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Zh. Vychisl. Mat. Mat. Fiz.:

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Zh. Vychisl. Mat. Mat. Fiz., 2014, Volume 54, Number 12, Pages 1904–1953 (Mi zvmmf10124)  

This article is cited in 6 scientific papers (total in 6 papers)

Singular Riemann–Hilbert problem in complex-shaped domains

S. I. Bezrodnykhab, V. I. Vlasova

a Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia
b Sternberg Astronomical Institute, Moscow State University, Universitetskii pr. 13, Moscow, 119992, Russia

Abstract: In simply connected complex-shaped domains $\mathcal{B}$ 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 $\mathcal{F}(z)$ is represented as the composition of a conformal mapping of $\mathcal{B}$ onto the half-plane $\mathbb{H}^+$ and the solution $\mathcal{P}^+$ of the corresponding Riemann–Hilbert problem in $\mathbb{H}^+$. Methods for finding this mapping are described, and a technique for constructing an analytic function $\mathcal{P}^+$ 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 $\mathcal{P}^+$ 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 Cauchy-type integrals.

Key words: Riemann–Hilbert problem, Cauchy-type integral, conformal mappings, Schwarz–Christoffel integral, hypergeometric functions.


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English version:
Computational Mathematics and Mathematical Physics, 2014, 54:12, 1826–1875

Bibliographic databases:

UDC: 519.642
Received: 10.06.2014

Citation: S. I. Bezrodnykh, V. I. Vlasov, “Singular Riemann–Hilbert problem in complex-shaped domains”, Zh. Vychisl. Mat. Mat. Fiz., 54:12 (2014), 1904–1953; Comput. Math. Math. Phys., 54:12 (2014), 1826–1875

Citation in format AMSBIB
\by S.~I.~Bezrodnykh, V.~I.~Vlasov
\paper Singular Riemann--Hilbert problem in complex-shaped domains
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2014
\vol 54
\issue 12
\pages 1904--1953
\jour Comput. Math. Math. Phys.
\yr 2014
\vol 54
\issue 12
\pages 1826--1875

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    This publication is cited in the following articles:
    1. S. I. Bezrodnykh, “Jacobi-Type Differential Relations for the Lauricella Function $F_D^{(N)}$”, Math. Notes, 99:6 (2016), 821–833  mathnet  crossref  crossref  mathscinet  isi  elib
    2. N. N. Nakipov, S. R. Nasyrov, “Parametricheskii metod nakhozhdeniya aktsessornykh parametrov v obobschennykh integralakh Kristoffelya–Shvartsa”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 158, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2016, 202–220  mathnet  mathscinet  elib
    3. S. I. Bezrodnykh, “Finding the Coefficients in the New Representation of the Solution of the Riemann–Hilbert Problem Using the Lauricella Function”, Math. Notes, 101:5 (2017), 759–777  mathnet  crossref  crossref  mathscinet  isi  elib
    4. S. I. Bezrodnykh, “Analytic continuation of the Appell function $F_1$ and integration of the associated system of equations in the logarithmic case”, Comput. Math. Math. Phys., 57:4 (2017), 559–589  mathnet  crossref  crossref  mathscinet  isi  elib
    5. S. I. Bezrodnykh, “The Lauricella hypergeometric function $F_D^{(N)}$, the Riemann–Hilbert problem, and some applications”, Russian Math. Surveys, 73:6 (2018), 941–1031  mathnet  crossref  crossref  adsnasa  isi  elib
    6. Bezrodnykh S., Bogatyrev A., Goreinov S., Grigor'ev O., Hakula H., Vuorinen M., “On Capacity Computation For Symmetric Polygonal Condensers”, J. Comput. Appl. Math., 361 (2019), 271–282  crossref  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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