Approximate solution of a dominant singular integral equation with conjugation
DOI:
https://doi.org/10.37863/umzh.v73i9.758Keywords:
singular integral equations Cauchy kernel successive approximationAbstract
UDC 517.5
In the present paper, the method of successive approximations and Faber polynomials are used to derive an approximate solution of a dominant singular integral equation with Holder continuous coefficients and conjugation on the Lyapunov curve.
Moreover, conditions of convergence in the L2 and H(α) spaces are presented.
References
I. Caraus, The numerical solution for system of singular integro-differential equations by Faber – Laurent polynomials, Numer. Anal. and Appl., 3401, ser. Lect. Notes Comput. Sci., 219 – 223 (2005). DOI: https://doi.org/10.1007/978-3-540-31852-1_25
V. D. Didenko, B. Silbermann, S. Roch, Some peculiarities of approximation methods for singular integral equations with conjugation, Methods and Appl. Anal., 7, № 4, 663 – 686 (2000), https://doi.org/10.4310/MAA.2000.v7.n4.a4 DOI: https://doi.org/10.4310/MAA.2000.v7.n4.a4
R. Duduchava, On general singular integral operators of the plane theory of elasticity, Rend. Sem. Mat. Univ. Politec. Torino, 42, № 3, 15 – 41 (1984).
R. Duduchava, General singular integral equations and basic problems of planar elasticity theory, Trudy Tbiliss. Mat. Inst. AN GSSR, 82, 45 – 89 (1986).
S. W. Ellacott, A survey of Faber methods in numerical approximation, Comput. and Math. Appl., 12, №5, 1103 – 1107 (1986). DOI: https://doi.org/10.1016/0898-1221(86)90234-8
F. D. Gakhov, On the Riemann boundary value problem, Mat. Sb., 44, № 4, 673 – 683 (1937).
F. D. Gakhov, Boundary value problems, Dover Publ., Mineloa (1990).
A. I. Kalandiya, Mathematical methods of two-dimensional elasticity, Mir, Moscow (1975).
E. Ladopoulos, G. Tsamasphyros, Approximations of singular integral equations on Lyapunov contours in Banach spaces, Comput. and Math. Appl., 50, № 3-4, 567 – 573 (2005), https://doi.org/10.1016/j.camwa.2005.01.025 DOI: https://doi.org/10.1016/j.camwa.2005.01.025
G. S. Litvinchiuk, The boundary problems and singular equations with displacement, Nauka, Moscow (1997).
N. I. Muskhelishvili, Singular integral equations: boundary problems of function theory and their application to mathematical physics, Dover Publ., Mineloa (2008).
N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Springer Sci. & Business Media (2013).
M. A. Sheshko, P. Karczmarek, D. Pylak, P. W´ojcik, Application of Faber polynomials to the approximate solution of a generalized boundary value problem of linear conjugation in the theory of analytic functions, Comput. Math. Appl., 67, № 8, 1474 – 1481 (2014), https://doi.org/10.1016/j.camwa.2014.02.012 DOI: https://doi.org/10.1016/j.camwa.2014.02.012
I. Spinei, V. Zolotarevschi, Direct methods for solving singular integral equations with complex conjugation, Comput. Sci., 6, № 1, 83 – 91 (1998).
P. K. Suetin, Series of Faber polynomials, Gordon and Breach Sci. Publ., New York (1998).
I. N. Vekua, On singular linear integral equations, Dokl. Akad. Nauk SSSR, 26, № 8, 335 – 338 (1940).
I. N. Vekua, Generalized analytic functions, Pergamon Press, Oxford (1962).
N. Vekua, Systems of singular integral equations and certain boundary value problems, Nauka, Moscow (1970).
P. W´ojcik, M. A. Sheshko, S. M. Sheshko, Application of Faber polynomials to the approximate solution of singular integral equations with the Cauchy kernel, Different. Equat., 49, № 2, 198 – 209 (2013), https://doi.org/10.1134/S0012266113020067 DOI: https://doi.org/10.1134/S0012266113020067
V. Zolotarevskii, Z. Li, I. Caraus, Approximate solution of singular integro-differential equations by reduction over Faber – Laurent polynomials, Different. Equat., 40, № 12, 1764 – 1769 (2004). DOI: https://doi.org/10.1007/s10625-005-0108-3
