Computational Mathematics and Mathematical Physics

, Volume 54, Issue 12, pp 1826–1875 | Cite as

Singular Riemann-Hilbert problem in complex-shaped domains

  • S. I. Bezrodnykh
  • V. I. Vlasov


In simply connected complex-shaped domains ℬ a Riemann-Hilbert problem with discontinuous data and growth condidions of a solution at some points of the boundary is considered. The desired analytic function ℱ(z) is represented as the composition of a conformal mapping of ℬ onto the half-plane \(\mathbb{H}^ + \) and the solution ℘ of the corresponding Riemann-Hilbert problem in \(\mathbb{H}^ + \). Methods for finding this mapping are described, and a technique for constructing an analytic function ℘+ in \(\mathbb{H}^ + \) in the terms of a modified Cauchy-type integral. In the case of piecewise constant data of the problem, a fundamentally new representation of ℘+ in the form of a Christoffel-Schwarz-type integral is obtained, which solves the Riemann problem of a geometric interpretation of the solution and is more convenient for numerical implementation than the conventional representation in terms of Cauchytype integrals.


Riemann-Hilbert problem Cauchy-type integral conformal mappings Schwarz-Christoffel integral hypergeometric functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse—Inauguraldissertation (Göttingen, 1851; Werke, (Leipzig, 1876).Google Scholar
  2. 2.
    Yu. V. Sokhotsky, Definite Integrals and Functions Used in Series Expansions (St. Petersburg, 1873) [in Russian].Google Scholar
  3. 3.
    V. Volterra, “Sopra alcune condizioni caratteristiche delle funzioni di una variabile complessa,” Ann. Math. Pura Appl. 11(2), 1–55 (1883).Google Scholar
  4. 4.
    D. Hilbert, “Uber eine Anwendung der Intergralgleichungen auf ein Problem der Functionentheorie,” Verhandl. des III Internat. Math. Kongr. (Heidelberg, 1904).Google Scholar
  5. 5.
    J. Plemelj, “Riemannshe funktionenscharen mit gegebener Monodromie-gruppe,” Monatsh. Math. Phys. 19, 205–210 (1908).zbMATHMathSciNetGoogle Scholar
  6. 6.
    D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Intergralgleichungen (Teubner, Leipzig, 1912).Google Scholar
  7. 7.
    F. Noether, “Über eine Klasse singulärer Intergralgleichungen,” Math. Ann. 82(1–2), 42–63 (1921).MathSciNetGoogle Scholar
  8. 8.
    T. Garleman, “Sur la résolution de certaines équations intégrals,” Ark. Math. Astron. Phys. 16(26), 1–19 (1922).Google Scholar
  9. 9.
    É. Picard, Lecons sur quelques types simples d’equations aux derives partielles avec des applications a la physique mathematique (Gauthier-Villars, Paris, 1927).Google Scholar
  10. 10.
    F. D. Gakhov, “On the Riemann boundary value problem,” Mat. Sb. 2(44), 673–683 (1937).Google Scholar
  11. 11.
    N. I. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1968; Wolters-Noordhoff, Groningen, 1972).zbMATHGoogle Scholar
  12. 12.
    S. G. Mikhlin, Integral Equations (Nauka, Moscow, 1966) [in Russian].Google Scholar
  13. 13.
    W. Wendland, Elliptic Systems in the Plane (Pitman, London, 1979).zbMATHGoogle Scholar
  14. 14.
    F. D. Gakhov, Boundary Value Problems (Nauka, Moscow, 1977) [in Russian].zbMATHGoogle Scholar
  15. 15.
    S. Prösdorf, Einige Klassen singularer Gleichungen (Akademie, Berlin, 1974).Google Scholar
  16. 16.
    E. Meister, Randweraufgaben der Funktionentheorie (Teubner, Stuttgart, 1983).Google Scholar
  17. 17.
    I. N. Vekua, Generalized Analytic Functions (Addison-Wesley, Reading, Mass., 1962; Nauka, Moscow, 1988).zbMATHGoogle Scholar
  18. 18.
