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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D D f(x) = \frac f(x)\,, and of the integration ...
in 1967.


Motivation

The Caputo fractional derivative is motivated from the Riemann–Liouville fractional integral. Let f be continuous on \left( 0,\, \infty \right), then the Riemann–Liouville fractional integral states that \left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac \, \operatornamet where \Gamma\left( \cdot \right) is the
Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
. Let's define \operatorname_^ := \frac, say that \operatorname_^ \operatorname_^ = \operatorname_^ and that \operatorname_^ = applies. If \alpha = m + z \in \mathbb \wedge m \in \mathbb_ \wedge 0 < z < 1 then we could say \operatorname_^ = \operatorname_^ = \operatorname_^ = \operatorname_^ = \operatorname_^\operatorname_^ = _^\operatorname_^. So if f is also C^\left( 0,\, \infty \right), then \left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac \, \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: \left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac \, \operatornamet where C^\left( 0,\, \infty \right) and m \in \mathbb_ \wedge 0 < z < 1. A popular equivalent definition is: \left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac\, \operatornamet where \alpha \in \mathbb_ \setminus \mathbb and \left\lceil \cdot \right\rceil is the
ceiling function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
. This can be derived by substituting \alpha = m + z so that \left\lceil \alpha \right\rceil = m + 1 would apply and \left\lceil \alpha \right\rceil + z = \alpha + 1 follows. Another popular equivalent definition is given by: \left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac\, \operatornamet where n - 1 < \alpha < n \in \mathbb. . The problem with these definitions is that they only allow arguments in \left( 0,\, \infty \right). This can be fixed by replacing the lower integral limit with a: \left f\left( x \right) \right= \frac \cdot \int\limits_^ \frac\, \operatornamet. The new domain is \left( a,\, \infty \right).


Properties and theorems


Basic properties and theorems

A few basic properties are:


Non-commutation

The index law does not always fulfill the property of commutation: \operatorname_^\operatorname_^ = \operatorname_^ \ne \operatorname_^\operatorname_^ where \alpha \in \mathbb_ \setminus \mathbb \wedge \beta \in \mathbb.


Fractional Leibniz rule

The Leibniz rule for the Caputo fractional derivative is given by: \operatorname_^\left g\left( x \right) \cdot h\left( x \right) \right= \sum\limits_^\left \binom \cdot g^\left( x \right) \cdot \operatorname_^\left[ h\left( x \right) \right\right] - \frac \cdot g\left( a \right) \cdot h\left( a \right) where \binom = \frac is the binomial coefficient.


Relation to other fractional differential operators

Caputo-type fractional derivative is closely related to the Riemann–Liouville fractional integral via its definition: \left f\left( x \right) \right= \left f\left( x \right) \right\right">\operatorname_^\left f\left( x \right) \right\right/math> Furthermore, the following relation applies: \left f\left( x \right) \right= \left f\left( x \right) \right- \sum\limits_^\left \frac \cdot f^\left( 0 \right) \right/math> where is the Riemann–Liouville fractional derivative.


Laplace transform

The
Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
of the Caputo-type fractional derivative is given by: \mathcal_\left\\left( s \right) = s^ \cdot F\left( s \right) - \sum\limits_^\left s^ \cdot f^\left( 0 \right) \right/math> where \mathcal_\left\\left( s \right) = F\left( s \right).


Caputo fractional derivative of some functions

The Caputo fractional derivative of a constant c is given by: \begin \left c \right&= \frac \cdot \int\limits_^ \frac\, \operatornamet = \frac \cdot \int\limits_^ \frac\, \operatornamet\\ \left c \right&= 0 \end The Caputo fractional derivative of a power function x^ is given by: \begin \left x^ \right&= \left x^ \right\right">\operatorname_^\left x^ \right\right= \frac \cdot \left x^ \right\ \left x^ \right&= \begin \frac \left( x^ - a^ \right),\, &\text \left\lceil \alpha \right\rceil - 1 < b \wedge b \in \mathbb\\ 0,\, &\text \left\lceil \alpha \right\rceil - 1 \geq b \wedge b \in \mathbb\\ \end \end The Caputo fractional derivative of an exponential function e^ is given by: \begin \left e^ \right&= \left e^ \right\right">\operatorname_^\left e^ \right\right= b^ \cdot \left e^ \right\ \left e^ \right &= b^ \cdot \left( E_\left( \left\lceil \alpha \right\rceil - \alpha,\, b \right) - E_\left( \left\lceil \alpha \right\rceil - \alpha,\, b \right) \right)\\ \end where E_\left( \nu,\, a \right) = \frac is the \operatorname_-function and \gamma \left( a,\, b \right) is the lower
incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, whic ...
.


References

{{reflist


Further reading

*Ricardo Almeida
A Caputo fractional derivative of a function with respect to another function
Fractional calculus