Abstract
Mixed boundary value problems for the two-dimensional Laplace’s equation in a domain D are reduced to the Riemann-Hilbert problem Re \(\overline {\lambda (t)}\psi (t) = 0\), t ∈ ∂D, with a given Hölder continuous function λ(t) on ∂D except at a finite number of points where a one-sided discontinuity is admitted. The celebrated Keldysh-Sedov formulae were used to solve such a problem for a simply connected domain. In this paper, a method of functional equations is developed to mixed problems for multiply connected domains. For definiteness, we discuss a problem having applications in composites with a discontinuous coefficient λ(t) on one of the boundary components. It is assumed that the domain D is a canonical domain, the lower half-plane with circular holes. A constructive iterative algorithm to obtain an approximate solution in analytical form is developed in the form of an expansion in the radius of the holes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
I.A. Aleksandrov, A.S. Sorokin, The problem of Schwarz for multiply connected domains. Sib. Math. Zh. 13, 971–1001 (1972)
R. Balu, T.K. DeLillo, Numerical methods for Riemann-Hilbert problems in multiply connected circle domains. J. Comput. Appl. Math. 307, 248–261 (2016)
T.K. DeLillo, A.R. Elcrat, E.H. Kropf, J.A. Pfaltzgraff, Efficient calculation of Schwarz-Christoffel transformations for multiply connected domains using Laurent series. Comput. Methods Funct. Theory 13, 307–336 (2013)
F.D. Gakhov, Boundary Value Problems, 3rd edn. (Nauka, Moscow, 1977, in Russian); Engl. transl. of 1st edn. (Pergamon Press, Oxford, 1966)
M.V. Keldysh, L.I. Sedov, Effective solution to some boundary value problems for harmonic functions. Dokl. Akad. Nauk SSSR 16, 7–10 (1937)
V. Mityushev, Solution of the Hilbert boundary value problem for a multiply connected domain. Slupskie Prace Mat-Przyr. 9a, 33–67 (1994)
V. Mityushev, Hilbert boundary value problem for multiply connected domains. Complex Var. 35, 283–295 (1998)
V. Mityushev, Riemann-Hilbert problems for multiply connected domains and circular slit maps. Comput. Methods Funct. Theory 11, 575–590 (2011)
V. Mityushev, Scalar Riemann-Hilbert problem for multiply connected domains, in Functional Equations in Mathematical Analysis, ed. by Th.M. Rassias, J. Brzdȩk. Springer Optimization and its Applications, vol. 52. (Springer Science+Business Media, LLC, New York, 2012), pp. 599–632. https://doi.org/10.1007/978-1-4614-0055-438
V. Mityushev, \(\mathbb R\) -Linear and Riemann-Hilbert Problems for Multiply Connected Domains, ed. by S.V. Rogosin, A.A. Koroleva. Advances in Applied Analysis (Birkhäuser, Basel, 2012), pp. 147–176
V. Mityushev, Poincare α-Series for Classical Schottky Groups and its Applications, ed. by G.V. Milovanović, M.Th. Rassias. Analytic Number Theory, Approximation Theory, and Special Functions (Springer, Berlin, 2014), pp. 827–852
V.V. Mityushev, S.V. Rogosin, Constructive Methods to Linear and Non-linear Boundary Value Problems of the Analytic Function. Theory and Applications. Monographs and Surveys in Pure and Applied Mathematics (Chapman & Hall/CRC, Boca Raton, 2000)
N. Rylko, Fractal local fields in random composites. Appl. Math. Comput. 69, 247–254 (2015)
N. Rylko, Edge effects for heat flux in fibrous composites. Appl. Math. Comput. 70, 2283–2291 (2015)
A.S. Sorokin, The homogeneous Keldysh-Sedov problem for circular multiply connected circular domains in Muskhelishvili’s class h 0. Differ. Uravn. 25, 283–293 (1989)
A.S. Sorokin, The Keldysh-Sedov problem for multiply connected circular domains. Sibirsk. Mat. Zh. 36, 186–202 (1995)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Mityushev, V. (2018). Mixed Problem for Laplace’s Equation in an Arbitrary Circular Multiply Connected Domain. In: Drygaś, P., Rogosin, S. (eds) Modern Problems in Applied Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-72640-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-72640-3_10
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-72639-7
Online ISBN: 978-3-319-72640-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)