A New Analysis of the AB-Fractal Operators Using the Generalized Gamma Function in Geometric Patterns

Abstract

Fractional calculus can be employed to precisely alter or control the fractal dimension of deterministic or random fractals with coordinates that can be denoted as functions of independent variables. Fractal geometry, enabling more accurate definition and measurement of the complicated nature of a shape, resorts to quantification, while gamma function refers to the generalization of the factorial function. This study has made a generalization of the Atangana-Baleanu (AB) fractal-fractional operators (derivative and integral), which is called p-AB-fractal fractional calculus through the utilization of the enhanced gamma function, which is called the p-gamma function. Additionally, we extend the proposed operators into a complex variable to discuss their properties geometrically in the open unit disk. Therefore, a normalization structure is provided with a set of examples. This leads to investigate the operators geometrically. In this direction, we introduce the sufficient conditions on these operators to get the starlikeness and convexity properties in the unit disk. As an application of these operators, we establish the existence and uniqueness solution of abstract fractal-fractional differential equation. Consequently, we illustrate a set of examples showing the validity of the new parameter p. The results provided in the study illustrate and demonstrate that the generalized and the extended operators can be considered in extensive applications and/or other geometric studies.