About Fractional Calculus and its Applications in Physics

TL;DR

This paper reviews the historical development of fractional calculus, focusing on Riemann-Liouville and Caputo definitions, and explores their applications in physics and engineering, emphasizing educational integration.

Contribution

It provides a comprehensive historical overview and practical examples of fractional derivatives, highlighting their potential for inclusion in physics curricula.

Findings

Classical fractional derivatives can be computed for basic functions.

Fractional calculus has broad applications in physics and engineering.

Educational implementation of fractional calculus is feasible and beneficial.

Abstract

Historically the fractional calculus concept works an extended idea based on the question asked by Guillaume de L'H\^opital to Gottfried Wilhelm Leibniz in 1695 about the notation ${d^nf}/{dx^n}$ for the derivative operator "What if $n=\frac{1}{2}$ ?" To which Leibiniz replied : "This is an apparent paradox, from which useful consequences will be established". Our work revisits the unfolds who followed this questions with some classical definitions of fractional derivative operators and fractional integral. We still point out possible applications in areas such as Engineering, Physics, among others. Among these definitions we will focus more on the Riemann-Liouville and Caputo definitions, however other definitions are also briefly commented. In this work we begin with a historical inspection of the birth of the fractional calculus, parallels with the differential calculus and some of its developments are traced. Always focusing on the definitions of Riemann-Liouville and Caputo, more commonly found in the bibliography of the area and more frequent in scientific works. Some examples of its operability are presented, such as the direct calculation of constant function derivatives, polynomial function and exponential function. Derivative operators and fractional integrals are defined as derivatives and noninteger-order integrals. Our work revisits these two classical definitions of derivative operators and fractional integral and points out possible applications in areas such as Engineering, Physics, among others. Our main goal is to address the feasibility of implementing this content in a degree form program in Physics, we strogly believe that this theme will aggregate a lot of content mainly because its multidisciplinary character.