# CaputoD

CaputoD[f,{x,α}]

gives the Caputo fractional differintegral of the function .

# Details and Options

• CaputoD is also known as the Caputo differintegral of f.
• CaputoD generalizes D to fractional order and unifies the notions of derivatives and integrals from calculus.
• CaputoD has found wide applications for modeling systems using initial value problems for fractional differential equations.
• The Caputo fractional derivative of order of 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 Caputo fractional derivative is connected with the FractionalD (RiemannLiouville fractional derivative) via the formula .
• The order α of a fractional derivative can be symbolic or an arbitrary real number.
• CaputoD[{array},{x,α}] threads CaputoD over each element of array.
• CaputoD 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

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## Basic Examples(4)

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

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

Plot these fractional derivatives for different 's:

The Caputo fractional derivative of a constant with respect to x for positive values of is 0:

Caputo fractional derivative of MittagLefflerE:

## Scope(4)

Caputo fractional derivative of the Exp function with respect to x:

Caputo fractional derivative of the Sin function with respect to x:

This expression can be further simplified:

Caputo fractional derivative of some BesselJ function:

Laplace transform of the CaputoD function in general form:

Apply the formula to Sin:

Get the same result applying LaplaceTransform to the CaputoD of Sin:

## Options(1)

### Assumptions(1)

CaputoD may return a ConditionalExpression:

Restricting parameters using Assumptions will simplify the output:

## Applications(8)

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

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

Recover the initial function using the Caputo fractional integration operation:

Solve a fractional DE containing a half-order Caputo derivative:

Add an initial condition:

Plot this solution:

Solve a fractional DE:

Plot the solution:

Solve a mixed fractional differential-integral equation:

Solve some fractional DE containing two different order Caputo derivatives:

Solve a system of two fractional DEs including CaputoD fractional derivatives of functions:

Verify the solution:

Parametric plot of this solution:

Solve a system of two fractional DEs in vector form:

Plot the solutions:

Parametrically plot the solution:

Solve a system of three fractional DEs in vector form:

Plot the solutions:

Parametrically plot the solution:

## Properties & Relations(7)

CaputoD is defined for all real :

0-order Caputo fractional derivative is the function itself:

CaputoD is not defined for complex order :

The Caputo fractional derivative of a constant is 0 for positive fractional orders:

While the general rule is:

CaputoD coincides with FractionalD for all negative orders :

Compare with the output of FractionalD:

Restricting the order to negative values, CaputoD will automatically generate the FractionalD output:

Calculate the Caputo fractional derivative of a function at some point:

Use the NCaputoD function for faster numerical calculations:

## Possible Issues(1)

CaputoD fractional derivative might not be defined for some fractional orders:

While for others it is defined:

## Neat Examples(1)

Create a table of half-order Caputo fractional derivatives for some special functions:

Wolfram Research (2022), CaputoD, Wolfram Language function, https://reference.wolfram.com/language/ref/CaputoD.html.

#### Text

Wolfram Research (2022), CaputoD, Wolfram Language function, https://reference.wolfram.com/language/ref/CaputoD.html.

#### CMS

Wolfram Language. 2022. "CaputoD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CaputoD.html.

#### APA

Wolfram Language. (2022). CaputoD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CaputoD.html

#### BibTeX

@misc{reference.wolfram_2024_caputod, author="Wolfram Research", title="{CaputoD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/CaputoD.html}", note=[Accessed: 15-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_caputod, organization={Wolfram Research}, title={CaputoD}, year={2022}, url={https://reference.wolfram.com/language/ref/CaputoD.html}, note=[Accessed: 15-September-2024 ]}