Caputo fractional derivative explained

In mathematics, the Caputo fractional derivative, also called Caputo-type fractional derivative, is a generalization of derivatives for non-integer orders named after Michele Caputo. Caputo first defined this form of fractional derivative in 1967.^[1]

Motivation

The Caputo fractional derivative is motivated from the Riemann–Liouville fractional integral. Let $f$ be continuous on

, then the Riemann–Liouville fractional integral $$ states that

$$\backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash frac\; \backslash cdot\; \backslash int\backslash limits\_^\; \backslash frac\; \backslash ,\; \backslash operatornamet$$

where $\backslash Gamma\backslash left(\backslash cdot\; \backslash right)$ is the Gamma function.

Let's define $\backslash operatorname\_^\; :=\; \backslash frac$, say that $\backslash operatorname\_^\; \backslash operatorname\_^\; =\; \backslash operatorname\_^$ and that $\backslash operatorname\_^\; =$ applies. If then we could say $\backslash operatorname\_^\; =\; \backslash operatorname\_^\; =\; \backslash operatorname\_^\; =\; \backslash operatorname\_^\; =\; \backslash operatorname\_^\backslash operatorname\_^\; =\; \_^\backslash operatorname\_^$. So if

is also , then

$$\backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash frac\; \backslash cdot\; \backslash int\backslash limits\_^\; \backslash frac\; \backslash ,\; \backslash operatornamet.$$

This is known as the Caputo-type fractional derivative, often written as $\_^$.

Definition

The first definition of the Caputo-type fractional derivative was given by Caputo as:

$$\backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash frac\; \backslash cdot\; \backslash int\backslash limits\_^\; \backslash frac\; \backslash ,\; \backslash operatornamet$$

where

and .^[2]

A popular equivalent definition is:

$$\backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash frac\; \backslash cdot\; \backslash int\backslash limits\_^\; \backslash frac\backslash ,\; \backslash operatornamet$$

where $\backslash alpha\; \backslash in\; \backslash mathbb\_\; \backslash setminus\; \backslash mathbb$ and $\backslash left\backslash lceil\; \backslash cdot\; \backslash right\backslash rceil$ is the ceiling function. This can be derived by substituting $\backslash alpha\; =\; m\; +\; z$ so that $\backslash left\backslash lceil\; \backslash alpha\; \backslash right\backslash rceil\; =\; m\; +\; 1$ would apply and $\backslash left\backslash lceil\; \backslash alpha\; \backslash right\backslash rceil\; +\; z\; =\; \backslash alpha\; +\; 1$ follows.^[3]

Another popular equivalent definition is given by:

$$\backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash frac\; \backslash cdot\; \backslash int\backslash limits\_^\; \backslash frac\backslash ,\; \backslash operatornamet$$

where .

The problem with these definitions is that they only allow arguments in $\backslash left(0,\backslash ,\; \backslash infty\; \backslash right)$. This can be fixed by replacing the lower integral limit with $a$: $\backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash frac\; \backslash cdot\; \backslash int\backslash limits\_^\; \backslash frac\backslash ,\; \backslash operatornamet$. The new domain is $\backslash left(a,\backslash ,\; \backslash infty\; \backslash right)$.^[4]

Properties and theorems

Basic properties and theorems

A few basic properties are:^[5]

A table of basic properties and theorems!Properties!

f\left(x\right)

!

	C
{
	a

	\alpha
\operatorname{D}
	x

}\left[f\left(x \right) \right]!Condition

Definition	f\left(x\right)	f\left(\left(x\right)-f\left(\left(a\right)
Linearity	b ⋅ g\left(x\right)+c ⋅ h\left(x\right)	b ⋅ C { a \alpha \operatorname{D} x }\left[g\left(x \right) \right] + c \cdot \left[h\left(x \right) \right]
Index law	\beta \operatorname{D} x	C { a \alpha+\beta \operatorname{D} x }	\beta\inZ
Semigroup property	C { a \beta \operatorname{D} x }	C { a \alpha+\beta \operatorname{D} x }	\left\lceil\alpha\right\rceil=\left\lceil\beta\right\rceil

Non-commutation

The index law does not always fulfill the property of commutation:

$$\backslash operatorname\_^\backslash operatorname\_^\; =\; \backslash operatorname\_^\; \backslash ne\; \backslash operatorname\_^\backslash operatorname\_^$$

where

.

