Regular, Singular and Hypersingular Integrals over Fractal Contours

Ilya Boykov (), Vladimir Roudnev and Alla Boykova
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Ilya Boykov: Department of Higher and Applied Mathematics, Penza State University, Krasnaya 40, 440026 Penza, Russia
Vladimir Roudnev: Department of Computational Physics, Saint Petersburg State University, 1 Ulyanovskaya Str., 198504 Saint Petersburg, Russia
Alla Boykova: Department of Computational Physics, Saint Petersburg State University, 1 Ulyanovskaya Str., 198504 Saint Petersburg, Russia

Mathematics, 2023, vol. 11, issue 23, 1-20

Abstract: The paper is devoted to the approximate calculation of Riemann definite integrals, singular and hypersingular integrals over closed and open non-rectifiable curves and fractals. The conditions of existence for the Riemann definite integrals over non-rectifiable curves and fractals are provided. We give a definition of a singular integral over non-rectifiable curves and fractals which generalizes the known one. We define hypersingular integrals over non-rectifiable curves and fractals. We construct quadratures for the calculation of Riemann definite integrals, singular and hypersingular integrals over non-rectifiable curves and fractals and the corresponding error estimates for various classes of functions. Singular and hypersingular integrals are defined up to an additive constant (or a combination of constants) that are subject to a convention that depends on the actual problem being solved. We illustrate our theoretical results with numerical examples for Riemann definite integrals, singular integrals and hypersingular integrals over fractals.

Keywords: singular integrals; hypersingular integrals; closed and open non-rectifiable curves; fractals; quadrature formulas (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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