Caputo fractional derivative

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 be continuous on , then the Riemann–Liouville fractional integral states that

where is the Gamma function.

Let's define , say that and that applies. If then we could say . So if is also , then

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:

where and .[2]

A popular equivalent definition is:

where and is the ceiling function. This can be derived by substituting so that would apply and follows.[3]

Another popular equivalent definition is given by:

where .

The problem with these definitions is that they only allow arguments in . This can be fixed by replacing the lower integral limit with : . The new domain is .[4]

Properties and theorems

Basic properties and theorems

A few basic properties are:[5]

A table of basic properties and theorems
Properties Condition
Definition
Linearity
Index law
Semigroup property

Non-commutation

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

where .

Fractional Leibniz rule

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

where 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:

Furthermore, the following relation applies:

where is the Riemann–Liouville fractional derivative.

Laplace transform

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

where .[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 is the -function and is the lower incomplete gamma function.[10]

References

  1. ^ Diethelm, Kai (2019). "General theory of Caputo-type fractional differential equations". Fractional Differential Equations. pp. 1–20. doi:10.1515/9783110571660-001. ISBN 978-3-11-057166-0. Retrieved 2023-08-10.
  2. ^ Caputo, Michele (1967). "Linear Models of Dissipation whose Q is almost Frequency Independent-II". ResearchGate. 13 (5): 530. Bibcode:1967GeoJ...13..529C. doi:10.1111/j.1365-246X.1967.tb02303.x.
  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. doi:10.3390/fractalfract7100750.
  5. ^ Sikora, Beata (2023). "Remarks on the Caputo fractional derivative" (PDF). 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. arXiv:2012.11360.
  7. ^ Weisstein, Eric W. (2024). "Binomial Coefficient". mathworld.wolfram.com. Retrieved 2024-05-20.
  8. ^ Sontakke, Bhausaheb Rajba; Shaikh, Amjad (2015). "Properties of Caputo Operator and Its Applications to Linear Fractional Differential Equations" (PDF). Journal of Engineering Research and Applications. 5 (5): 23–24. ISSN 2248-9622.
  9. ^ Weisstein, Eric W. "Fractional Derivative". mathworld.wolfram.com. Retrieved 2024-05-20.
  10. ^ Weisstein, Eric W. (2024). "E_t-Function". mathworld.wolfram.com. Retrieved 2024-05-20.

Further reading

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