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Numerical Approximations of the Riemann–Liouville and Riesz Fractional Integrals
Volume 35, Issue 1 (2024), pp. 21–46
Pub. online: 22 November 2023      Type: Research Article      Open accessOpen Access
Received
1 May 2023
Accepted
1 November 2023
Published
22 November 2023

Abstract

In this paper, the numerical algorithms for calculating the values of the left- and right-sided Riemann–Liouville fractional integrals and the Riesz fractional integral using spline interpolation techniques are derived. The linear, quadratic and three variants of cubic splines are taken into account. The estimation of errors using analytical methods are derived. We show four examples of numerical evaluation of the mentioned fractional integrals and determine the experimental rate of convergence for each derived algorithm. The high-precision calculations are executed using the 128-bit floating-point numbers and arithmetic routines.

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Biographies

Ciesielski Mariusz
https://orcid.org/0000-0002-3695-9569
mariusz.ciesielski@icis.pcz.pl

M. Ciesielski received his MSc degree in computer science and PhD degree in mechanical engineering from the Czestochowa University of Technology, in 2000 and 2005, respectively. He has been working in the Faculty of Mechanical Engineering and Computer Science there since 2000. Currently, he is an assistant professor in the Department of Computer Science. He is an author and a co-author of over 100 scientific papers, 42 of them were published in Web of Science journals. His research interests include the numerical modelling of heat transfer processes, computational methods of mechanics, numerical algorithms, fractional calculus and its applications.

Grodzki Grzegorz
https://orcid.org/0000-0003-1206-6923
grzegorz.grodzki@icis.pcz.pl

G. Grodzki received his MSc and PhD degree in mechanic science from the Czestochowa University of Technology, Poland, in 1992 and 2000, respectively. At this time, he was conducting research measurements with use LDA (Laser Doppler Anemometry), and digital analysis and processing of randomly sampled signals. Since 2000, he worked at the Institute of Mathematics and Computer Science. Currently, he is an assistant professor at the Department of Computer Science, Czestochowa University of Technology, Poland. He is an author of over 30 papers in refereed journal and conference papers. He is also the co-author of 2 scientific monographs. His current interests include security informatics systems and computer network, optimization of organization strategy.


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Keywords
fractional calculus numerical integration numerical algorithms spline interpolation

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