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.
The Grunwald–Letnikov differintegralThe 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 differintegralIn 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 transformsedit
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.
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:
^Herrmann, Richard (2011). Fractional Calculus: An Introduction for Physicists. ISBN 9789814551076.
^See Herrmann, Richard (2011). Fractional Calculus: An Introduction for Physicists. p. 16. ISBN 9789814551076.
^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.
Miller, Kenneth S. (1993). Ross, Bertram (ed.). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley. ISBN 0-471-58884-9.
Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering. Vol. V. Academic Press. ISBN 0-12-525550-0.
Podlubny, Igor (1998). 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. Academic Press. ISBN 0-12-558840-2.
Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag. ISBN 3-211-82913-X.
Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press. ISBN 978-1-84816-329-4. Archived from the original on 2012-05-19.
Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Nonlinear Physical Science. Springer. ISBN 978-3-642-14003-7.
Uchaikin, V.V. (2012). Fractional Derivatives for Physicists and Engineers. Nonlinear Physical Science. Springer. Bibcode:2013fdpe.book.....U. ISBN 978-3-642-33910-3.
West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003). Physics of Fractal Operators. Springer Verlag. ISBN 0-387-95554-2.
External linksedit
MathWorld – Fractional calculus
MathWorld – Fractional derivative
Specialized journal: Fractional Calculus and Applied Analysis (1998-2014) and Fractional Calculus and Applied Analysis (from 2015)
Igor Podlubny's collection of related books, articles, links, software, etc.
Podlubny, I. (2002). "Geometric and physical interpretation of fractional integration and fractional differentiation" (PDF). Fractional Calculus and Applied Analysis. 5 (4): 367–386. arXiv:math.CA/0110241. Bibcode:2001math.....10241P. Archived from the original (PDF) on 2006-04-07. Retrieved 2004-05-18.
Zavada, P. (1998). "Operator of fractional derivative in the complex plane". Communications in Mathematical Physics. 192 (2): 261–285. arXiv:funct-an/9608002. Bibcode:1998CMaPh.192..261Z. doi:10.1007/s002200050299. S2CID 1201395.