Fractional calculus - Wikipedia, the free encyclopedia

From: Roger Bagula (tftn_at_earthlink.net)
Date: 09/16/04 

Date: Thu, 16 Sep 2004 19:37:26 GMT

  Fractional calculus

       From Wikipedia, the free encyclopedia.

In mathematics <http://en.wikipedia.org/wiki/Mathematics>, fractional
calculus is a branch of mathematical analysis
<http://en.wikipedia.org/wiki/Mathematical_analysis>, studying the
possibility of taking real number
<http://en.wikipedia.org/wiki/Real_number> powers of the differential
operator <http://en.wikipedia.org/wiki/Differential_operator>

D = d/dx

and the integration operator I. For example, one may pose the question
of interpreting meaningfully

?D = D½

as a square root <http://en.wikipedia.org/wiki/Square_root> of the
differentiation operator, qua operator
<http://en.wikipedia.org/wiki/Operator> (an operator half iterate
<http://en.wikipedia.org/wiki/Half_iterate>). That means, some operator
that when applied twice to a function, will have the same effect as
differentiation <http://en.wikipedia.org/wiki/Differentiation>. More
generally, one can look at the question of defining

Ds

for real number values of s, in such a way that when s takes an integer
<http://en.wikipedia.org/wiki/Integer> value n, the usual power of
n-fold differentiation is recovered for n > 0, and the -nth power of I
when n < 0.

There are various possible reasons for looking at this question. One is
that in this way the semigroup <http://en.wikipedia.org/wiki/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
<http://en.wikipedia.org/wiki/Rational_number>, but the fractional
calculus name has become traditional.

As far as the existence of such a theory is concened, the foundations of
the subject were laid by Liouville
<http://en.wikipedia.org/wiki/Liouville> in a paper from 1832. The
fractional derivative of a function to order a is often now defined by
means of the Fourier transform
<http://en.wikipedia.org/wiki/Fourier_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 can't 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 <http://en.wikipedia.org/wiki/Boundary_condition>, involving
information on the function further out. To use a metaphor, the
fractional derivative requires some peripheral vision
<http://en.wikipedia.org/wiki/Peripheral_vision>.

The classical form of fractional calculus is given by the
Riemann-Liouville differintegral
<http://en.wikipedia.org/wiki/Riemann-Liouville_differintegral>. The
theory for periodic functions
<http://en.wikipedia.org/wiki/Periodic_function>, therefore including
the 'boundary condition' of repeating after a period, is the Weyl
differintegral
<http://en.wikipedia.org/w/wiki.phtml?title=Weyl_differintegral&action=edit>.
It is defined on Fourier series
<http://en.wikipedia.org/wiki/Fourier_series>, and requires the constant
Fourier coefficient to vanish (so, applies to functions on the unit
circle <http://en.wikipedia.org/wiki/Unit_circle> integrating to 0).

In the context of functional analysis
<http://en.wikipedia.org/wiki/Functional_analysis>, functions f(D) more
general than powers are studied in the functional calculus
<http://en.wikipedia.org/wiki/Functional_calculus> of spectral theory
<http://en.wikipedia.org/wiki/Spectral_theory>. The theory of
pseudo-differential operators
<http://en.wikipedia.org/wiki/Pseudo-differential_operator> also allows
one to consider powers of D. The operators arising are examples of
singular integral operators
<http://en.wikipedia.org/w/wiki.phtml?title=Singular_integral_operator&action=edit>;
and the generalisation of the classical theory to higher dimensions is
called the theory of Riesz potientials
<http://en.wikipedia.org/w/wiki.phtml?title=Riesz_potiential&action=edit>.
So there are a number of contemporary theories available, within which
fractional calculus can be discussed.

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 <http://www.diogenes.bg/fcaa/>
      (http://www.diogenes.bg/fcaa/), vol. 5, no. 4, 2002, 367386.
      (available as original article
      <http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf>
      (http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf), or preprint at
      Arxiv.org <http://arxiv.org/abs/math.CA/0110241>
      (http://arxiv.org/abs/math.CA/0110241))

See also:

    * differintegral <http://en.wikipedia.org/wiki/Differintegral>

http://en.wikipedia.org/wiki/Fractional_calculus 

-- 
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/