    V. I. Smirnov, A Course of Higher Mathematics (Nauka, Moscow, 1969/1958; Addison-Wesley, Reading, Mass., 1964), Vols. 3, 4.Google Scholar
  19. 19.
    M. A. Lavrent’ev and B. V. Shabat, Methods of Complex Analysis (Nauka, Moscow, 1973) [in Russian].Google Scholar
  20. 20.
    P. Henrici, Applied and Computational Complex Analysis (Wiley, New York, 1991), Vols. 1–3.zbMATHGoogle Scholar
  21. 21.
    F. D. Gakhov, “Linear boundary value problems in the theory of functions of a complex variable,” Izv. Kazan. Fiz-Mat. Ob-va, Nauchno-Issled. Inst. Mat. Mekh. Kazan. Univ., Ser. 3 10, 39–79 (1938).zbMATHGoogle Scholar
  22. 22.
    G. Giraud, “Sur une classe d’équations linéaires o figurant des valeurs principales d’intégrales simples,” Ann. Sci. Ecolé Norm. Supér., Sér. 3 56, 119–172 (1939).Google Scholar
  23. 23.
    F. D. Gakhov, “Boundary value problems in the theory of analytic functions and singular integral equations—Doctoral dissertation, Tbilisi, 1941,” Izv. Kazan. Fiz-Mat. Ob-va, Ser. 3 14, 75–160 (1949).Google Scholar
  24. 24.
    N. I. Muskhelishvili and D. A. Kveselava, “Singular integral equations with Cauchy-type kernels on unclosed contours,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR 11, 141–172 (1942).Google Scholar
  25. 25.
    N. I. Muskhelishvili and N. P. Vekua, “Riemann boundary value problem for several unknown functions and its applications to systems of singular integral equations,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR 12, 1–46 (1943).Google Scholar
  26. 26.
    B. V. Khvedelidze, “Linear discontinuous boundary value problem in function theory, singular integral equations, and some applications,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR 23, 3–158 (1956).Google Scholar
  27. 27.
    B. V. Bojarski, “A special case of the Riemann-Hilbert problem,” Dokl. Akad. Nauk SSSR 119(3), 411–414 (1958).MathSciNetGoogle Scholar
  28. 28.
    F. D. Gakhov and B. V. Khvedelidze, “Boundary value problems in the theory of analytic functions of a complex variable,” in Mathematics in the USSR over 40 years: 1917–1957 (Fizmatgiz, Moscow, 1959), Vol. 1, pp. 498–510 [in Russian].Google Scholar
  29. 29.
    B. Bojarski, “On the index problem for systems of singular integral equations,” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 11(10) (1963).Google Scholar
  30. 30.
    B. V. Khvedelidze, “Method of Cauchy-type integrals as applied to discontinuous boundary value problems in the theory of holomorphic functions,” in Advances in Science and Engineering: Modern Problems in Mathematics (VINITI Akad. Nauk SSSR, Moscow, 1975), Vol. 7, pp. 5–162 [in Russian].Google Scholar
  31. 31.
    F. D. Gakhov, “Riemann’s boundary value problem for a system of n pairs of functions,” Usp. Mat. Nauk 7(4), 3–54 (1952).zbMATHGoogle Scholar
  32. 32.
    I. Ts. Gokhberg and M. G. Krein, “Systems of integral equations on the half-line with kernels depending on the difference of the arguments,” Usp. Mat. Nauk 13(2), 3–72 (1958).Google Scholar
  33. 33.
    R. T. Seely, “The index of elliptic systems of singular integral operators,” J. Math. Anal. Appl. 7(2), 289–309 (1963).MathSciNetGoogle Scholar
  34. 34.
    M. I. Vishik and G. I. Eskin, “Convolution equations in a bounded domain in spaces with weight norms,” Mat. Sb. 69(1), 65–110 (1966).MathSciNetGoogle Scholar
  35. 35.
    N. P. Vekua, Systems of Singular Integral Equations (Noordhoff, Groningen, 1967; Nauka, Moscow, 1970).zbMATHGoogle Scholar
  36. 36.
