Advertisement

Mathematical Notes

, Volume 101, Issue 5–6, pp 759–777 | Cite as

Finding the coefficients in the new representation of the solution of the Riemann–Hilbert problem using the Lauricella function

  • S. I. Bezrodnykh
Volume 101, Number 5, May, 2017
  • 28 Downloads

Abstract

The solution of the Riemann–Hilbert problem for an analytic function in a canonical domain for the case in which the data of the problem is piecewise constant can be expressed as a Christoffel–Schwartz integral. In this paper, we present an explicit expression for the parameters of this integral obtained by using a Jacobi-type formula for the Lauricella generalized hypergeometric function F D (N). The results can be applied to a number of problems, including those in plasma physics and the mechanics of deformed solids.

Keywords

Riemann–Hilbert problem with piecewise constant data Lauricella function FD(N) Jacobi-type formula Christoffel–Schwartz integral 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, Inauguraldissertation (Göttingen, 1851; Gostekhizdat, 1948).Google Scholar
  2. 2.
    D. Hilbert, Über eine Anwendung der Integralgleichungen auf ein Problem der Functionentheorie, Verhandl. des III Internat. Math. Kongr. (Heidelberg, 1904).Google Scholar
  3. 3.
    D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (B. G. Teubner, Leipzig, 1912).zbMATHGoogle Scholar
  4. 4.
    F. Noether, “Über eine Klasse singulärer Integralgleichungen,” Math. Ann. 82 (1–2), 42–63 (1921).zbMATHGoogle Scholar
  5. 5.
    T. Carleman, “Sur la résolution de certaines équations intégrales,” Ark. Mat. Astron. Fys. 16 (26), 1–19 (1922).zbMATHGoogle Scholar
  6. 6.
    N. Muskhelishvili, Applications des intégrales analogues à celles de Cauchy à quelques pronlémes de la physique mathématique (Édition de l’Universitéde Tiflis, Tiflis, 1922).zbMATHGoogle Scholar
  7. 7.
    É. Picard, Leçons sur quelques types simples d’équations aux dérivées partielles avec des applications ála physique mathématique (Gauthier-Villars, Paris, 1927).zbMATHGoogle Scholar
  8. 8.
    F. D. Gakhov, “Linear boundary-value problems of the theory of functions of a complex variable,” Izv. Kazan. Fiz.-Mat. Obshch. 10 (3), 39–79 (1938).Google Scholar
  9. 9.
    N. I. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1968) [in Russian].zbMATHGoogle Scholar
  10. 10.
    F. D. Gakhov, Boundary-Value Problems (Nauka, Moscow, 1977) [in Russian].zbMATHGoogle Scholar
  11. 11.
    W. Wendland, Elliptic Systems in the Plane, in Monogr. Stud. in Math. (Pitman, London, 1979), Vol. 3.Google Scholar
  12. 12.
    F. Frank and R. Mieses, Differential and Integral Equations of Mathematical Physics (ONTI, Moscow–Leningrad, 1937) [Russian transl.].Google Scholar
  13. 13.
    L. N. Trefethen and R. J. Williams, “Conformal mapping solution Laplace’s equation for a polygon with oblique derivative boundary condition,” J. Comput. Appl. Math. 14 (1–2), 227–249 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S. A. Markovskii and B. V. Somov, “A model of magnetic field line reconnection in a current layer with shock waves,” in Physics of Sun Plasma (Nauka, Moscow, 1989), pp. 456–472 [in Russian].Google Scholar
  15. 15.
    P. A. Krutitskii, “Flow of an electric current from rectilinear electrodes in a magnetized semiconducting film,” Zh. Vychisl. Mat. i Mat. Fiz. 30 (11), 1689–1701 (1990) [U. S. S. R. Comput. Math. and Math. Phys. 30 (6), 64-73 (1992)].MathSciNetGoogle Scholar
  16. 16.
    A. I. Aptekarev, V. Van Assshe, and S. P. Suetin, “A scalar Riemann problem approach to the strong asymptotics of Padéapproximates and orthogonal polynomials,” Keldysh Institute Preprints, No. 026 (2001).Google Scholar
  17. 17.
    A. Aptekarev, A. Cachafeiro, and F. Marcellán, “A scalar Riemann boundary value problem approach orthogonal polynomials on the circle,” J. Approx. Theory 141 (2), 174–181 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. S. Demidov, “Functional geometric method for solving free boundary problems for harmonic functions,” UspekhiMat. Nauk 65 (1), 3–96 (2010) [RussianMath. Surveys 65 (1), 1–94 (2010)].MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    S. I. Bezrodnykh and V. I. Vlasov, “Singular Riemann-Hilbert problem in complex-shaped domains,” Zh. Vychisl. Mat. i Mat. Fiz. 54 (12), 1904–1953 (2014) [Comput. Math. Math. Phys. 54 (12), 1826–1875 (2014)].MathSciNetzbMATHGoogle Scholar
  20. 20.
    A. P. Soldatov, Boundary-Value Problems of Function Theory in Domains with a Piecewise-Smooth Boundary (Izd. Tbilis. Gos. Univ., Tbilisi, 1991) [in Russian].zbMATHGoogle Scholar
  21. 21.
    A. P. Soldatov, “Weighted Hardy classes of analytic functions,” Differ. Uravn. 38 (6), 809–817 (2002) [Differ. Equations 38 (6), 855–864 (2002)].MathSciNetzbMATHGoogle Scholar
