REVIEW OF THE CONFORMABLE CALCULUS AND SOME APPLICATIONS

Abstract

The most important characteristics of conformable integrals and derivatives have recently been explored in the literature [1][2]. In this work, we delve into the newly emerging field of fractional calculus, with a particular focus on the novel class of fractional derivatives known as conformable fractional derivatives. We provide an extensive examination of various definitions and formulations of fractional derivatives, emphasizing the ”new conformable fractional derivative” introduced in [13]. This new derivative is defined as ( D α G ) ( z ) = lim x −→ 0 G ( z + x e ( α − 1 ) z ) − G ( z ) x , which is distinct from traditional fractional derivatives in its formulation and application. We investigate the implications and utility of this new definition in proving certain results related to conformable fractional derivatives, as established in [14]. Our study includes a comparative analysis of the new conformable fractional derivative with existing fractional calculus frameworks, and we demonstrate how it simplifies and enhances the understanding of fractional calculus problems. Additionally, we explore its potential applications in various fields, including mathematical modeling and engineering. The findings contribute to a deeper understanding of fractional calculus and offer new tools for researchers and practitioners in the field.