# NFractionalD

NFractionalD[f,{x,α},x0]

gives a numerical approximation to the RiemannLiouville fractional derivative of order α of the function f at the point x0.

# Details and Options • NFractionalD is the numerical analog of FractionalD, also known as the RiemannLiouville differintegral of f.
• The RiemannLiouville fractional derivative of of order is defined as , where .
• NFractionalD is typically used in cases when symbolic evaluation with FractionalD fails.
• The following options can be given to NFractionalD:
•  Method Automatic method to use AccuracyGoal Infinity digits of absolute accuracy sought PrecisionGoal Automatic digits of precision sought WorkingPrecision MachinePrecision the precision used in internal computations
• The following Method option settings may be used:
•  Automatic automatically selected method "GrunwaldLetnikov" Grünwald-Letnikov fractional differintegration scheme "RiemannLiouville" Riemann-Liouville fractional differintegration scheme

# Examples

open allclose all

## Basic Examples(5)

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

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

Plot the -order fractional derivative of an exponential function:

Numerical fractional derivative of MittagLefflerE at a point:

Generate a list of numerical values of the fractional derivative of the MittagLefflerE function:

Plot this fractional derivative:

Numerical fractional integral:

## Scope(9)

Plot the -order fractional derivative of the square function with respect to x:

Calculate the 0.23-order fractional derivative of the Exp function numerically and symbolically:

Calculate the value of the half-order derivative of the ArcTan function at some point:

Plot this function:

The -order fractional derivative of the Sin function:

The -order fractional integral of the Sin function:

The half-order fractional derivative of the AiryAi function:

Numerically calculate the fractional derivative of the BesselJ function:

Numerically calculate the fractional derivative of the MeijerG function:

Plot the numerically calculated fractional derivatives and integrals of a trigonometric product:

## Options(2)

### Method(2)

NFractionalD has two built-in methods, the RiemannLiouville and the GrünwaldLetnikov methods:

If the Method is not specified, NFractionalD automatically uses the RiemannLiouville approach:

In some cases, the GrünwaldLetnikov method is able to calculate the numerical fractional derivative:

It is not possible to calculate this fractional derivative using the RiemannLiouville method:

## Applications(3)

NFractionalD is able to numerically calculate fractional derivatives when FractionalD fails:

FractionalD outputs may contain DifferenceRoot sequences:

For a specific order of fractional differentiation, this is a cumbersome sum of HypergeometricPFQ functions:

However, the numerically calculated fractional derivative plot might be insightful:

Effectively generate a list of numerical values of fractional derivatives of a complex function using NFractionalD:

The timing for the calculated 151 values:

Plot this fractional derivative:

## Properties & Relations(6)

NFractionalD is defined for all real : In general, the fractional derivative of a constant is not 0:

NFractionalD gives the output in WorkingPrecision (if not specified, it is MachinePrecision):

If the Method option is not specified, NFractionalD uses the "RiemannLiouville" method:

Plot the fractional derivative using the numerical NFractionalD approach:

Compare with the symbolic FractionalD approach:

For negative orders , NCaputoD coincides with NFractionalD:

## Possible Issues(3)

NFractionalD will generate an error message if the Method option is not correct: NFractionalD will generate an error message if the precision of input is less than the WorkingPrecision: NFractionalD takes only numeric points of evaluation: ## Neat Examples(1)

Plot the Sin function and its half, first and -order derivatives: