Relationship of Upper Box Dimension Between Continuous Fractal Functions and Their Riemann–liouville Fractional Integral

Abstract

This paper considers the relationship of box dimension between a continuous fractal function and its Riemann–Liouville fractional integral. For an arbitrary fractal function [Formula: see text] it is proved that the upper box dimension of the graph of Riemann–Liouville fractional integral [Formula: see text] does not exceed the upper box dimension of [Formula: see text], i.e. [Formula: see text]. This estimate shows that [Formula: see text]order Riemann–Liouville fractional integral [Formula: see text] does not increase the fractal dimension of the integrand [Formula: see text], which means that Riemann–Liouville fractional integration does not decrease the smoothness at least that is obvious known result for classic integration. Our result partly answers fractal calculus conjecture in [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217–229] and [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of one-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438].

A Technique for Estimation of Box Dimension about Fractional Integral

“…However, it is generally believed that (1.3) holds although no theory proof appears. Xiao [1] proved that (1.4) is true for all fractal continuous functions, which is the first discussion the conjecture 1.1 for an arbitrary fractal functions. This paper tries to improve Rfs.…”

“…

This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function

…”

Partially Explore the Differences and Similarities between Riemann-Liouville Integral and Mellin Transform

“…Both the two previous transformations can be regarded as integral transformations. Compared with M(x), the smoothness of its Riemann-Liouville fractional integral can only be improved [4], while its Mellin transformed function is differentiable.…”

“…Through further research, researchers can discuss the relationship between fractal dimension of fractal or fractal functions and any order of fractional integrals. Xiao [4] has concluded upper box dimension of continuous fractal functions is equal to or more than their Riemann-Liouville integral, and Gao [5] has also concluded upper box dimension of continuous fractal functions is equal to or more than their Weyl integral. Both of them partly answer Fractal Calculus Conjecture [4].…”

“…Xiao [4] has concluded upper box dimension of continuous fractal functions is equal to or more than their Riemann-Liouville integral, and Gao [5] has also concluded upper box dimension of continuous fractal functions is equal to or more than their Weyl integral. Both of them partly answer Fractal Calculus Conjecture [4]. In addition, Noreen Azhar [6] recently gives solution of fuzzy fractional order differential equations by fractional Mellin transform method.…”