in all journals with MSC: 45B05, 31B20, 35J05, 35J25, 76Q05


Additional Information

DOI: https://doi.org/10.1090/qam/1848531
Article copyright: © Copyright 2001 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website