DOI: https://doi.org/10.26089/NumMet.v20r433

Projection algorithms for calculating the Roben potential

Authors

  • V.G. Lezhnev
  • A.N. Markovsky

Keywords:

Roben potential
projection algorithm
complete systems of potentials
method of fundamental solutions
Roben problem of the classical potential theory

Abstract

Several projection algorithms of the basic potential method are proposed for calculating the density of the Roben potential. The completeness of a special system of potentials is proved.

New computational technologies

Downloads

Published

2019-10-29

Issue

Vol. 20 (2019): Issue 4.

Section

Section 1. Numerical methods and applications

Author Biographies

V.G. Lezhnev

Kuban State University,
Faculty of Mathematics and Computer Sciences
• Professor

A.N. Markovsky

Kuban State University,
Faculty of Mathematics and Computer Sciences
• Associate Professor

References

  1. V. G. Lezhnev, “Stream Function of the Two-Dimensional Flow Problem, Robin Potential, and the Exterior Dirichlet Problem,” Dokl. Akad. Nauk 394 (5), 615-617 (2004) [Dokl. Phys. 49 (2), 116-118 (2004)].
  2. V. G. Lezhnev and A. N. Markovsky, “Projection Algorithms for 2D Vortex Flows in Complicated Domains,” Taurida Vestn. Inform. Mat., No. 1, 42-49 (2015).
  3. V. A. Morozov, V. G. Lezhnev, and N. M. Tokarev, “A Variational Problem for the Biharmonic Equation,” Vychisl. Metody Programm. 13, 409-412 (2012).
  4. R. Russo and A. Tartaglione, “On the Robin Problem in Classical Potential Theory,” Math. Models Methods Appl. Sci. 11 (8), 1343-1347 (2001).
  5. A. N. Markovsky, “Integral Representation of a Linear Combination of the Fundamental Solutions to The Laplace Equation,” Ekologich. Vestn. Nauchnykh Tsentrov, No. 4, 49-54 (2011).
  6. J. F. Ahner, V. V. Dyakin, V. Ya. Raevskii, and St. Ritter, “Spectral Properties of Operators of the Theory of Harmonic Potential,” Mat. Zametki 59 (1), 3-11 (1996) [Math. Notes 59 (1), 3-9 (1996)].
  7. P. N. Vabishchevich, “Approximate Solution of the Modified Dirichlet Problem,” Zh. Vychisl. Mat. Mat. Fiz. 31 (11), 1655-1669 (1991) [USSR Comput. Math. Math. Phys. 31 (11), 37-47 (1991)].
  8. V. V. Dyakin, Yu. G. Lebedev, and V. Ya. Rayevskii, “Investigation of a Magnetostatic Model in TsMD Theory,” Fiz. Met. Metalloved. 56 (2), 246-248 (1983).
  9. I. A. Chegis, “An Algorithm for Numerical Solution of an Integral Equation for the Density of a Simple Layer Potential,” Zh. Vychisl. Mat. Mat. Fiz. 29 (12), 1904-1907 (1989) [USSR Comput. Math. Math. Phys. 29 (6), 213-216 (2001)].
  10. I. A. Chegis, “Unique Solvability of an Integral Equation and a Computer Algorithm for an Inner Neumann Problem,” Zh. Vychisl. Mat. Mat. Fiz. 41 (10), 1557-1565 (2001) [Comput. Math. Math. Phys. 41 (10), 1480-1488 (2001)].
  11. V. N. Belykh, “On the Problem of Numerical Solution of the Dirichlet Problem by a Harmonic Single-Layer Potential,” Dokl. Akad. Nauk 329 (4), 392-395 (1993) [Dokl. Math. 47 (2), 252-256 (1993)].
  12. G. Fairweather and R. L. Johnston, “The Method of Fundamental Solutions for Problems in Potential Theory,” in Treatment of Integral Equations by Numerical Methods (Academic Press, London, 1982), pp. 349-359.
  13. R. L. Johnston and G. Fairweather, “The Method of Fundamental Solutions for Problems in Potential Flow,” Appl. Math. Model. 8 (4), 265-270 (1984).
  14. M. Katsurada and H. Okamoto, “The Collocation Points of the Fundamental Solution Method for the Potential Problem,” Comput. Math. Appl. 31 (1), 123-137 (1996).
  15. C. S. Chen, A. Karageorghis, and Y. Li, “On Choosing the Location of the Sources in the MFS,” Numer. Algor. 72 (1), 107-130 (2016).
  16. V. G. Lezhnev, “Systems of Potentials that are Complete on the Boundary of a Domain,” in Proc. Int. Conf., “Mathematical Physics. Vladimirov-90’’ Dedicated to the 90th Anniversary of Academician V. S. Vladimirov, Moscow, Russia, November 13-15, 2013
    |
    http://www.mathnet.ru/php/conference.phtml?confid=364&option_lang=eng|. Cited September 19, 2019.
  17. V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1971; M. Dekker, New York, 1971).
  18. M. Hazewinkel (Ed.), Encyclopaedia of Mathematics , Vol. 3 (Sovetskaya Entsiclopediya, Moscow, 1984; Springer, New York, 1995).
  19. V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1976; Mir, Moscow, 1978).
  20. V. G. Lezhnev and A. N. Markovskiy, “The Basic Potential Method for an Inhomogeneous Biharmonic Equation,” Vestn. Samar. Gos. Univ., No. 1, 127-139 (2008).
  21. V. G. Lezhnev and A. N. Markovskiy, “Projective Algorithm of Boundary Value Problem for Inhomogeneous Lame’s Equation,” Vestn. Samar. Gos. Tekh. Univ., Ser.: Fiz. Mat., No. 1, 236-240 (2011).
  22. M. A. Aleksidze, Fundamental Functions in Approximate Solutions of Boundary Value Problems (Nauka, Moscow, 1991) [in Russian].
  23. V. A. Morozov, Regular Methods for Solving Ill-Posed Problems (Mosk. Gos. Univ., Moscow, 1987) [in Russian].