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Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator

D = \frac{d}{dx} \,

and the integration operator J. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator (Usually J is used in favor of I to avoid confusion with other I-like glyphs and identities)

In this context powers refer to iterative application or composition, in the same sense that f2(x) = f(f(x)). In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of
For example, one may pose the question of interpreting meaningfully

\sqrt{D} = D^{1/2} \,

as a square root of the differentiation operator (an operator half iterate), i. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, an operator is a function which operates on (or modifies another function e. , an expression for some operator that when applied twice to a function will have the same effect as differentiation. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change More generally, one can look at the question of defining

D^s \,

for real-number values of s in such a way that when s takes an integer value n, the usual power of n-fold differentiation is recovered for n > 0, and the −nth power of J when n < 0. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

There are various reasons for looking at this question. One is that in this way the semigroup of powers Dn in the discrete variable n is seen inside a continuous semigroup (one hopes) with parameter s which is a real number. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that fraction is then a misnomer for the exponent, since it need not be rational, but the term fractional calculus has become traditional. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

Contents

Fractional derivative

As far as the existence of such a theory is concerned, the foundations of the subject were laid by Liouville in a paper from 1832. Joseph Liouville ( March 24 1809 &ndash September 8 1882) was a French Mathematician. The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In Mathematics, the Mellin transform is an Integral transform that may be regarded as the multiplicative version of the Two-sided Laplace transform An important point is that the fractional derivative at a point x is a local property only when a is an integer; in non-integral cases we cannot say that the fractional derivative at x of a function f depends only on the graph of f very near x, in the way that integer-power derivatives certainly do. Therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints To use a metaphor, the fractional derivative requires some peripheral vision. Peripheral vision is a part of vision that occurs outside the very center of gaze

For the history of the subject, see the thesis (in French): Stéphane Dugowson, Les différentielles métaphysiques (histoire et philosophie de la généralisation de l'ordre de dérivation), Thèse, Université Paris Nord (1994)

Heuristics

A fairly natural question to ask is, does there exist an operator H, or half-derivative, such that

H^2 f(x) = D f(x) = \frac{d}{dx} f(x) = f'(x) ?

It turns out that there is such an operator, and indeed for any a > 0, there exists an operator P such that

(P ^ a f)(x) = f'(x) \,,

or to put it another way, \frac{d^ny}{dx^n} is well-defined for all real values of n > 0. In Mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc A similar result applies to integration.

To delve into a little detail, start with the Gamma function \Gamma \,, which extends factorials to non-integer values. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function Definition The factorial function is formally defined by n!=\prod_{k=1}^n k This is defined such that

n! = \Gamma(n+1) \,.

Assuming a function f(x) that is well defined where x > 0, we can form the definite integral from 0 to x. Let's call this

( J f ) ( x ) = \int_0^x f(t) \; dt .

Repeating this process gives

( J^2 f ) ( x ) = \int_0^x ( J f ) ( t ) dt = \int_0^x \left( \int_0^t f(s) \; ds \right) \; dt,

and this can be extended arbitrarily.

The Cauchy formula for repeated integration, namely

(J^n f) ( x ) = { 1 \over (n-1) ! } \int_0^x (x-t)^{n-1} f(t) \; dt,

leads to a straightforward way to a generalization for real n. The Cauchy formula for repeated integration allows one to compress n Antidifferentiations of a function into a single integral

Simply using the Gamma function to remove the discrete nature of the factorial function (recalling that \Gamma\left(n+1\right)\,=\,n!, or equivalently \Gamma\left(n\right)\,=\,(n-1)!) gives us a natural candidate for fractional applications of the integral operator.

(J^\alpha f) ( x ) = { 1 \over \Gamma ( \alpha ) } \int_0^x (x-t)^{\alpha-1} f(t) \; dt

This is in fact a well-defined operator.

It can be shown that the J operator is both commutative and additive. That is,

(J^\alpha) (J^\beta) f = (J^\beta) (J^\alpha) f = (J^{\alpha+\beta} ) f = { 1 \over \Gamma ( \alpha + \beta) } \int_0^x (x-t)^{\alpha+\beta-1} f(t) \; dt

This property is called the Semi-Group property of fractional differintegral operators. In Mathematics, the differintegral is the combined differentiation / integration operator used in Fractional calculus. Unfortunately the comparable process for the derivative operator D is significantly more complex, but it can be shown that D is neither commutative, nor additive in general. In Mathematics, commutativity is the ability to change the order of something without changing the end result

Half derivative of a simple function

The half derivative (maroon curve) of the function y=x (blue curve) together with the first derivative (red curve).
The half derivative (maroon curve) of the function y=x (blue curve) together with the first derivative (red curve).

Let us assume that f(x) is a monomial of the form

f(x) = x^k\;.

The first derivative is as usual

f'(x) = {d \over dx } f(x) = k x^{k-1}\;.

Repeating this gives the more general result that

{d^a \over dx^a } x^k = { k! \over (k - a) ! } x^{k-a}\;,

Which, after replacing the factorials with the Gamma function, leads us to

{d^a \over dx^a } x^k = { \Gamma(k+1) \over \Gamma(k - a + 1) } x^{k-a}\;.

