Katugampola fractional integral and fractal dimension of bivariate functions

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].

19 pages, 2 figures. This work is a part of the Ph.D. thesis of the first author submitted to IIT Delhi. The thesis is defended successfully in October 2020. This paper is submitted to a journal for possible publication