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Fractional calculus

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In mathematics, fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator

:[D = \frac \, ]
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 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^ \,]
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

:[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.

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.

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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.

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 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] is well-defined for all real values of n > 0. A similar result applies to integration.

To delve into a little detail, start with the Gamma function [\Gamma \,], which 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 ) = \int_0^x (x-t)^ 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 ) = \int_0^x (x-t)^ 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^ ) f = \int_0^x (x-t)^ 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.

Half derivative of a simple function

The half derivative (maroon curve) of the function y=x.
Enlarge
The half derivative (maroon curve) of the function y=x.

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

:[ f(x) = x^k\;.]
The first derivative is as usual

:[ f'(x) = f(x) = k x^\;.]
Repeating this gives the more general result that

:[ x^k = x^\;,]
Which, after replacing the factorials with the Gamma function, leads us to

:[ x^k = x^\;.]
So, for example, the half-derivative of [x] is

:[ \over dx^ } x = + 1 ) } x^} = ) } x^ = }} x^\;.]
Repeating this process gives

:[ \over dx^ } }} x^ = }} ) \over \Gamma ( - + 1 ) } x^ = }} ) \over \Gamma ( 1 ) } x^0 = = 1\;,]

which is indeed the expected result of

:[ \left( \frac}} \frac}} \right) x = x = 1 \,]

Laplace transform

One can also come at the question via the Laplace transform. 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(\mathcal Lf)(s)]
etc., we assert
[J^\alpha f=\mathcal L^\left(s\mapsto s^(\mathcal Lf)(s)\right)].
For example
[J^\alpha\left(t\mapsto t^k\right)][=\mathcal L^\left(s\mapsto}\right)][=t\mapstot^]
as expected. Indeed, given the convolution rule [\mathcal L(f*g)=(\mathcal Lf)(\mathcal Lg)] (and shorthanding [p(x)=x^] for clarity) we find that
[J^\alpha f=\frac1\mathcal L^\left(\left(\mathcal Lp\right)(\mathcal Lf)\right)][=\frac1(p*f)][=x\mapsto\frac1\int_0^xp(x-t)f(t)\,dt][=x\mapsto\frac1\int_0^x(x-t)^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).

By contrast the Grunwald-Letnikov differintegral starts with the derivative.

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.

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

References

See also

External links

 

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