So, for example, the half-derivative of x is

{ d^{1 \over 2} \over dx^{1 \over 2} } x = { \Gamma(1 + 1) \over \Gamma ( 1 - {1 \over 2} + 1 ) } x^{1-{1 \over 2}} = { \Gamma( 2 ) \over \Gamma ( { 3 \over 2 } ) } x^{1 \over 2} = {2 \pi^{-{1 \over 2}}} x^{1 \over 2}\; = \frac{2\,x^{1 \over 2}}{\sqrt{\pi}}.

Repeating this process gives

{ d^{1 \over 2} \over dx^{1 \over 2} } {2 \pi^{-{1 \over 2}}} x^{1 \over 2} = {2 \pi^{-{1 \over 2}}} { \Gamma ( 1 + {1 \over 2} ) \over \Gamma ( {1 \over 2} - { 1 \over 2 } + 1 ) } x^{{1 \over 2} - {1 \over 2}} = {2 \pi^{-{1 \over 2}}} { \Gamma( { 3 \over 2 } ) \over \Gamma ( 1 ) } x^0 = { 1 \over \Gamma (1) } = 1\;,

which is indeed the expected result of

\left( \frac{d^{1/2}}{dx^{1/2}} \frac{d^{1/2}}{dx^{1/2}} \right) x = { d \over dx } x = 1 \,

This extension of the above differential operator need not be constrained only to real powers. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function For example, the (1+i)th derivative of the (1-i)th derivative yields the 2nd derivative. Also notice that setting negative values for a yields integrals.

Laplace transform

We can also come at the question via the Laplace transform. In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic Noting that

\mathcal L\left(t\mapsto\int_0^t f(\tau)\,d\tau\right)=\mathcal LJf=s\mapsto\frac1s(\mathcal Lf)(s)

and

\mathcal LJ^2f=s\mapsto\frac1s(\mathcal LJf)(s)=s\mapsto\frac1{s^2}(\mathcal Lf)(s)

etc. , we assert

J^\alpha f=\mathcal L^{-1}\left(s\mapsto s^{-\alpha}(\mathcal Lf)(s)\right).

For example

J^\alpha\left(t\mapsto t^k\right) =\mathcal L^{-1}\left(s\mapsto{\Gamma(k+1)\over s^{\alpha+k+1}}\right) =t\mapsto{\Gamma(k+1)\over\Gamma(\alpha+k+1)}t^{\alpha+k}

as expected. Indeed, given the convolution rule \mathcal L(f*g)=(\mathcal Lf)(\mathcal Lg) (and shorthanding p(x) = xα − 1 for clarity) we find that

J^\alpha f=\frac1{\Gamma(\alpha)}\mathcal L^{-1}\left(\left(\mathcal Lp\right)(\mathcal Lf)\right) =\frac1{\Gamma(\alpha)}(p*f) =x\mapsto\frac1{\Gamma(\alpha)}\int_0^xp(x-t)f(t)\,dt =x\mapsto\frac1{\Gamma(\alpha)}\int_0^x(x-t)^{\alpha-1}f(t)\,dt

which is what Cauchy gave us above. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and

Laplace transforms "work" on relatively few functions, but they are often useful for solving fractional differential equations.

Riemann-Liouville differintegral

The classical form of fractional calculus is given by the Riemann-Liouville differintegral, essentially what has been described above. The theory for periodic functions, therefore including the 'boundary condition' of repeating after a period, is the Weyl differintegral. In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable In Mathematics, the Weyl differentintegral is an operator defined as an example of Fractional calculus, on functions f on the Unit circle having It is defined on Fourier series, and requires the constant Fourier coefficient to vanish (so, applies to functions on the unit circle integrating to 0). In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions In Mathematics, a unit circle is

By contrast the Grunwald-Letnikov differintegral starts with the derivative. In Mathematics, the combined differentiation / integration operator used in Fractional calculus is called the Differintegral.

Functional calculus

In the context of functional analysis, functions f(D) more general than powers are studied in the functional calculus of spectral theory. For functional analysis as used in psychology see the Functional analysis (psychology article In Mathematics, a functional calculus is a theory allowing one to apply Mathematical functions to Mathematical operators The term was also used previously In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators The theory of pseudo-differential operators also allows one to consider powers of D. In Mathematical analysis a pseudo-differential operator is an extension of the concept of Differential operator. The operators arising are examples of singular integral operators; and the generalisation of the classical theory to higher dimensions is called the theory of Riesz potentials. In Mathematics, singular integrals are central to abstract Harmonic analysis and are intimately connected with the study of partial differential equations In Mathematics, a Riesz potential is a scalar function V_{\alpha}: \mathbb{R}^{n} \to \mathbb{R} n \geq 2 of the form V_{\alpha} So there are a number of contemporary theories available, within which fractional calculus can be discussed. See also Erdélyi-Kober operator, important in special function theory. Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the Mathematical analysis

For possible geometric and physical interpretation of fractional-order integration and fractional-order differentiation, see:

References

See also

External links

The University of Notre Dame du Lac (or simply Notre Dame) (ˌnoʊtɚˈdeɪm is a private Roman Catholic Research university located in
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