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Keywords = closed and open non-rectifiable curves

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20 pages, 667 KB  
Article
Regular, Singular and Hypersingular Integrals over Fractal Contours
by Ilya Boykov, Vladimir Roudnev and Alla Boykova
Mathematics 2023, 11(23), 4752; https://doi.org/10.3390/math11234752 - 24 Nov 2023
Viewed by 1838
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Convolution Equations: Theory, Numerical Methods and Applications)
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