    E. I. Zverovich, “Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces,” Russ. Math. Surv. 26(1), 117–192 (1971).Google Scholar
  37. 37.
    B. V. Pal’tsev, “On the canonical solution matrix of the linear matching problem with a piecewise continuous matrix coefficient on an elementary piecewise smooth curve,” Dokl. Akad. Nauk SSSR 297(5), 1054–1058 (1987).Google Scholar
  38. 38.
    B. V. Pal’tsev, “Conditions under which solutions of the homogeneous linear matching problem with a piecewise continuous matrix coefficient are continuous up to the boundary and have power-law growth near nodal points,” Dokl. Akad. Nauk SSSR 299(3), 558–562 (1988).Google Scholar
  39. 39.
    A. P. Soldatov, Boundary Value Problems in Function Theory in Domains with a Piecewise Smooth Boundary (Tbilis. Univ., Tbilisi, 1991) [in Russian].Google Scholar
  40. 40.
    L. Wolfersdorf, “On the theory of the nonlinear Hilbert problem for holomorphic functions,” in Partial Differential Equations with Complex Analysis, Ed. by H. Begehr and A. Jeffrey (Addison Wesley, Harlow, 1992), pp. 134–149.Google Scholar
  41. 41.
    A. P. Soldatov, “A function theory method in elliptic problems in the plane. II: The piecewise smooth case,” Russ. Acad. Sci. Izv. Math. 40(3), 529–563 (1992).zbMATHMathSciNetGoogle Scholar
  42. 42.
    E. Wegert, Nonlinear Boundary Value Problems for Holomorphic Functions and Singular Integral Equations (Akademie, Berlin, 1992).zbMATHGoogle Scholar
  43. 43.
    M. A. Efendiev and W. L. Wendland, “Nonlinear Riemann-Hilbert problem for multiply connected domains,” J. Nonlinear Anal. Theory Methods Appl. 27, 37–58 (1996).zbMATHMathSciNetGoogle Scholar
  44. 44.
    H. Begehr and G. C. Wen, Nonlinear Elliptic Boundary Value Problems and Their Applications (Addison Wesley, Harlow, 1996).zbMATHGoogle Scholar
  45. 45.
    A. P. Soldatov, “Weighted Hardy classes of analytic functions,” Differ. Equations 38(6), 855–864 (2002).zbMATHMathSciNetGoogle Scholar
  46. 46.
    B. V. Pal’tsev, “Asymptotic behavior of the spectra of integral convolution operators on a finite interval with homogeneous polar kernel,” Izv. Math. 67(4), 695–779 (2003).zbMATHMathSciNetGoogle Scholar
  47. 47.
    I. N. Vekua, New Methods for Solving Elliptic Equations (GITTL, Moscow, 1948) [in Russian].Google Scholar
  48. 48.
    L. Bers, Theory of Pseudo-Analytic Functions (New York Univ., New York, 1953).zbMATHGoogle Scholar
  49. 49.
    A. V. Bitsadze, Equations of Mixed Type (Akad. Nauk SSSR, Moscow, 1959) [in Russian].zbMATHGoogle Scholar
  50. 50.
    L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics (New York Univ., New York, 1952; Inostrannaya Literatura, Moscow, 1961).Google Scholar
  51. 51.
    S. Bergman, Integral Operators in the Theory of Linear Partial Differential Equations (New York, Springer, 1962; Mir, Moscow, 1964).Google Scholar
  52. 52.
    I. Ya. Shtaerman, Contact Problem in Elasticity (Gostekhizdat, Moscow, 1949) [in Russian].Google Scholar
  53. 53.
    D. I. Sherman, “Relation between the fundamental elasticity problem and a special case of the Poincaré problem,” Prikl. Mat. Mekh. 17, 685–692 (1953).zbMATHGoogle Scholar
  54. 54.
    N. I. Muskhelishvili, Some Basic Problems in the Mathematical Theory of Elasticity (Nauka, Moscow, 1966) [in Russian].Google Scholar
  55. 55.
