Dimensional Analysis of Mixed Riemann–Liouville Fractional Integral of Vector-Valued Functions

Abstract

In this paper, we attempt to develop the concept of fractal dimension of the continuous bivariate vector-valued maps. We give few fundamental concepts of the dimension of the graph of bivariate vector-valued functions and prove some basic results. We prove that upper bound of the Hausdorff dimension of Hölder continuous function is $$3-\sigma $$ . Because of its wide applications in many important areas, fractal dimension has become one of the most interesting parts of fractal geometry. Estimating the fractal dimension is one of the most fascinating works in fractal theory. It is not always easy to estimate the fractal dimension even for elementary real-valued functions. However, in this paper, an effort is made to find the fractal dimension of continuous bivariate vector-valued maps and, in particular, the fractal dimension of the Riemann–Liouville fractional integral of a continuous vector-valued bivariate map of bounded variation defined on a rectangular domain is also found.