Numerical Approximations of Fractional Derivatives with Applications

TL;DR

This paper introduces two numerical approximation methods for fractional derivatives based on series expansions, enabling the transformation of fractional differential problems into classical ones for easier computation and analysis.

Contribution

It presents novel approximation techniques derived from series expansions of Riemann-Liouville derivatives, facilitating numerical solutions of fractional differential equations.

Findings

Effective in transforming fractional problems into classical ones

Demonstrated advantages in numerical computation of fractional derivatives

Discussed limitations and potential applications of the approximations

Abstract

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains fractional derivatives into a classical problem in which only derivatives of integer order are present. Corresponding approximations provide useful numerical tools to compute fractional derivatives of functions. Application of such approximations to fractional differential equations and fractional problems of the calculus of variations are discussed. Illustrative examples show the advantages and disadvantages of each approximation.