    I. N. Vekua, “On Prandtl’s integro-differential equation,” Prikl. Mat. Mekh. 9(2), 143–150 (1945).zbMATHMathSciNetGoogle Scholar
  56. 56.
    D. I. Sherman, “Prandtl’s equation in finite wing theory,” Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, No. 5, 595–600 (1948).Google Scholar
  57. 57.
    G. W. Veltkamp, “The drag on a vibrating aerofoil in incompressible flow I, II,” Indagationes Math. 61(3), 278–297 (1958).zbMATHMathSciNetGoogle Scholar
  58. 58.
    C. Jacob, Introduction mathématique a la mécaniqe des fluides (Gauthier-Villars, Paris, 1959).Google Scholar
  59. 59.
    L. C. Woods, The Theory of Supersonic Plane Flow (Cambridge Univ. Press, Cambridge, 1961).Google Scholar
  60. 60.
    L. Wolfersdorf, “Abelsche integralgleichungen und Randwertprobleme für die verallgemeinerte Tricomi-Gleichung,” Math. Nachr. 32(3–4), 161–178 (1965).Google Scholar
  61. 61.
    L. I. Sedov, Two-Dimensional Problems in Fluid Dynamics and Aerodynamics (Nauka, Moscow, 1966) [in Russian].Google Scholar
  62. 62.
    V. N. Monakhov, Boundary Value Problems with Free Boundaries for Elliptic Systems of Equations (Nauka, Novosibirsk, 1978) [in Russian].Google Scholar
  63. 63.
    W. Wick, “Solution of the field problem of the germanium gyrator,” J. Appl. Phys. 25, 731–756 (1954).Google Scholar
  64. 64.
    W. Vesnel, “The geometrical correction factor for a rectangular Hall plate,” J. Appl. Phys. 53, 4980–4986 (1982).Google Scholar
  65. 65.
    L. N. Trefethen and R. J. Williams, “Conformal mapping solution of Laplace’s equation on a polygon with oblique derivative boundary condition,” J. Comput. Appl. Math. 14, 227–249 (1986).zbMATHMathSciNetGoogle Scholar
  66. 66.
    E. Wegert and D. Oestreich, “On a nonsymmetric problem in electrochemical machining,” Math. Meth. Appl. Sci. 20, 841–854 (1997).zbMATHMathSciNetGoogle Scholar
  67. 67.
    R. Wegmann, “Keplerian discs around magnetized neutron stars—a free boundary problem,” Meth. Verfahren Math. Phys. 37, 233–253 (1991).MathSciNetGoogle Scholar
  68. 68.
    V. Isakov, “Prospecting discontinuities by boundary measurements in inverse problems,” in Principles and Applications in Geophysics, Technology, and Medicine (Akademie, Berlin, 1993), pp. 215–223.Google Scholar
  69. 69.
    J. Powell, “On small perturbation in the two-dimensional inverse conductivity problem,” J. Math. Anal. Appl. 175(1), 292–304 (1993).zbMATHMathSciNetGoogle Scholar
  70. 70.
    S. L. Sobolev, “On a limit problem in the theory of logarithmic potential and its applications to elastic wave reflection,” Tr. Seism. Inst., No. 11, 1–18 (1930).Google Scholar
  71. 71.
    M. M. Fridman, “Diffraction of a plane elastic wave on a tension-free straight cut,” Dokl. Akad. Nauk SSSR 66(1), 21–24 (1949).zbMATHMathSciNetGoogle Scholar
  72. 72.
    G. G. Tumashev and M. T. Nuzhin, Inverse Boundary Value Problems and Applications (Kazan. Univ., Kazan, 1965) [in Russian].Google Scholar
  73. 73.
    V. E. Zakharov, S. V. Monakov, S. P. Novikov, and L. P. Pitaevskii, Soliton Theory—Inverse Scattering Method (Nauka, Moscow, 1980) [in Russian].Google Scholar
  74. 74.
    A. V. Latyshev, “The vector Riemann-Hilbert boundary value problem in boundary value problems of the scattering of polarized light,” Comput. Math. Math. Phys. 35(7), 885–900 (1995).zbMATHMathSciNetGoogle Scholar
  75. 75.
