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, Methods of Mathematical Physics, Nauka, Moscow, 1974 MR 0363045
  • [2] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley and Sons, New York, 1983 MR 700400
  • [3] J. Giroire and J. C. Nedelec, Numerical solution of an exterior Neumann problem using a double layer potential, Math. Comp. 32, 973-990 (1978) MR 0495015
  • [4] D. S. Jones, Integral equations for the exterior acoustic problem, Quart. J. Mech. Appl. Math. 27, 129-142 (1974) MR 0339669
  • [5] D. S. Jones, Methods in Electromagnetic Wave Propagation, Oxford University Press, Oxford, 1994 MR 1369090
  • [6] P. A. Krutitskii, Dirichlet problem for the Helmholtz equation outside cuts in a plane, Comput. Math. Math. Phys. 34, 1073-1090 (1994) MR 1300397
  • [7] P. A. Krutitskii, Neumann problem for the Helmholtz equation outside cuts in a plane, Comput. Math. Math. Phys. 34, 1421-1431 (1994) MR 1307611
  • [8] P. A. Krutitskii, The mixed problem for the Helmholtz equation in a multiply connected region, Comput. Math. Math. Phys. 36, 1087-1095 (1996) 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, 439-452 (1998) MR 1661204
  • [10] P. A. Krutitskii, Wave propagation in a 2-D external domain bounded by closed and open curves, Nonlinear Analysis, Theory, Methods and Applications 32, 135-144 (1998) MR 1491619
  • [11] R. Kussmaul, Ein numerische Verfahren zur Lösung des Neumannschen Aussenraumproblems für die Helmholtzsche Schwingungsgleichung, Computing 4, 246-273 (1969) 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. Nedelec, Curved finite element methods for the solution of singular integral equations on surfaces in $ {\mathbb{R}^{3}}$, Comput. Math. Appl. Mech. Engrg. 8, 61-80 (1976) MR 0455503
  • [16] A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhäuser, Basel, 1988 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] S. Prössdorf and B. Silbermann, Numerical Analysis for Integral and Related Operator Equations, Academie-Verlag, Berlin, 1991 MR 1206476
  • [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] C. Ruland, Ein Verfahren zur Lösung von $ \left( \Delta + {k^2} \right)U = 0$ in Assengebieten mit Ecken, Applicable Analysis 7, 69-79 (1978) MR 0474895
  • [21] F. Ursell, On the exterior problems of acoustics, Proc. Cambridge Philos. Soc. 74, 117-125 (1973) MR 0315985
  • [22] F. Ursell, On the exterior problems of acoustics II, Proc. Cambridge Philos. Soc. 84, 545-548 (1978) MR 503014
  • [23] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963
  • [24] V. S. Vladimirov, Equations of Mathematical Physics, Marcel Dekker, New York, 1971 MR 0268497
  • [25] C. H. Wilcox, Scattering theory for the d'Alembert equation in the exterior domains, Lecture Notes in Math. 422, Springer, Berlin, 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

American Mathematical Society