Fractional Leibniz rule

The Leibniz rule for the Caputo fractional derivative is given by:

$$\backslash operatorname\_^\backslash left[g\backslash left(x\; \backslash right)\; \backslash cdot\; h\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash sum\backslash limits\_^\backslash left[\backslash binom\{a\}\{k\}\; \backslash cdot\; g^\{\backslash left(k\; \backslash right)\}\backslash left(x\; \backslash right)\; \backslash cdot\; \backslash operatorname\{\_\{a\}^\{\backslash text\{RL\}\}D\}\_\{x\}^\{\backslash alpha\; -\; k\}\backslash left[\; h\backslash left(x\; \backslash right)\; \backslash right]\; \backslash right]\; -\; \backslash frac\; \backslash cdot\; g\backslash left(a\; \backslash right)\; \backslash cdot\; h\backslash left(a\; \backslash right)$$

where $\backslash binom\; =\; \backslash frac$ is the binomial coefficient.^{[6] [7]}

Relation to other fractional differential operators

Caputo-type fractional derivative is closely related to the Riemann–Liouville fractional integral via its definition:

$$\backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash left[\backslash operatorname\{D\}\_\{x\}^\{\backslash left\backslash lceil\; \backslash alpha\; \backslash right\backslash rceil\}\backslash left[\; f\backslash left(x\; \backslash right)\; \backslash right]\; \backslash right]$$

Furthermore, the following relation applies:

$$\backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; =\; \backslash left[f\backslash left(x\; \backslash right)\; \backslash right]\; -\; \backslash sum\backslash limits\_^\backslash left[\backslash frac\{x^\{k\; -\; \backslash alpha\}\}\{\backslash Gamma\backslash left(k\; -\; \backslash alpha\; +\; 1\; \backslash right)\}\; \backslash cdot\; f^\{\backslash left(k\; \backslash right)\}\backslash left(0\; \backslash right)\; \backslash right]$$

where

} is the Riemann–Liouville fractional derivative.

Laplace transform

The Laplace transform of the Caputo-type fractional derivative is given by:

$$\backslash mathcal\_\backslash left\backslash \backslash left(s\; \backslash right)\; =\; s^\; \backslash cdot\; F\backslash left(s\; \backslash right)\; -\; \backslash sum\backslash limits\_^\backslash left[s^\{\backslash alpha\; -\; k\; -\; 1\}\; \backslash cdot\; f^\{\backslash left(k\; \backslash right)\}\backslash left(0\; \backslash right)\; \backslash right]$$

where $\backslash mathcal\_\backslash left\backslash \backslash left(s\; \backslash right)\; =\; F\backslash left(s\; \backslash right)$.^[8]

Caputo fractional derivative of some functions

The Caputo fractional derivative of a constant

is given by:

The Caputo fractional derivative of a power function

is given by:^[9]

The Caputo fractional derivative of a exponential function

is given by:

where $E\_\backslash left(\backslash nu,\backslash ,\; a\; \backslash right)\; =\; \backslash frac$ is the $\backslash operatorname\_$-function and $\backslash gamma\; \backslash left(a,\backslash ,\; b\; \backslash right)$ is the lower incomplete gamma function.^[10]

Further reading

• Ricardo Almeida, A Caputo fractional derivative of a function with respect to another function

Notes and References

1. Book: Diethelm, Kai . Fractional Differential Equations . 2019 . General theory of Caputo-type fractional differential equations . 1–20 . 10.1515/9783110571660-001 . 978-3-11-057166-0 . https://opus4.kobv.de/opus4-fhws/frontdoor/index/index/year/2023/docId/3212 . 2023-08-10.
2. Caputo . Michele . 1967 . Linear Models of Dissipation whose Q is almost Frequency Independent-II . ResearchGate . 13 . 5 . 530. 10.1111/j.1365-246X.1967.tb02303.x . free . 1967GeoJ...13..529C .
3. Lazarević . Mihailo . Rapaić . Milan Rade . Šekara . Tomislav . 2014 . Introduction to Fractional Calculus with Brief Historical Background . ResearchGate . 8.
4. Dimitrov . Yuri . Georgiev . Slavi . Todorov . Venelin . 2023 . Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations . Fractal and Fractional. 7 . 10 . 750 . 10.3390/fractalfract7100750 . free .
5. Sikora . Beata . 2023 . Remarks on the Caputo fractional derivative . Matematyka I Informatyka Na Uczelniach Technicznych . 5 . 78–79.
6. Huseynov . Ismail . Ahmadova . Arzu . Mahmudov . Nazim . 2020 . Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications . ResearchGate . 1. 2012.11360 .
7. Web site: Weisstein . Eric W. . 2024 . Binomial Coefficient . 2024-05-20 . mathworld.wolfram.com . en.
8. Sontakke . Bhausaheb Rajba . Shaikh . Amjad . 2015 . Properties of Caputo Operator and Its Applications to Linear Fractional Differential Equations . Journal of Engineering Research and Applications . 5 . 5 . 23–24 . 2248-9622.
9. Web site: Weisstein . Eric W. . Fractional Derivative . 2024-05-20 . mathworld.wolfram.com . en.
10. Web site: Weisstein . Eric W. . 2024 . E_t-Function . 2024-05-20 . mathworld.wolfram.com . en.