Katugampola fractional integral and fractal dimension of bivariate functions

TL;DR

This paper extends the Katugampola fractional integral to bivariate functions, analyzing its properties and estimating the fractal dimension of the resulting function graphs, with implications for fractional calculus and fractal analysis.

Contribution

It introduces the mixed Katugampola fractional integral for bivariate functions and studies its properties and fractal dimension, extending previous univariate fractional calculus work.

Findings

Preserves boundedness, continuity, and bounded variation.

Estimates fractal dimension of the integral's graph.

Constructs examples with non-bounded variation but box dimension 2.

Abstract

The subject of this note is the mixed Katugampola fractional integral of a bivariate function defined on a rectangular region in the Cartesian plane. This is a natural extension of the Katugampola fractional integral of a univariate function - a concept well-received in the recent literature on fractional calculus and its applications. It is shown that the mixed Katugampola fractional integral of a prescribed bivariate function preserves properties such as boundedness, continuity and bounded variation of the function. Furthermore, we estimate fractal dimension of the graph of the mixed Katugampola integral of a continuous bivariate function. Some examples for bivariate functions that are not of bounded variation but with graphs having box dimension $2$ are constructed. The findings in the current note may be viewed as a sequel to our work reported in [Appl. Math. Comp., 339, 2018, pp. 220-230].