Abstract
In this paper, we consider problems of discrete approximation of special integral operators with the Calderon–Zygmund kernel. We introduce discrete spaces and bounded discrete operators acting in these spaces; then we use these operators for the search for approximate solutions of the corresponding equations. We state theorems on the solvability of equations with discrete operators, compare integral operators with their discrete analogs, and obtain estimates of errors of approximate solutions.
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S. K. Abdullaev, “Multidimensional singular integral equations in weighted Hölder spaces,” Dokl. Akad. Nauk SSSR, 292, No. 4, 777–779 (1987).
S. M. Belotserkovsky and I. K. Lifanov, Numerical Methods in Singular Integral Equations and Their Application in Aerodynamics, Elasticity Theory, and Electrodynamics [in Russian], Nauka, Moscow (1985).
V. Didenko and B. Silbermann, Approximation of Additive Convolution-Like Operators. Real C∗-Algebra Approach, Birkhäuser, Basel (2008).
B. G. Gabdulkhaev, Numerical Analysis of Singular Integral Equations [in Russian], Izd. Kazan. Univ., Kazan (1995).
I. K. Lifanov, Method of Singular Integral Equations and Numerical Experiment [in Russian], Yanus, Moscow (1995).
S. G. Mikhlin and S. Prössdorf, Singular Integral Operators, Springer-Verlag, Berlin (1986).
G. Vainikko, Multidimensional Weakly Singular Integral Equations, Springer-Verlag, Berlin–Heidelberg (1993).
A. V. Vasilyev and V. B. Vasilyev, “Numerical analysis for some singular integral equations,” Neural Parallel Sci. Comput., 20, No. 3-4, 313–326 (2012).
A. V. Vasilyev and V. B. Vasilyev, “On the error estimate for calculating some singular integrals,” Proc. Appl. Math. Mech., 12, 665–666 (2012).
A. V. Vasilyev and V. B. Vasilyev, “Discrete singular operators and equations in a half-space,” Azerb. J. Math., 3, No. 1, 84–93 (2013).
A. V. Vasilyev and V. B. Vasilyev, “Approximation rate and invertibility for some singular integral operators,” Proc. Appl. Math. Mech., 13, No. 1, 373–374 (2013).
A. V. Vasilyev and V. B. Vasilyev, “Approximate solutions of multidimensional singular integral equations and fast algorithms for finding them,” Vladikavkaz. Mat. Zh., 16, No. 1, 3–11 (2014).
A. V. Vasilyev and V. B. Vasilyev, “Some common properties of certain continual and discrete convolutions,” Proc. Appl. Math. Mech., 14, 845–846 (2014).
A. V. Vasilyev and V. B. Vasilyev, “The periodic Riemann problem and discrete convolution equations,” Differ. Uravn., 51, No. 5, 642–649 (2015).
A. V. Vasilyev and V. B. Vasilyev, “Discrete singular integrals in a half-space,” in: Current Trends in Analysis and Their Applications. Research Perspectives, Birkhäuser, Basel (2015), pp. 663–670.
A. V. Vasilyev and V. B. Vasilyev, “On the solvability of some discrete equations and related estimates of discrete operators,” Dokl. Ross. Akad. Nauk, 464, No. 6, 651–655 (2015).
A. V. Vasilyev and V. B. Vasilyev, “On finite discrete operators and equations,” Proc. Appl. Math. Mech., 16, No. 1, 771–772 (2016).
A. V. Vasilyev and V. B. Vasilyev, “Discrete approximations for multidimensional singular integral operators,” Lect. Notes Comput. Sci., 10187, 706–712 (2017).
A. V. Vasilyev and V. B. Vasilyev, “Two-scale estimates for special finite discrete operators,” Math. Model. Anal., 22, No. 3, 300–310 (2017).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 160, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences ICMMAS’17, Saint Petersburg, July 24–28, 2017, 2019.
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Vasilyev, A.V. On Approximate Solution of Certain Equations. J Math Sci 257, 8–16 (2021). https://doi.org/10.1007/s10958-021-05464-6
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DOI: https://doi.org/10.1007/s10958-021-05464-6