On the Katugampola fractional integral and dimensional analysis of the fractal basin boundary for a random dynamical system

In this paper, we mainly investigate the geometric based relationship between the Katugampola fractional calculus and a Weierstrass-type function whose graph can be characterized as a fractal basin boundary for a random dynamical system. Using the potential-theoretic approach with some classical analytical tools, we have derived some kinds of fractal dimensions of the graph of the Katugampola fractional integral of this fractal function. It has been shown that there is a linear relationship between the order of the Katugampola fractional integral and the fractal dimension of the graph of this generalized Weierstrass function. Numerical results have also been provided to corroborate such linear connection.