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

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.

DOI: https://doi.org/10.7868/S0044466914120096

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

Bibliographic databases:

UDC: 519.642

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
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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
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
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
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
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