Fractional calculus generalizes the operations of differentiation and integration by unifying them into a single fractional derivative of arbitrary order. Fractional calculus is used in finance, engineering, science and other fields. The Wolfram Language provides tools for computing fractional derivatives using the Riemann–Liouville and Caputo definitions, as well as for using the popular Laplace transform technique to solve systems of linear fractional differential equations with constant coefficients in terms of the Mittag–Leffler and related functions.

Fractional Derivatives

FractionalD — Riemann–Liouville fractional derivative

CaputoD — Caputo fractional derivative

NFractionalD — numerical Riemann–Liouville derivative

NCaputoD — numerical Caputo derivative

Fractional Integral Transforms

LaplaceTransform — Laplace transforms of fractional derivatives

InverseLaplaceTransform — inverse Laplace transforms of fractional rational functions

Fractional Differential Equations

DSolve, DSolveValue — solutions of fractional differential equations

AsymptoticDSolveValue — asymptotic solutions of fractional differential equations

Fractional Special Functions

MittagLefflerE — solution representations for fractional differential equations