Wang, B.; Xiao, W. The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral. Symmetry2023, 15, 2158.
Wang, B.; Xiao, W. The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral. Symmetry 2023, 15, 2158.
Wang, B.; Xiao, W. The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral. Symmetry2023, 15, 2158.
Wang, B.; Xiao, W. The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral. Symmetry 2023, 15, 2158.
Abstract
This article is a study on the (k,s)-Riemann-Liouville fractional integral, a generalization of the Riemann-Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It presents that the upper Box dimension of the (k,s)-Riemann-Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval I=[0,1], which indicates that the upper Box dimension of the integrand f(x) will not be increased by the σ-order (k,s)-Riemann-Liouville fractional integral ksD−σf(x) where σ>0 on I. Additionally, we prove that the fractal dimension of ksD−σf(x) of one-dimensional continuous functions f(x) is still one.
Keywords
the fractal dimension; continuous functions; the (k,s)-Riemann-Liouville fractional integral
Subject
Computer Science and Mathematics, Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
