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Regular, Singular and Hypersingular Integrals over Fractal Contours

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ilya Boykov & Vladimir Roudnev & Alla Boykova, 2023. "Regular, Singular and Hypersingular Integrals over Fractal Contours," Mathematics, MDPI, vol. 11(23), pages 1-20, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:23:p:4752-:d:1287038
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