Overview of fractional calculus and its computer implementation in Wolfram Mathematica

Abstract

This survey aims to present various approaches to non-integer integrals and derivatives and their practical implementation within Wolfram Mathematica. It begins by short discussion of historical moments and applications related to fractional calculus. Different methods for handling non-integer powers of differentiation operators are presented, along with generalizations of fractional integrals and derivatives. The survey also delves into the diverse applications of fractional calculus in physics, engineering, medicine, and numerical calculations. Essential details of fractional integro-differentiation implemented in Wolfram Mathematica are highlighted. The Hadamard regularization of Riemann-Liouville operator is utilized as the foundation for creating the arbitrary order of integro-differential operator in Mathematica. The survey describes the application of fractional integro-differentiation to Taylor series expansions near zero using Hadamard regularization and the use of the Meijer G-function for evaluating derivatives of complex orders. We conclude with a discussion on applying fractional integro-differentiation to “differential constants” and provide generic formulas for fractional differentiation. The extensive list of references underscores the vast body of works on fractional calculus.