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

Document Type : Regular research papers

Authors

1 Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt.

2 Department of Mathematics, Faculty of Science, Alexandria University, Egypt.

Abstract

In this paper, we study some properties of the fractal and fractal-fractional integral
and differential operators and define the linear first and second kinds Abel’s
fractal and fractal-fractional integral equations. The existence of solutions of these
kinds 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 differential
and integral operators, which serves as a crucial tool for studying various phenomena.
Specifically, we examine the theoretical foundation of the fractal-fractional integral operator
and 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 the
fractal and fractal-fractional Abel’s integral equations of the first and second kinds. By exploring
these equations, we aim to expand the understanding of how fractal and fractional
operators interact and their potential applications in fields such as physics, engineering,
and mathematics.

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