    T. L. Saaty, Elements of Queuing Theory with Applications (McGraw-Hill, New York, 1961; Sov. Radio, Moscow, 1971).Google Scholar
  76. 76.
    K. L. Chung and R. J. Williams, An Introduction to Stochastic Integration (Birkhäuser, Boston, 1983).Google Scholar
  77. 77.
    G. M. Goluzin, Geometric Theory of Functions of a Complex Variable (GITTL, Moscow, 1952; Am. Math. Soc., Providence, R.I., 1969).zbMATHGoogle Scholar
  78. 78.
    A. Hurwitz and R. Courant, Funktionentheorie (Springer-Verlag, Berlin, 1929; Nauka, Moscow, 1968).zbMATHGoogle Scholar
  79. 79.
    W. Koppenfels and F. Stallmann, Praxis der Konformen Abbildung (Springer-Verlag, Berlin, 1959; Inostrannaya Literatura, Moscow, 1963).zbMATHGoogle Scholar
  80. 80.
    L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis (Fizmatgiz, Moscow, 1962; Wiley, New York, 1964).Google Scholar
  81. 81.
    D. Gaier, Konstructive Methoden der konformen Abbildung (Springer, Berlin, 1964).Google Scholar
  82. 82.
    L. N. Trefethen, “Numerical computation of the Schwarz-Christoffel transformation,” SIAM J. Sci. Stat. Comput. 1, 82–102 (1980).zbMATHMathSciNetGoogle Scholar
  83. 83.
    L. N. Trefethen, “Numerical construction of conformal maps,” Appendix to E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis for Mathematics, Science, and Engineering (Prentice Hall, New York, 1993).Google Scholar
  84. 84.
    S. Bezrodnykh and V. I. Vlasov, “The Riemann-Hilbert problem in a complicated domain for the model of magnetic reconnection in plasma,” Comput. Math. Math. Phys. 42(3), 263–298 (2002).MathSciNetGoogle Scholar
  85. 85.
    L. N. Trefethen and T. A. Driscoll, Schwarz-Christoffel Transformation (Cambridge Univ. Press, Cambridge, 2005).Google Scholar
  86. 86.
    S. I. Bezrodnykh and V. I. Vlasov, “The Riemann-Hilbert problem in domains of complex geometry and applications,” Spectral Evolution Probl. 16, 51–61 2006.Google Scholar
  87. 87.
    A. B. Bogatyrev, “Conformal mapping of rectangular heptagons,” Sb. Math. 203(12), 1715–1735 (2012).zbMATHMathSciNetGoogle Scholar
  88. 88.
    H. A. Schwarz, “Uber diejenjgen Fälle in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt,” Gesammelte Math. Abhandlungen Berlin 2, 211–259 (1890).Google Scholar
  89. 89.
    V. V. Golubev, Lectures on the Analytic Theory of Differential Equations (Gostekhizdat, Moscow, 1950) [in Russian].zbMATHGoogle Scholar
  90. 90.
    Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdelyi (McGraw-Hill, New York, 1953; Nauka, Moscow, 1973), Vol. 1.Google Scholar
  91. 91.
    V. I. Smirnov, Master’s Dissertation (Petrograd, 1918).Google Scholar
  92. 92.
    V. A. Fok, “Conformal map of quadrangle with zero angles in the plane,” Zh. Leningr. Fiz-Mat. Ob-va 1(2), 147–167 (1927).Google Scholar
  93. 93.
    A. V. Venkov, “On accessory coefficients of second order Fuchsian equation with real singular points,” Zap. Semin. LOMI 129, 17–29 (1983).zbMATHMathSciNetGoogle Scholar
  94. 94.
    P. G. Zograf and L. A. Takhtadzhyan, “On Liouville’s equation, accessory parameters, and the geometry of Teichmüller space for Riemann surfaces of genus 0,” Math. USSR Sb. 60(1), 143–161 (1988).zbMATHMathSciNetGoogle Scholar
  95. 95.
