CaputoD
✖
CaputoD
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 (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
open allclose allBasic Examples (4)Summary of the most common use cases
Calculate the half-order Caputo fractional derivative of a quadratic function with respect to x:
https://wolfram.com/xid/0g7e1ltwjzz-2mv3pc
Arbitrary-order Caputo fractional derivative of a quadratic function with respect to x:
https://wolfram.com/xid/0g7e1ltwjzz-06lozd
Plot these fractional derivatives for different 's:
https://wolfram.com/xid/0g7e1ltwjzz-pggxbv
The Caputo fractional derivative of a constant with respect to x for positive values of is 0:
https://wolfram.com/xid/0g7e1ltwjzz-y2w9t3
Caputo fractional derivative of MittagLefflerE:
https://wolfram.com/xid/0g7e1ltwjzz-gl9r59
Scope (4)Survey of the scope of standard use cases
Caputo fractional derivative of the Exp function with respect to x:
https://wolfram.com/xid/0g7e1ltwjzz-napkzk
Caputo fractional derivative of the Sin function with respect to x:
https://wolfram.com/xid/0g7e1ltwjzz-tvyo03
This expression can be further simplified:
https://wolfram.com/xid/0g7e1ltwjzz-5hk10q
Caputo fractional derivative of some BesselJ function:
https://wolfram.com/xid/0g7e1ltwjzz-e3yxgu
Laplace transform of the CaputoD function in general form:
https://wolfram.com/xid/0g7e1ltwjzz-wq7gti
Apply the formula to Sin:
https://wolfram.com/xid/0g7e1ltwjzz-cx0jn8
Get the same result applying LaplaceTransform to the CaputoD of Sin:
https://wolfram.com/xid/0g7e1ltwjzz-hje8jp
Options (1)Common values & functionality for each option
Assumptions (1)
CaputoD may return a ConditionalExpression:
https://wolfram.com/xid/0g7e1ltwjzz-bais0b
Restricting parameters using Assumptions will simplify the output:
https://wolfram.com/xid/0g7e1ltwjzz-mu7kqu
Applications (8)Sample problems that can be solved with this function
Calculate the half-order Caputo fractional derivative of the cubic function:
https://wolfram.com/xid/0g7e1ltwjzz-il7axk
Get the ordinary derivative of the cubic function repeating the half-order Caputo fractional differentiation:
https://wolfram.com/xid/0g7e1ltwjzz-2aq2lc
Recover the initial function using the Caputo fractional integration operation:
https://wolfram.com/xid/0g7e1ltwjzz-t34o6x
Solve a fractional DE containing a half-order Caputo derivative:
https://wolfram.com/xid/0g7e1ltwjzz-uyu64e
https://wolfram.com/xid/0g7e1ltwjzz-c5jec4
https://wolfram.com/xid/0g7e1ltwjzz-7vjm35
https://wolfram.com/xid/0g7e1ltwjzz-l33f4t
https://wolfram.com/xid/0g7e1ltwjzz-x7j2ys
Solve a mixed fractional differential-integral equation:
https://wolfram.com/xid/0g7e1ltwjzz-uswg8t
Solve some fractional DE containing two different order Caputo derivatives:
https://wolfram.com/xid/0g7e1ltwjzz-0s8scw
Solve a system of two fractional DEs including CaputoD fractional derivatives of functions:
https://wolfram.com/xid/0g7e1ltwjzz-ptt6xv
https://wolfram.com/xid/0g7e1ltwjzz-i8hs7n
Parametric plot of this solution:
https://wolfram.com/xid/0g7e1ltwjzz-6d7cdd
Solve a system of two fractional DEs in vector form:
https://wolfram.com/xid/0g7e1ltwjzz-bar9ov
https://wolfram.com/xid/0g7e1ltwjzz-ta7tm
Parametrically plot the solution:
https://wolfram.com/xid/0g7e1ltwjzz-d7tqvr
Solve a system of three fractional DEs in vector form:
https://wolfram.com/xid/0g7e1ltwjzz-i1xpko
https://wolfram.com/xid/0g7e1ltwjzz-e9s42t
Parametrically plot the solution:
https://wolfram.com/xid/0g7e1ltwjzz-bh47ib
Properties & Relations (7)Properties of the function, and connections to other functions
CaputoD is defined for all real :
https://wolfram.com/xid/0g7e1ltwjzz-z2ojjm
https://wolfram.com/xid/0g7e1ltwjzz-n1trsx
0-order Caputo fractional derivative is the function itself:
https://wolfram.com/xid/0g7e1ltwjzz-vm2in3
CaputoD is not defined for complex order :
https://wolfram.com/xid/0g7e1ltwjzz-ri9o7g
The Caputo fractional derivative of a constant is 0 for positive fractional orders:
https://wolfram.com/xid/0g7e1ltwjzz-heuuox
https://wolfram.com/xid/0g7e1ltwjzz-zohye7
CaputoD coincides with FractionalD for all negative orders :
https://wolfram.com/xid/0g7e1ltwjzz-w0ohos
Compare with the output of FractionalD:
https://wolfram.com/xid/0g7e1ltwjzz-0aku20
Restricting the order to negative values, CaputoD will automatically generate the FractionalD output:
https://wolfram.com/xid/0g7e1ltwjzz-2kotxh
https://wolfram.com/xid/0g7e1ltwjzz-uk31s
Calculate the Caputo fractional derivative of a function at some point:
https://wolfram.com/xid/0g7e1ltwjzz-rpoch9
Use the NCaputoD function for faster numerical calculations:
https://wolfram.com/xid/0g7e1ltwjzz-s892kc
Possible Issues (1)Common pitfalls and unexpected behavior
CaputoD fractional derivative might not be defined for some fractional orders:
https://wolfram.com/xid/0g7e1ltwjzz-z88rqj
While for others it is defined:
https://wolfram.com/xid/0g7e1ltwjzz-n2ltq3
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.
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.
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
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: 22-December-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: 22-December-2024
]}