  22. 22.
    S. B. Klimentov, Boundary Properties of Generalized Analytic Functions (YuMI VNTs RAN and RSO-A, Vladikavkaz, 2014) [in Russian].zbMATHGoogle Scholar
  23. 23.
    S. I. Bezrodnykh, V. I. Vlasov, and B. V. Somov, “Generalized analytic models of the Syrovatskii current layer,” Pis’ma Astronom. Zh. 37 (2), 133–150 (2011).Google Scholar
  24. 24.
    B. V. Somov, Plasma Astrophysics Part II: Reconnection and Flares (Springer, New York, 2013).CrossRefzbMATHGoogle Scholar
  25. 25.
    E. Treffz, “Über die Wirkung einer Abrundung auf die Torsionsspannungen in der inneren Ecke eines Winkeleisens,” Z. Angew. Math. Mech. 2 (4), 263–267 (1922).CrossRefzbMATHGoogle Scholar
  26. 26.
    S. I. Bezrodnykh and V. I. Vlasov, “On a new representation for the solution of the Riemann–Hilbert problem,” Math. Notes 99 (6), 932–937 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. I. Bezrodnykh and V. I. Vlasov, “A singular Riemann–Hilbert problem in complex-shaped domains,” Spectral and Evolution Problems 16, 51–62 (2006).zbMATHGoogle Scholar
  28. 28.
    S. I. Bezrodnykh and V. I. Vlasov, “The Riemann–Hilbert problem in a complicated domain for a model of magnetic reconnection in a plasma,” Zh. Vychisl. Mat. iMat. Fiz. 42 (3), 277–312 (2002) [Comput. Math. Math. Phys. 42 (3), 263–298 (2002)].MathSciNetzbMATHGoogle Scholar
  29. 29.
    S. I. Bezrodnykh, “A Jacobi-type formula for the generalized hypergeometric function,” in TheMathematical Ideas of P. L. Chebyshev and Their Application toModernProblems in theNatural Sciences, Abstracts from the 3rd International Conference held inObninsk, May 14–18, 2006 (Obninsk. Gos. Tekhn. Univ. Atom. Énergetiki, Obninsk, 2006), pp. 18–19 [in Russian].Google Scholar
  30. 30.
    S. I. Bezrodnykh, “Analytic continuation formulas and relations of Jacobi type for a Lauricella function,” Dokl. Ross. Akad. Nauk 467 (1), 7–12 (2016) [Dokl. Math. 93 (2), 129–134 (2016)].MathSciNetzbMATHGoogle Scholar
  31. 31.
    S. I. Bezrodnykh, “Jacobi-type differential relations for the Lauricella function F D (N),” Mat. Zametki 99 (6), 832–847 (2016) [Math. Notes 99 (5–6), 821–833 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    G. Lauricella, “Sulle funzioni ipergeometriche a piu variabili,” Rend. Circ. Math. Palermo 7, 111–158 (1893), Suppl. 1.CrossRefzbMATHGoogle Scholar
  33. 33.
    H. Exton, Multiple Hypergeometric Functions and Applications (JohnWiley & Sons, New York, 1976).zbMATHGoogle Scholar
  34. 34.
    K. Aomoto and M. Kita, Theory of Hypergeometric Functions (Springer-Verlag, Tokyo, 2011) [in Russian].CrossRefzbMATHGoogle Scholar
  35. 35.
    P. P. Kufarev, “Torsion and bending of rods of polygonal section,” Prikl. Matem. Mekh. 1 (1), 43–76 (1937).zbMATHGoogle Scholar
  36. 36.
    W. von Koppenfels and F. Stallmann, Praxis der konformen Abbildung, in Die Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin–Göttingen–Heidelberg, 1959; Inostr. Lit., Moscow, 1963), Bd. 100.Google Scholar
  37. 37.
    V. I. Vlasov, S. A. Markovskii, and B. V. Somov, On the analytic model of magnetic reconnection in a plasma, Available from VINITI, No. 769-V89 (Moscow, 1989) [in Russian].Google Scholar
  38. 38.
    V. I. Vlasov, Boundary-Value Problems in Domains with Curvilinear Boundary (Izd. VTs AN SSSR, Moscow, 1987) [in Russian].zbMATHGoogle Scholar
  39. 39.
    V. I. Vlasov and S. L. Skorokhodov, “On the development of the Trefftz method,” Dokl. Ross. Akad. Nauk 337 (6), 713–717 (1994) [Dokl. Math. 50 (1), 157–163 (1995)].zbMATHGoogle Scholar
  40. 40.
    W. C. Hassenpflug, “Torsion uniform bars with polygon cross-section,” Comput. Math. Appl. 46 (2-3), 313–392 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    S. I. Bezrodnykh and B. V. Somov, “Analysis of magnetic field and magnetosphere of neutron star under effect of a shock wave,” Adv. in Space Res. 56 (5), 964–969 (2015).CrossRefGoogle Scholar
  42. 42.
    S. I. Bezrodnykh, The Singular Riemann–Hilbert Problem and Its Application, Cand. Sci. (Phys.–Math.) Dissertation (VTs RAN, Moscow, 2006) [in Russian].Google Scholar
  43. 43.
    C. G. J. Jacobi, “Untersuchungen über die Differentialgleichungen der hypergeometrischen Reihe,” J. Reine Angew. Math. 56, 149–165 (1859).MathSciNetCrossRefGoogle Scholar
  44. 44.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1: The Hypergeometric Function, Legendre Functions (McGraw–Hill, New York–Toronto–London, 1953; Nauka, Moscow, 1965 and 1973 (2nd ed.)).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Federal Research Center “Computer Science and Control,”Russian Academy of SciencesMoscowRussia
  2. 2.RUDN UniversityMoscowRussia
  3. 3.Sternberg Astronomical InstituteLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations