Differential Equations

, Volume 54, Issue 4, pp 539–550 | Cite as

Constructive Method for Solving a Boundary Value Problem with Impedance Boundary Condition for the Helmholtz Equation

  • E. H. Khalilov
Numerical Methods


We substantiate the collocation method for the singular integral equation of a boundary value problem with impedance condition for the Helmholtz equation. We construct a sequence converging to the exact solution of the original problem and estimate the error.


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  1. 1.
    Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, New York: Wiley, 1983. Translated under the title Metody integral’nykh uravnenii v teorii rasseyaniya, Moscow: Mir, 1987.zbMATHGoogle Scholar
  2. 2.
    Gunter, N.M., Teoriya potentsialov i ee primenenie k osnovnym zadacham matematicheskoi fiziki (Potential Theory and Its Application to Basic Problems of Mathematical Physics), Moscow: Gostekhizdat, 1953.Google Scholar
  3. 3.
    Abdullaev, F.A. and Khalilov, E.G., Justification of the collocation method for a class of boundary integral equations, Differ. Equations, 2004, vol. 40, no. 1, pp. 89–93.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kashirin, A.A., On conditionally well-posed integral equations and numerical solution of stationary problems of acoustic wave diffraction, Vestnik Tikhookeanskogo Gos. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 3 (26), pp. 33–40.Google Scholar
  5. 5.
    Kashirin, A.A. and Smagin, S.I., Potential-based numerical solution of the Dirichlet problem for the Helmholtz equation, Comput. Math. Math. Phys., 2012, vol. 52, no. 8, pp. 1173–1185.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Medvedik, M.Yu., Smirnov, Yu.G., and Tsupak, A.A., Scalar problem of plane wave diffraction by a system of nonintersecting screens and inhomogeneous bodies, Comput. Math. Math. Phys., 2014, vol. 54, no. 8, pp. 1280–1292.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Khalilov, E.H., Justificaiton of the collocation method the integral equation of a mixed boundary value problem for the Helmholtz equation, Comput. Math. Math. Phys., 2016, vol. 56, no. 7, pp. 1310–1318.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Khalilov, E.H., On an approximate solution of a class of boundary integral equations of the first kind, Differ. Equations, 2016, vol. 52, no. 9, pp. 1234–1240.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Harris, P.J. and Chen, K., On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem, J. Comput. Appl. Math., 2003, vol. 156, pp. 303–318.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Khalilov, E.H., On approximate solution of external Dirichlet boundary value problem for Laplace equation by collocation method, Azerb. J. Math., 2015, vol. 5, no. 2, pp. 13–20.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kress, R., Boundary integral equations in time-harmonic acoustic scattering, Math. Comput. Modeling, 1991, vol. 15, no. 3–5, pp. 229–243.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1967.Google Scholar
  13. 13.
    Kustov, Yu.A., Musaev, Yu. A., and Musaev, B.I., Kubaturnaya formula dlya dvumernogo integrala i ee prilozheniya (Cubature Formula for a Two-Dimensional Singular Integral and Its Applications), Moscow: VINITI, 1981, Depon. no. 4281-81.Google Scholar
  14. 14.
    Khalilov, E.H., Cubic formula for class of weakly singular surface integrals, Proc. Inst. Math. Mech. Natz. Akad. Sci. Azerb., 2013, vol. 39 (47), pp. 69–76.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Vainikko, G.M., Regular convergence of operators and approximate solution of equations, in Itogi nauki i tekhniki. Matematicheskii analiz (Progress in Science and Technology. Mathematical Analysis), Moscow: VINITI, 1979, vol. 16, pp. 5–53.MathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Azerbaijan State Oil and Industry UniversityBakuAzerbaijan

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