# NCaputoD

NCaputoD[f,{x,α},x0]

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

# Details and Options • NCaputoD is the numerical analog of CaputoD, also known as the Caputo differintegral of f.
• The Caputo fractional derivative of order of is defined as , where .
• NCaputoD is typically used in cases when symbolic evaluation with CaputoD fails.
• The following options can be given to NCaputoD:
•  AccuracyGoal Infinity digits of absolute accuracy sought PrecisionGoal Automatic digits of precision sought WorkingPrecision MachinePrecision the precision used in internal computations

# Examples

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## Basic Examples(4)

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

Plot the -order Caputo fractional derivative of an exponential function:

Numerical Caputo fractional derivative of MittagLefflerE at some point:

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

Plot this fractional derivative:

Numerical Caputo fractional integral:

## Scope(8)

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

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

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

Plot this function:

The -order Caputo fractional derivative of the Sin function:

The -order Caputo fractional integral of the Sin function:

The half-order Caputo fractional derivative of the AiryAi function:

Numerically calculate the Caputo fractional derivative of the BesselJ function:

Plot the numerically calculated fractional derivatives and integrals of complex functions:

## Applications(3)

NCaputoD is able to numerically calculate fractional derivatives when CaputoD fails:

CaputoD outputs might be cumbersome:

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

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

The timing for the calculated 101 values:

Plot this fractional derivative:

## Properties & Relations(4)

NCaputoD is defined for all real : The Caputo fractional derivative of a constant is 0:

Plot the Caputo fractional derivative using the numerical NCaputoD approach:

Compare with the symbolic CaputoD approach:

For negative orders , NCaputoD coincides with NFractionalD:

## Possible Issues(2)

NCaputoD will generate an error message if the precision of input is less than the WorkingPrecision: NCaputoD takes only numeric points of evaluation: ## Neat Examples(1)

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