A Comprehensive Study of Fractional-Order Derivative and Their Interplay with Basic Functions

Fractional calculus from the nineteenth century to date has gained considerable attention due to its versatile applications in various scientific and engineering domains. This work examines the complex relationship between fractional-order derivative and basic functions, unraveling the profound interplay between mathematics and simulation. In this study, we illustrate the Mittag-Leffler function, Grunwald-Letnikov’s, Riemann-Liouville’s, and Caputo’s fractional derivative and integral are presented with examples of basic functions and their graphical presentations. The purpose of this study is to examine the features of fractional derivatives from the perspective of researchers’ motivations and interests.