Fractional calculus is a branch of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
that studies the several different possibilities of defining
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
powers or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
powers of the
differentiation operator
:
and of the
integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
operator
[The symbol is commonly used instead of the intuitive in order to avoid confusion with other concepts identified by similar –like glyphs, such as identities.]
:
and developing a
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
for such operators generalizing the classical one.
In this context, the term ''powers'' refers to iterative application of a
linear operator to a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, that is, repeatedly
composing with itself, as in
.
For example, one may ask for a meaningful interpretation of
:
as an analogue of the
functional square root
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function is a function satisfying ...
for the differentiation operator, that is, an expression for some linear operator that, when applied ''twice'' to any function, will have the same effect as
differentiation. More generally, one can look at the question of defining a linear operator
:
for every real number
in such a way that, when
takes an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
value
, it coincides with the usual
-fold differentiation
if
, and with the
-th power of
when
.
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator
is that the
sets of operator powers
defined in this way are ''continuous''
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s with parameter
, of which the original ''discrete'' semigroup of
for integer
is a
denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, also known as extraordinary differential equations,
are a generalization of differential equations through the application of fractional calculus.
Historical notes
In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
and mathematical analysis, a fractional derivative is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to
Guillaume de l'Hôpital
Guillaume François Antoine, Marquis de l'Hôpital (; sometimes spelled L'Hospital; 1661 – 2 February 1704), also known as Guillaume-François-Antoine Marquis de l'Hôpital, Marquis de Sainte-Mesme, Comte d'Entremont, and Seigneur d'Ouques-la- ...
by
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...
in 1695.
Around the same time, Leibniz wrote to one of the Bernoulli brothers describing the similarity between the binomial theorem and the Leibniz rule for the fractional derivative of a product of two functions. Fractional calculus was introduced in one of
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
's early papers where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized operation, and even the unified notation for differentiation and integration of arbitrary real order.
Independently, the foundations of the subject were laid by
Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
in a paper from 1832.
The
autodidact Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed ...
introduced the practical use of
fractional differential operators in electrical transmission line analysis circa 1890. The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives and integrals.
Nature of the fractional derivative
The
-th derivative of a function
at a point
is a ''local property'' only when
is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of
at
depends on all values of
, even those far away from
. Therefore, it is expected that the fractional derivative operation involves some sort of
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s, involving information on the function further out.
The fractional derivative of a function of order
is nowadays often defined by means of the
Fourier or
Mellin
Mellin is a village and a former municipality in the district Altmarkkreis Salzwedel, in Saxony-Anhalt, Germany. Since 1 January 2009, it is part of the municipality Beetzendorf
Beetzendorf is a municipality in the district Altmarkkreis Salzwe ...
integral transforms.
Heuristics
A fairly natural question to ask is whether there exists a
linear operator , or half-derivative, such that
:
It turns out that there is such an operator, and indeed for any , there exists an operator such that
:
or to put it another way, the definition of can be extended to all real values of .
Let be a function defined for . Form the definite integral from 0 to . Call this
:
Repeating this process gives
:
and this can be extended arbitrarily.
The
Cauchy formula for repeated integration The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress ''n'' antidifferentiations of a function into a single integral (cf. Cauchy's formula).
Scalar case
Let ''f'' be a continuous function on the r ...
, namely
:
leads in a straightforward way to a generalization for real .
Using the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
to remove the discrete nature of the factorial function gives us a natural candidate for fractional applications of the integral operator.
:
This is in fact a well-defined operator.
It is straightforward to show that the operator satisfies
:
This relationship is called the semigroup property of fractional
differintegral
In fractional calculus, an area of mathematical analysis, the differintegral (sometime also called the derivigral) is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by ...
operators. Unfortunately, the comparable process for the derivative operator is significantly more complex, but it can be shown that is neither
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
nor
additive
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with f ...
in general.
Fractional integrals
Riemann–Liouville fractional integral
The classical form of fractional calculus is given by the
Riemann–Liouville integral
In mathematics, the Riemann–Liouville integral associates with a real function f: \mathbb \rightarrow \mathbb another function of the same kind for each value of the parameter . The integral is a manner of generalization of the repeated antide ...
, which is essentially what has been described above. The theory of fractional integration for
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
s (therefore including the "boundary condition" of repeating after a period) is given by the
Weyl integral
In mathematics, the Weyl integral (named after Hermann Weyl) is an operator defined, as an example of fractional calculus, on functions ''f'' on the unit circle having integral 0 and a Fourier series. In other words there is a Fourier series for ...
. It is defined on
Fourier series, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
whose integrals evaluate to zero). The Riemann–Liouville integral exists in two forms, upper and lower. Considering the interval , the integrals are defined as
:
:
Where the former is valid for and the latter is valid for .
By contrast the
Grünwald–Letnikov derivative
In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from P ...
starts with the derivative instead of the integral.
Hadamard fractional integral
The ''Hadamard fractional integral'' was introduced by
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...
and is given by the following formula,
:
Atangana–Baleanu fractional integral
The Atangana–Baleanu fractional integral of a continuous function is defined as:
:
Fractional derivatives
Unlike classical Newtonian derivatives, fractional derivatives can be defined in a variety of different ways that often do not all lead to the same result even for smooth functions. Some of these are defined via a fractional integral. Because of the incompatibility of definitions, it is frequently necessary to be explicit about which definition is used.
Riemann–Liouville fractional derivative
The corresponding derivative is calculated using Lagrange's rule for differential operators. Computing th order derivative over the integral of order , the order derivative is obtained. It is important to remark that is the smallest integer greater than (that is, ). Similar to the definitions for the Riemann–Liouville integral, the derivative has upper and lower variants.
:
:
Caputo fractional derivative
Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by
Michele Caputo
Michele (), is an Italian male given name, akin to the English male name Michael (given name), Michael.
Michele (pronounced ), is also an English female given name that is derived from the French Michèle. It is a variant spelling of the more c ...
in his 1967 paper. In contrast to the Riemann–Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Caputo's definition is illustrated as follows, where again :
:
There is the Caputo fractional derivative defined as:
: