Fractal-fractional differential and integral operators: Definitions, some properties and applications

Abstract

In this paper, we study some properties of the fractal and fractal-fractional integraland differential operators and define the linear first and second kinds Abel’sfractal and fractal-fractional integral equations. The existence of solutions of thesekinds of Abel’s integral equations will be studied.Two initial-value problems of fractal integro-differential Abel’s equations will be studied.This paper focuses on exploring the properties of the fractal and the fractal-fractional differentialand integral operators, which serves as a crucial tool for studying various phenomena.Specifically, we examine the theoretical foundation of the fractal-fractional integral operatorand delve into its application to the study of Abel’s integral equations.The objective of this study is to investigate some of the key problems associated with thefractal and fractal-fractional Abel’s integral equations of the first and second kinds. By exploringthese equations, we aim to expand the understanding of how fractal and fractionaloperators interact and their potential applications in fields such as physics, engineering,and mathematics.