Ukrainian Mathematical Journal

, Volume 68, Issue 12, pp 1965–1974 | Cite as

Application of Faber Polynomials to the Approximate Solution of the Riemann Problem

  • M. A. Sheshko
  • D. Pylak
  • P. Wójcik
Article
  • 31 Downloads

The Faber polynomials are used to obtain approximate solutions of the Riemann problem on a Lyapunov curve. Moreover, an estimate for the error of the approximated solution is presented and proved.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. V. Andrievskii, “Approximation of functions by partial sums of series of Faber polynomials on continua with nonzero local geometric characteristic,” Ukr. Math. Zh., 32, No. 1, 3–10; English translation: Ukr. Math. J., 32, No. 1, 1–6 (1980).Google Scholar
  2. 2.
    A. V. Batyrev, “Approximate solution of the Riemann–Privalov problem,” Uspekhi Mat. Nauk, 11, No. 5, 71–76 (1956).MathSciNetMATHGoogle Scholar
  3. 3.
    T. A. Chakhmianok and S. V. Rogosin, “On solvability of inhomogeneous nonlinear power-type boundary value problem,” Complex Var. Elliptic Equat., 52, No. 10-11, 933–943 (2007).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. H. Curtiss, “Solutions of the Dirichlet problem in the plane by approximation with Faber polynomials,” SIAM J. Numer. Anal., 3, No. 2, 204–228 (1966).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    S. W. Ellacott, “Computation of Faber series with application to numerical polynomial approximation in the complex plane,” Math. Comp., 40, No. 162, 575–587 (1983).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    R. Estrada and R. P. Kanwal, Singular Integral Equations, Birkhäuser, Boston (2000).CrossRefMATHGoogle Scholar
  7. 7.
    F. D. Gakhov, “On the Riemann boundary-value problem,” Mat. Sb., 44, No. 4, 673–683 (1937).Google Scholar
  8. 8.
    F. D. Gakhov, Boundary Value Problems, Dover Publ., Mineloa (1990).MATHGoogle Scholar
  9. 9.
    S. G. Gal, “Approximation in compact sets by q-Stancu–Faber polynomials, q > 1,” Comput. Math. Appl., 61, No. 10, 3003–3009 (2011).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    A. Guven and D. Israfilov, “Polynomial approximation in Smirnov-Orlicz classes,” Comput. Methods Funct. Theory, 2, No. 2, 509–517 (2002).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    D. Hilbert, Grundz¨uge einer Allgemeinen Theorie der Linearen Integralgleichungen, Berlin, Leipzig, Leipzig B.G. Teubner (1912).Google Scholar
  12. 12.
    D. M. Israfilov, “Approximation by p-Faber polynomials in the weighted Smirnov class E p(G, ω) and the Bieberbach polynomials,” Constr. Approx., 17, No. 3, 335–351 (2001).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    B. A. Kats, “The refined metric dimension with applications,” Comput. Methods Funct. Theory, 7, No. 1, 77–89 (2007).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    B. A. Kats, “The Riemann boundary-value problem on nonrectifiable curves and related questions,” Complex Var. Elliptic Equat., 1053–1069 (2013).Google Scholar
  15. 15.
    T. Kövari and C. Pommerenke, “On Faber polynomials and Faber expansions,” Math. Z., 99, No. 3, 193–206 (1967).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    D. Kumar, “Faber polynomial approximation of entire functions of slow growth over Jordan domains,” Ann. Univ. Ferrara Sez. VII Sci. Mat., 1–21 (2013).Google Scholar
  17. 18.
    E. G. Ladopoulos and G. Tsamasphyros, “Approximations of singular integral equations on Lyapunov contours in Banach spaces,” Comput. Math. Appl., 50, No. 3-4, 567–573 (2005).MathSciNetCrossRefMATHGoogle Scholar
  18. 19.
    P. Lassre and E. Mazzilli, “Bohr’s phenomenon on a regular condenser in the complex plane,” Comput. Methods Funct. Theory, 12, No. 1, 31–43 (2012).MathSciNetCrossRefMATHGoogle Scholar
  19. 20.
    F. D. Lesley, V. S. Vinge, and S. E. Warschawski, “Approximation by Faber polynomials for a class of Jordan domains,” Math. Z., 138, No. 3, 225–237 (1974).MathSciNetCrossRefMATHGoogle Scholar
  20. 21.
    J. K. Lu, Boundary Value Problems for Analytic Functions, World Scientific, Singapore, 16 (1993).Google Scholar
  21. 22.
    V. V. Mityushev and S. V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary-Value Problems for Analytic Functions: Theory and Applications, CRC Press (2000).Google Scholar
  22. 23.
    N. I. Muskhelishvili, Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics, Dover Publ., Mineloa (2008).Google Scholar
  23. 24.
    N. I. Privalov, “On a boundary-value problem,” Mat. Sb., 41, No. 4, 519–526 (1934).MATHGoogle Scholar
  24. 25.
    M. A. Sheshko, Singular Integral Equations with Cauchy and Hilbert Kernels and Their Approximated Solutions, Catholic University of Lublin, Lublin (2003).Google Scholar
  25. 26.
    P. K. Suetin, Series of Faber Polynomials, Gordon & Breach Sci. Publ., New York (1998).MATHGoogle Scholar
  26. 27.
    P. Wójcik, M. A. Sheshko, and S. M. Sheshko, “Application of Faber polynomials to the approximate solution of singular integral equation with the Cauchy kernel,” Differ. Equat., 49, No. 2, 198–209 (2013).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • M. A. Sheshko
    • 1
  • D. Pylak
    • 1
  • P. Wójcik
    • 1
  1. 1.John Paul II Catholic UniversityLublinPoland

Personalised recommendations