    E. L. Ince, Ordinary Differential Equations (Dover, New York, 1927; ONTI, Kharkov, 1939).zbMATHGoogle Scholar
  96. 96.
    L. Fuchs, “Lineare differentialgleihungen mit veranderlichen Koefficienten,” J. Reine Angew. Math. 66, 60–121 (1866).Google Scholar
  97. 97.
    E. G. C. Poole, Introduction to the Theory of Linear Differential Equations (Clarendon, Oxford, 1936).Google Scholar
  98. 98.
    A. Kratzer and W. Franz, Transzendente Funktionen (Akademische Verlagsgesellschaft, Leipzig, 1960; Inostrannaya Literatura, Moscow, 1963).zbMATHGoogle Scholar
  99. 99.
    V. I. Vlasov, Boundary Value Problems in Domains with a Curved Boundary (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1987) [in Russian].Google Scholar
  100. 100.
    V. I. Vlasov and D. B. Volkov, “Inversion problem for an equation of Fuchs class,” Differ. Uravn. 22(2), 1854–1864 (1986).MathSciNetGoogle Scholar
  101. 101.
    G. Goluzin, L. Kantorovich, V. Krylov, P. Melent’ev, M. Muratov, N. Stenin, Conformal Mappings of Simply and Multiply Connected Domains (Nauka, Leningrad, 1937) [in Russian].Google Scholar
  102. 102.
    T. Theodorsen and I. E. Garrick, “General potential theory of arbitrary wing sections,” NACA Rep., No. 452 (1933).Google Scholar
  103. 103.
    I. E. Garrick, “Conformal mapping in aerodynamics with emphasis on the method of successive conjugates,” Proceedings of Symposium on Construction and Applications of Conformal Maps (Natl. Bureau of Standards, Los Angeles, 1952), pp. 137–147.Google Scholar
  104. 104.
    A. Ostorvski, “On the convergence of Theodorsen’s and Garrick’s method of conformal mapping,” Proceedings of Symposium on Construction and Applications of Conformal Maps (Natl. Bureau of Standards, Los Angeles, 1952), pp. 149–163.Google Scholar
  105. 105.
    G. T. Symm, “An integral equation method in conformal mapping,” Numer. Math. 9, 250–258 (1966).zbMATHMathSciNetGoogle Scholar
  106. 106.
    S. Chakravarthy and D. Anderson, “Numerical conformal mapping,” Math. Comp. 33, 953–969 (1979).zbMATHMathSciNetGoogle Scholar
  107. 107.
    R. Menikoff and C. Zemach, “Methods for numerical conformal mapping,” J. Comput. Phys. 36(3), 366–410 (1980).zbMATHMathSciNetGoogle Scholar
  108. 108.
    B. Fornberg, “Numerical method for conformal mapping,” SIAM J. Sci. Stat. Comput. 1(3), 386–400 (1980).zbMATHMathSciNetGoogle Scholar
  109. 109.
    V. I. Vlasov, Doctoral Dissertation (Computing Center, USSR Acad. Sci., Moscow, 1990).Google Scholar
  110. 110.
    L. Bieberbach, “Zur Theorie und Praxis der konformen Abbildung,” Rend. Circ. Mat. Palermo 38(1), 98–112 (1914).zbMATHGoogle Scholar
  111. 111.
    G. Julia, “Sur une suite double de polynomes lièe à la reprèsentation conforme des airs planes simplement connexes,” Liouville’s J. Ser. 9 7, 381–407 (1928).zbMATHGoogle Scholar
  112. 112.
    M. V. Keldysh and M. A. Lavrntieff, “Sur la representation conforme des domains limites par les courbes rectifiable,” Ann. Sci. Ecole Norm. Super. 54(1), 1–38 (1937).zbMATHGoogle Scholar
  113. 113.
    C. Caratheodory, “Untersuchengen über die konformen Abbildungen von festen und veränderlichen Gebieten,” Math. Ann. 72, 107–144 (1912).MathSciNetGoogle Scholar
  114. 114.
