Elements of fractional calculus. Fractional integrals

Authors

  • Yuliya S. Mishura Taras Shevchenko National University of Kyiv https://orcid.org/0000-0002-6877-1800
  • Olha M. Hopkalo Taras Shevchenko National University of Kyiv
  • Hanna S. Zhelezniak Taras Shevchenko National University of Kyiv

DOI:

https://doi.org/10.17721/1812-5409.2022/1.1

Keywords:

fractional integral, Hölder property, continuity according to the fractional index

Abstract

The paper is devoted to the basic properties of fractional integrals. It is a survey of the well-known properties of fractional integrals, however, the authors tried to present the known information about fractional integrals as short and transparently as possible. We introduce fractional integrals on the compact interval and on the semi-axes, consider the famous Hardy-Littlewood theorem and other properties of integrability of fractional integrals. Among other basic properties, we consider Holder continuity and establish to what extent fractional integration increases the smoothness of the integrand. Also, we establish continuity of fractional integrals according to the index of fractional integration, both at strictly positive value and at zero. Then we consider properties of restrictions of fractional integrals from semi-axes on the compact interval. Generalized Minkowsky inequality is applied as one of the important tools. Some examples of calculating fractional integrals are provided.

Pages of the article in the issue: 11 - 19

Language of the article: Ukrainian

References

BAUDOIN, F., NUALART, D. (2003) Equivalence of Volterra processes // Stochastic Processes and their Applications. Vol. 107, No. 2, p. 327–350.

MISHURA, YU., SHEVCHENKO, G., SHKLYAR, S. Gaussian processes with Volterra kernels. URL: https://arxiv.org/pdf/2001.03405.pdfmath.

MISHURA, YU., SHKLYAR, S. (2022) Gaussian Volterra processes with power-type kernels. // Mod. Stoch. Theory Appl.

SAMKO, S. G., KILBAS, A. A., MARICHEV, O. I. (1993) Fractional integrals and derivatives (Vol. 1). Yverdon-les-Bains, Switzerland: Gordon and Breach science.

SOTTINEN, T., VIITASAARI, L. (2016) Stochastic Analysis of Gaussian Processes via Fredholm Representation // International Journal of Stochastic Analysis. DOI: 10.1155/2016/8694365, URL: http://dx.doi.org/10.1155/2016/8694365.

STEIN, E. M., WEISS, G. (2016) Introduction to Fourier Analysis on Euclidean Spaces (PMS-32), Volume 32. Princeton University press.

DAS S. (2011) Functional Fractional Calculus. – Springer-Verlag Berlin Heidelberg, Springer, Berlin, Heidelberg, 2011. URL: https://doi.org/10.1007/978-3-642-20545-3

BUTZER, P.L., WESTPHAL, U. (2000) An introduction to fractional calculus // Applications of Fractional Calculus in Physics, pp. 1-85. URL: https://doi.org/10.1142/9789812817747.0001

ANASTASSIOUS, G. A.(2021) Constructive Fractional Analysis with Applications. Springer Nature.

ANASTASSIOUS, G. A.(2021) Generalized Fractional Calculus: New Advancements and Applications. Springer.

Downloads

Published

2022-04-26

How to Cite

Mishura, Y. S., Hopkalo, O. M., & Zhelezniak, H. S. (2022). Elements of fractional calculus. Fractional integrals. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (1), 11–19. https://doi.org/10.17721/1812-5409.2022/1.1

Issue

Section

Algebra, Geometry and Probability Theory