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Encyclopedia > Fractional calculus

In mathematics, fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator Euclid, detail from The School of Athens by Raphael. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

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

and the integration operator J. (Usually not I, to avoid confusion with other I-like glyphs, or identities; but J must not be confused with Bessel functions, which often come up in study of differential equations.) In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ... In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

In this context powers refer to iterative application, in the same sense that f²(x) = f(f(x)).
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.e., an expression for some operator that when applied twice to a function will have the same effect as differentiation. More generally, one can look at the question of defining In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... The half iterate of a function f (denoted by f1/2) is any function which satisfies the following: f1/2(f1/2(x)) = f(x) The half iterate is a special case of fractional iteration of functions. ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ...

$D^s ,$

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 consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...

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. 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 semigroup is a set with an associative binary operation on it. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

1 Fractional derivative
2 Heuristics
3 Half derivative of a simple function
4 Laplace transform
5 Riemann-Liouville differintegral
6 Functional calculus
7 References
8 See also
9 External links

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. The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms. 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. To use a metaphor, the fractional derivative requires some peripheral vision. Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... Peripheral vision is a part of vision that occurs outside the very center of gaze. ...

About 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. A similar result applies to integration. In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...

To delve into a little detail, start with the Gamma function $Gamma ,$ , which is defined such that The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

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

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 The Cauchy formula for repeated integration allows one to compress antidifferentiations of a function into a single integral. ...

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

leads to a straightforward way to a generalization for real n.

Simply using the Gamma function to remove the discrete nature of the factorial function 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. 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, the differintegral is the combined differentiation/integration operator used in fractional calculus. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... Look up Additive in Wiktionary, the free dictionary When used as a noun, additive refers to something that is introduced to a larger quantity of something else, usually to alter characteristics of the larger quantity. ...

Half derivative of a simple function

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 In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

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

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

${ d^{1 over 2} over dx^{1 over 2} } x = { Gamma(1 + 1) over Gamma ( 1 - {1 over 2} + 1 ) } x^{1-{1 over 2}} = { Gamma( 2 ) over Gamma ( { 3 over 2 } ) } x^{1 over 2} = {2 pi^{-{1 over 2}}} x^{1 over 2};.$

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

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

Laplace transform

One can also come at the question via the Laplace transform. Noting that In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...

$mathcal Lleft(tmapstoint_0^t f(tau),dtauright)=mathcal LJf=smapstofrac1s(mathcal Lf)(s)$

and

$mathcal LJ^2f=smapstofrac1s(mathcal LJf)(s)=smapstofrac1{s^2}(mathcal Lf)(s)$

etc., we assert

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

For example

$J^alphaleft(tmapsto t^kright)$ $=mathcal L^{-1}left(smapsto{Gamma(k+1)over s^{alpha+k+1}}right)$ $=tmapsto{Gamma(k+1)overGamma(alpha+k+1)}t^{alpha+k}$

as expected. Indeed, given the convolution rule $mathcal L(f*g)=(mathcal Lf)(mathcal Lg)$ (and shorthanding $p (x) = x α - 1$ for clarity) we find that For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...

$J^alpha f=frac1{Gamma(alpha)}mathcal L^{-1}left(left(mathcal Lpright)(mathcal Lf)right)$ $=frac1{Gamma(alpha)}(p*f)$ $=xmapstofrac1{Gamma(alpha)}int_0^xp(x-t)f(t),dt$ $=xmapstofrac1{Gamma(alpha)}int_0^x(x-t)^{alpha-1}f(t),dt$

which is what Cauchy gave us above.

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. 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, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ... Illustration of a unit circle. ...

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, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly. ...

Functional calculus

In the context of functional analysis, functions f(D) more general than powers are studied in the functional calculus of spectral theory. The theory of pseudo-differential operators also allows one to consider powers of D. 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. So there are a number of contemporary theories available, within which fractional calculus can be discussed. See also Erdelyi-Kober operator, important in special function theory. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. ... In mathematics, several functions are important enough to deserve their own name. ...

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

Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002, 367–386. (available as original article, or preprint at Arxiv.org)

References

An Introduction to the Fractional Calculus and Fractional Differential Equations, by Kenneth S. Miller, Bertram Ross (Editor). Hardcover: 384 pages. Publisher: John Wiley & Sons; 1 edition (May 19, 1993). ISBN 0471588849
The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V), by Keith B. Oldham, Jerome Spanier. Hardcover. Publisher: Academic Press; (November 1974). ISBN 0125255500
Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications., (Mathematics in Science and Engineering, vol. 198), by Igor Podlubny. Hardcover. Publisher: Academic Press; (October 1998) ISBN 0125588402
Fractals and Fractional Calculus in Continuum Mechanics, by A. Carpinteri (Editor), F. Mainardi (Editor). Paperback: 348 pages. Publisher: Springer-Verlag Telos; (January 1998). ISBN 321182913X
Physics of Fractal Operators, by Bruce J. West, Mauro Bologna, Paolo Grigolini. Hardcover: 368 pages. Publisher: Springer Verlag; (January 14, 2003). ISBN 0387955542
Fractional Calculus and the Taylor-Riemann Series, Rose-Hulman Undergrad. J. Math. Vol.6(1) (2005).

External links

MathWorld - Fractional calculus
MathWorld - Fractional derivative
Specialized journal: Fractional Calculus and Applied Analysis
[1]
[2]
Igor Podlubny's collection of related books, articles, links, software, etc.
[3]
History, Definitions, and Applications for the Engineer (PDF), by Adam Loverro, University of Notre Dame
Fractional Calculus Modelling Organisation

Category: Mathematical analysis PDF is an abbreviation with several meanings: Portable Document Format Post-doctoral fellowship Probability density function There also is an electronic design automation company named PDF Solutions. ... The University of Notre Dame is a leading Roman Catholic institution of higher learning located immediately northeast of South Bend, Indiana, United States. ...

Results from FactBites:

Fractional calculus - Wikipedia, the free encyclopedia (945 words)
	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.
	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.
	The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms.

Differintegral - Wikipedia, the free encyclopedia (347 words)
	In mathematics, the differintegral is the combined differentiation/integration operator used in fractional calculus.
	The operator does not define a seperate function, but is a notation style for taking both the fractional derivative and the fractional integral of the same expression.
	See the page on fractional calculus for the general context.

More results at FactBites »

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