    A. M. Markushevich, Theory of Functions of a Complex Variable (Prentice Hall, Englewood Cliffs, N.J., 1965; Nauka, Moscow, 1968), Vol. 2.Google Scholar
  115. 115.
    M. Shiffer, “Some new results in the theory of conformal mappings,” appendix to R. Courant, Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces (Interscience, New York, 1950; Inostrannaya Literatura, Moscow, 1953).Google Scholar
  116. 116.
    V. I. Vlasov and A. B. Pal’tsev, “An analytical-numerical method for conformal mappings of complex-shaped domains,” Dokl. Math. 80, 790–792 (2009).zbMATHMathSciNetGoogle Scholar
  117. 117.
    V. I. Vlasov, “On a method for solving mixed problems for Laplace’s equation,” Dokl. Akad. Nauk SSSR 237(5), 1012–1015 (1977).MathSciNetGoogle Scholar
  118. 118.
    V. I. Vlasov, “Variation in a mapping function under domain deformation,” Dokl. Akad. Nauk SSSR 275(6), 1299–1302 (1984).MathSciNetGoogle Scholar
  119. 119.
    S. A. Markovskii and B. V. Somov, “Some properties of magnetic reconnection in a current sheet with shock waves,” Proceedings of the 6th Annular Seminar on Problems in Solar Flare Physics (Nauka, Moscow, 1988), pp. 93–110.Google Scholar
  120. 120.
    B. V. Somov, Plasma Astrophysics, Part I: Fundamentals and Practice, Part II: Reconnection and Flares (Springer Science, New York, 2006).Google Scholar
  121. 121.
    S. I. Akasofu and S. Chapmen, Solar Terrestrial Physics (Pergamon, Oxford, 1972; Mir, Moscow, 1975), Part. 2.Google Scholar
  122. 122.
    L. M. Zelenyi, “Dynamics of plasma and magnetic fields in the Earth’s magnetotail,” in Advances in Science and Engineering: Space Research (VINITI, Moscow, 1986), Vol. 24 [in Russian].Google Scholar
  123. 123.
    B. V. Somov, Physical Processes in Solar Flares (Kluwer, Dordrecht, 1992).Google Scholar
  124. 124.
    B. V. Somov, Space Electrodynamics and Solar Physics (Mosk. Gos. Univ., Moscow, 1993) [in Russian].Google Scholar
  125. 125.
    E. Priest and T. Forbes, Magnetic Reconnection (Cambridge Univ. Press, Cambridge, 2000; Fizmatlit, Moscow, 2005).zbMATHGoogle Scholar
  126. 126.
    S. I. Syrovatskii, “Emergence of current sheets in plasma with a strong frozen magnetic field,” Zh. Eksp. Teor. Fiz. 60, 1721–1741 (1971).Google Scholar
  127. 127.
    B. V. Somov, V. I. Vlasov, and S. I. Bezrodnykh, “Mathematical aspects of the theory of reconnection in strong magnetic fields,” Bull. Russ. Acad. Sci. Phys. 70(1), 13–28 (2006).Google Scholar
  128. 128.
    S. I. Bezrodnykh, V. I. Vlasov, and B. V. Somov, “Analytical model of magnetic reconnection in the presence of shock waves attached to a current sheet,” Astron. Lett. 33(2), 130–136 (2007).Google Scholar
  129. 129.
    S. I. Bezrodnykh, V. I. Vlasov, and B. V. Somov, “Generalized analytical models of Syrovatskii’s current sheet,” Astron. Lett. 37(2), 113–130 (2011).Google Scholar
  130. 130.
    B. V. Somov, S. I. Bezrodnykh, and L. S. Ledentsov, “Overview of open issues in the physics of large solar flares,” Astron. Astrophys. Trans. 27(1), 69–81 (2011).Google Scholar
  131. 131.
    S. I. Bezrodnykh, V. I. Vlasov, and B. V. Somov, “Generalized models of Syrovatskii’s current sheet with attached MHD shock waves,” Nauchn. Vedom. Belaruss. Gos. Univ. Ser. Mat. Fiz. 25(24), 127–143 (2011).Google Scholar
  132. 132.
