FRACTAL DIMENSIONS FOR THE MIXED (κ,s)-RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL OF BIVARIATE FUNCTIONS

Abstract

The research object of this paper is the mixed $(\kappa ,s)$-Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed $(\kappa ,s)$-Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under $\sigma =({\sigma }_1,{\sigma }_2)$ order of the mixed integral is $3-min\{\frac{{\sigma }_1}{\kappa },\frac{{\sigma }_2}{\kappa }\}$ where $\kappa >0$.