In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by

is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.

Standard definitions

The four most common forms are:

  • The Riemann–Liouville differintegral
    This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the Cauchy formula for repeated integration to arbitrary order. Here, .
  • The Grunwald–Letnikov differintegral
    The Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot.
  • The Weyl differintegral
    This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period.
  • The Caputo differintegral
    In opposite to the Riemann-Liouville differintegral, Caputo derivative of a constant is equal to zero. Moreover, a form of the Laplace transform allows to simply evaluate the initial conditions by computing finite, integer-order derivatives at point .

Definitions via transforms

The definitions of fractional derivatives given by Liouville, Fourier, and Grunwald and Letnikov coincide.[1] They can be represented via Laplace, Fourier transforms or via Newton series expansion.

Recall the continuous Fourier transform, here denoted :

Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:

So,

which generalizes to

Under the bilateral Laplace transform, here denoted by and defined as , differentiation transforms into a multiplication

Generalizing to arbitrary order and solving for , one obtains

Representation via Newton series is the Newton interpolation over consecutive integer orders:

For fractional derivative definitions described in this section, the following identities hold:

[2]

Basic formal properties

  • Linearity rules
  • Zero rule
  • Product rule

In general, composition (or semigroup) rule is a desirable property, but is hard to achieve mathematically and hence is not always completely satisfied by each proposed operator;[3] this forms part of the decision making process on which one to choose:

  • (ideally)
  • (in practice)

See also

References

  1. Herrmann, Richard (2011). Fractional Calculus: An Introduction for Physicists. ISBN 9789814551076.
  2. See Herrmann, Richard (2011). Fractional Calculus: An Introduction for Physicists. p. 16. ISBN 9789814551076.
  3. See Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). "2. Fractional Integrals and Fractional Derivatives §2.1 Property 2.4". Theory and Applications of Fractional Differential Equations. Elsevier. p. 75. ISBN 9780444518323.
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