    S. I. Bezrodnykh, V. I. Vlasov, and B. V. Somov, “Analytical models of generalized Syrovatskii’s current layer with MHD shock waves,” Astronomic and Space Science Proceedings (Springer, Berlin, 2012), Vol. 30, pp. 133–144.Google Scholar
  133. 133.
    S. I. Bezrodnykh, “On the Riemann-Hilbert problem with growth conditions,” Spectral Evolution Probl. 15, 112–118 (2005).Google Scholar
  134. 134.
    S. I. Bezrodnykh, “Jacobi-type relation for a generalized hypergeometric function,” Abstracts of Papers of the 3rd International Conference on Chebyshev’s Mathematical Ideas and Their Applications to Modern Problems in Natural Sciences (Obninsk, 2006), pp. 18–19.Google Scholar
  135. 135.
    S. I. Bezrodnykh, Candidate’s Dissertation in Mathematics and Physics (Computing Center, Russ. Acad. Sci., Moscow, 2006).Google Scholar
  136. 136.
    S. I. Bezrodnykh and V. I. Vlasov, “Singular Riemann-Hilbert problem and its application to plasma physics,” Abstracts of Papers of the International Conference on Differential Equations and Related Topics, Moscow, May 21–26, 2007 (Moscow, 2007), p. 36.Google Scholar
  137. 137.
    S. I. Bezrodnykh and V. I. Vlasov, “Singular Riemann-Hilbert problem in domains of complex geometry and its applications,” Abstracts of Papers of the International Conference on Differential Equations, Function Theory, and Applications, Novosibirsk, May 28–June 2, 2007 (Novosibirsk, 2007), pp. 420–421.Google Scholar
  138. 138.
    S. I. Bezrodnykh and V. I. Vlasov, “A computational problem of two-dimensional harmonic mappings,” Nauchn. Vedom. Belaruss. Gos. Univ. Ser. Mat. Fiz. 70(15), 31–45 (2009).Google Scholar
  139. 139.
    S. Bezrodnykh and V. I. Vlasov, “On a problem in the constructive theory of harmonic mappings,” Sovrem. Mat. Fundam. Napravl. 46, 5–30 (2012).Google Scholar
  140. 140.
    S. I. Bezrodnykh and V. I. Vlasov, “On a certain problem in constructive theory of harmonic mappings,” J. Math. Sci. 201(6), 705–732 (2014).Google Scholar
  141. 141.
    P. J. Roache and S. Steinberg, “A new approach to grid generation using a variational formulation,” Proceedings of AIAA 7th CFD Conference (Cincinnati, 1985), pp. 360–370.Google Scholar
  142. 142.
    P. Knupp and R. Luczak, “Truncation error in grid generation: A case study,” Numer. Methods Partial Differ. Equations 11, 561–571 (1995).zbMATHMathSciNetGoogle Scholar
  143. 143.
    S. A. Ivanenko, “Control of cell shapes in the course of grid generation,” Comput. Math. Math. Phys. 40(11), 1596–1616 (2000).zbMATHMathSciNetGoogle Scholar
  144. 144.
    B. N. Azarenok, “Generation of structured difference grids in two-dimensional nonconvex domains using mappings,” Comput. Math. Math. Phys. 49(5), 797–809 (2009).MathSciNetGoogle Scholar
  145. 145.
    S. I. Bezrodnykh and B. V. Somov, “Analytical solution to the problem of interaction between a shock wave and a neutron star’s magnetosphere,” Dokl. Phys. 59(8), 355–359 (2014).Google Scholar
  146. 146.
    M. A. Evgrafov, Asymptotic Estimates and Entire Functions (Nauka, Moscow, 1979) [in Russian].zbMATHGoogle Scholar
  147. 147.
    G. V. Kolosov, Application of Complex Variables in Elasticity (ONTI, Moscow, 1935) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Dorodnicyn Computing CenterRussian Academy of SciencesMoscowRussia
  2. 2.Sternberg Astronomical InstituteMoscow State UniversityMoscowRussia

Personalised recommendations