# FractionalD

FractionalD[f,{x,α}]

gives the RiemannLiouville fractional derivative of order α of the function f.

# Details • FractionalD is also known as the RiemannLiouville differintegral of f.
• FractionalD generalizes D to fractional order and unifies the notions of derivatives and integrals from calculus.
• FractionalD plays a foundational role in fractional calculus since other types of fractional derivatives such as CaputoD can be defined in terms of it.
• The RiemannLiouville fractional derivative of of order is defined as , where .
• The derivatives of fractional order "interpolate" between the derivatives of integer orders, as shown below for the function and its fractional derivatives of order given by for :
• • The order α of a fractional derivative can be symbolic or an arbitrary real number.
• FractionalD[array,{x,α}] threads FractionalD over each element of array.
• FractionalD takes different Assumptions on the parameters of input functions.
• All expressions that do not explicitly depend on the given variable are interpreted as constants.

# Examples

open allclose all

## Basic Examples(4)

Calculate the half-order fractional derivative of a quadratic function with respect to x:

Arbitrary-order fractional derivative of a quadratic function with respect to x:

Plot these fractional derivatives for different 's:

Calculate the -order fractional derivative of a constant with respect to x:

Fractional derivative of MittagLefflerE:

## Scope(8)

Fractional derivative of the power function with respect to x:

0.23-order fractional derivative of the Exp function with respect to x:

For positive integer , the fractional RiemannLiouville derivative coincides with the ordinary derivative:

For negative integer , FractionalD differs from the ordinary indefinite integral by a constant:

Fractional derivatives of Sin function are written in terms of HypergeometricPFQ:

Fractional derivatives of BesselJ function:

Fractional derivatives of MeijerG function are given in terms of another MeijerG function:

Laplace transform of a fractional integral in general form:

Substitute the exponential function:

Get the same result by applying LaplaceTransform to the FractionalD of Exp:

## Options(1)

### Assumptions(1)

FractionalD may return a ConditionalExpression:

Restricting parameters using Assumptions will simplify the output:

## Applications(2)

Calculate the half-order fractional derivative of the cubic function:

Get the ordinary derivative of the cubic function repeating the half-order fractional differentiation:

Recover the initial function using fractional integration:

Consider the following fractional order integral equation:

Solve it for the Laplace transform:

Find the inverse transform:

Verify this solution:

## Properties & Relations(5)

FractionalD is defined for all real :

The 0-order fractional derivative of a function is the function itself:

FractionalD is not defined for complex order :

In general, the fractional derivative of a constant is not 0:

FractionalD results may contain DifferenceRoot sequences:

This general expression is simplified to a finite sum of HypergeometricPFQ instances if is a given real number:

## Neat Examples(1)

Create a table of α fractional and n ordinary derivatives of a few special functions: