A NEW ESTIMATION OF BOX DIMENSION OF RIEMANN–LIOUVILLE FRACTIONAL CALCULUS OF CONTINUOUS FUNCTIONS

Jun-Ru Wu, Zhe Ji and Kai-Chao Zhang
Additional contact information
Jun-Ru Wu: Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, P. R. China
Zhe Ji: ��College of Science, China University of Petroleum, Qingdao 266580, P. R. China‡School of the Gifted Young, University of Science and Technology of China, Hefei 230026, P. R. China
Kai-Chao Zhang: Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, P. R. China

FRACTALS (fractals), 2024, vol. 32, issue 02, 1-13

Abstract: This paper establishes a linear relationship between the order of the Riemann–Liouville fractional calculus and the exponent of the Hölder condition, whether the Hölder condition is global, local, or at a single point. We propose and prove a control inequality between the Hölder derivative (Hf(x,α) as defined in Proposition 12) of a continuous function and the Hölder derivative of the Riemann–Liouville fractional calculus of this function. In addition, this paper provides a more accurate estimation of the Box dimension of the graph of the Riemann–Liouville fractional integral of an arbitrary continuous function. More specifically, it establishes the result that whenever there is a continuous function whose graph has the upper Box dimension s with 1 < s ≤ 2, the graph of its Riemann–Liouville fractional integral of order ν, with 0 < ν < 1, has the upper Box dimension not greater than s − (s − 1)ν.

Keywords: Riemann–Liouville Fractional Calculus; Box Dimension; Fractal Function; Hölder Condition (search for similar items in EconPapers)
Date: 2024
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.worldscientific.com/doi/abs/10.1142/S0218348X2440005X
Access to full text is restricted to subscribers

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:fracta:v:32:y:2024:i:02:n:s0218348x2440005x

Ordering information: This journal article can be ordered from

DOI: 10.1142/S0218348X2440005X

Access Statistics for this article

FRACTALS (fractals) is currently edited by Tara Taylor

More articles in FRACTALS (fractals) from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().