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 n=max(0,TemplateBox[{alpha}, Ceiling]).
  • 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 2/TemplateBox[{{3, -, alpha}}, Gamma] x^(2-alpha) for :
  • The Caputo fractional derivative is connected with the FractionalD (Riemann–Liouville 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_2025_caputod, author="Wolfram Research", title="{CaputoD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/CaputoD.html}", note=[Accessed: 20-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_caputod, organization={Wolfram Research}, title={CaputoD}, year={2022}, url={https://reference.wolfram.com/language/ref/CaputoD.html}, note=[Accessed: 20-February-2025 ]}