Analysis of Fractal Dimension of Mixed Riemann-Liouville Fractional Integral

TL;DR

This paper studies the fractal dimensions of graphs of mixed Riemann-Liouville fractional integrals for various continuous functions, establishing bounds and exact dimensions for different classes.

Contribution

It provides new bounds and exact values for the fractal dimensions of these fractional integrals' graphs across multiple function classes.

Findings

Box and Hausdorff dimensions are two for continuous functions.

Dimensions are also two for unbounded variational continuous functions.

Bounds for the dimensions are estimated for Holder continuous functions.

Abstract

In this article, we investigate the fractal dimension of the graph of the mixed Riemann-Liouville fractional integral for various choice of continuous functions on a rectangular region. We estimate bounds for the box dimension and the Hausdorff dimension of the graph of the mixed Riemann-Liouville fractional integral of the functions which belong to the class of continuous functions and the class of Holder continuous functions. We also show that the box dimension of the graph of the mixed Riemann-Liouville fractional integral of two-dimensional continuous functions is also two. Furthermore, we give construction of unbounded variational continuous functions. Later, we prove that the box dimension and the Hausdorff dimension of the graph of the mixed Riemann-Liouville fractional integral of unbounded variational continuous functions are also two.