Home > Fractional calculus

D = d/dx

and the integration operator I. By powers we refer to iteration, for example in the sense that f²(x) = f(f(x)). For example, one may pose the question of interpreting meaningfully

√D = D^1/2

as a square root of the differentiation operator, qua operator (an operator half iterate). That means, 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 I when n < 0.

There are various possible 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 fractional calculus name 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 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, involving information on the function further out. To use a metaphor, the fractional derivative requires some peripheral vision.

The classical form of fractional calculus is given by the Riemann-Liouville differintegralIn 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. It is noted: : and i. The theory for periodic functionIn mathematics, a periodic function is a function that repeats its values, after adding some definite period to the variable. Everyday examples are seen when the variable is time for instance the hands of a clock or the phases of the moon show periodic bes, therefore including the 'boundary condition' of repeating after a period, is the Weyl differintegral . It is defined on Fourier seriesIn mathematics, a Fourier series named in honor of Joseph Fourier ( 1768- 1830), is a representation of a periodic function (often taken to have period 2π in a sense, the simplest case) as a sum of periodic functions of the form : which are harmonics o, and requires the constant Fourier coefficient to vanish (so, applies to functions on the unit circleIllustration of a unit circle. t is an angle measure. In mathematics, a unit circle is a circle with unit radius, i. a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, integrating to 0).

In the context of functional analysisFunctional analysis Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. It has its historical roots in the study of transformations such as the Fourier transform and in t, functions f(D) more general than powers are studied in the functional calculusIn mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression of spectral theoryLinear algebra Operator theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. The name was introduced by David Hilbert in his original formulation of Hilbert spa. The theory of pseudo-differential operatorIn mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator to include non-integer orders of differentiation (see Differintegral). Pseudo-differential operators are used extensively in the theory of parts also allows one to consider powers of D. The operators arising are examples of singular integral operator s; and the generalisation of the classical theory to higher dimensions is called the theory of Riesz potential s. 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:

• 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)