Geodesic
Mathematical Modelling in Piecewise-Uniform Invironment Based on the Solution of the Markushevich Boundary Problem in the Class of Automorphic Functions
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 2, pp. 49-61 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

An algorithm for the explicit solution of the Markushevich boundary value problem in the class of automorphic functions with respect of Fuchsian group $\Gamma$ of the second kind is suggested. The boundary condition of the problem is given on the main circle. The coefficients of the tasks are Holder functions. The alqorithm is based on a reduction of the problem to the Hilbert boundary problem. The solution is found in a closed form under additional restriction on the coefficient $b(t)$ of the problem: if $\chi_{+}(t), \chi_{-}(t)$ are factorization multipliers of coefficient $a(t)$, the product of the function $b(t)$ on the quotient of $\overline{\chi_{+}(t)}$ and $\chi_{+}(t)$ is analytic in the domain $D_{-}$ and automorphic with respect to $\Gamma$ in this the domain.
Keywords: boundary problems for analytic functions, the Markushevich boundary problem, automorphic functions. 
@article{VYURU_2013_6_2_a3,
     author = {A. A. Patrushev},
     title = {Mathematical {Modelling} in {Piecewise-Uniform} {Invironment} {Based} on the {Solution} of the {Markushevich} {Boundary} {Problem} in the {Class} of {Automorphic} {Functions}},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {49--61},
     year = {2013},
     volume = {6},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2013_6_2_a3/}
}
TY  - JOUR
AU  - A. A. Patrushev
TI  - Mathematical Modelling in Piecewise-Uniform Invironment Based on the Solution of the Markushevich Boundary Problem in the Class of Automorphic Functions
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2013
SP  - 49
EP  - 61
VL  - 6
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VYURU_2013_6_2_a3/
LA  - ru
ID  - VYURU_2013_6_2_a3
ER  - 
%0 Journal Article
%A A. A. Patrushev
%T Mathematical Modelling in Piecewise-Uniform Invironment Based on the Solution of the Markushevich Boundary Problem in the Class of Automorphic Functions
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2013
%P 49-61
%V 6
%N 2
%U http://geodesic.mathdoc.fr/item/VYURU_2013_6_2_a3/
%G ru
%F VYURU_2013_6_2_a3
A. A. Patrushev. Mathematical Modelling in Piecewise-Uniform Invironment Based on the Solution of the Markushevich Boundary Problem in the Class of Automorphic Functions. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 2, pp. 49-61. http://geodesic.mathdoc.fr/item/VYURU_2013_6_2_a3/

[1] Vekua I. N., Generalized Analytic Functions, Fizmatgiz, M., 1959, 560 pp. | MR | Zbl

[2] Ford R., Automorphic Functions, ONTI, M., 1936, 340 pp.

[3] Patrushev A. A., Adukov V. M., “Algorithm of Exact Solution of Generalized Four-Element Riemann–Hilbert Boundary Problem with Rational Coefficiets and Its Programm Realization”, Bulletin of the South Ural State University. Series «Mathematical Modelling, Programming Computer Software». Issue 6, 2010, no. 35 (211), 4–12

[4] Patrushev A. A., “The Generalized Hilbert–Riemann Boundery Problem on the Singular Circle”, News of Smolensk State University, 2010, no. 4, 82–97

[5] Patrushev A. A., Adukov V. M., “On Explicit and Exact Solution of the Markushevich Boundary Problem for Circle”, Proceedings of Saratov State University. Series«Mathematics. Mechanics. Informatics», 11:2 (2011), 9–20

[6] Patrushev A. A., “The Markushevich problem in the class of automorphic functions for arbitrary circle”, Bulletin of the South Ural State University. Series «Mathematics. Physics. Chemistry». Issue 4, 2011, no. 10(227), 29–37 | Zbl

[7] Silvestrov V. V., Chibrikova L. I., “The Issue of Efficiency of Solving the Boundary Value the Tasks of the Riemann for Automorphic Functions”, Russian Mathematics (Izvestiya VUZ. Matematika), 1978, no. 12, 117–121 | MR | Zbl

[8] Zverovich E. I., “Boundary Value Problems in the Theory of Analytic Functions in Holder Classes on Riemann Surfaces”, Russian Mathematical Surveys, 26:1 (1971), 117–192 | DOI | MR | Zbl

[9] Chibrikova L. I., “Boundary Value Problem of Riemann for Automorphic Functions in the the Case of Groups with Two Invariants”, Russian Mathematics (Izvestiya VUZ. Matematika), 1961, no. 6, 121–131

[10] Silvestrov V. V., “Boundary Value Problem of Hilbert's for One Infinite the Field in the Class of Automorphic Functions”, Proceedings of the Seminar on Boundary Value Problems, Publisher Kazan Gos. Univ., Kazan', 1980, 180–194 | Zbl

[11] Gakhov F. D., “Degenerate Cases of Singular Integral Equations with Cauchy's Kernel”, Differential Equations, 2:2 (1966), 533–544