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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 four most common forms are:
- 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 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.
- This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period.
- 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
Recall the continuous Fourier transform, here denoted :
Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:
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 Dqf(t), one obtains
Basic formal properties
- 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. 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. 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.
- MathWorld – Fractional calculus
- MathWorld – Fractional derivative
- Specialized journal: Fractional Calculus and Applied Analysis (1998-2014) and Fractional Calculus and Applied Analysis (from 2015)
- Specialized journal: Fractional Differential Equations (FDE)
- Specialized journal: Communications in Fractional Calculus (ISSN 2218-3892)
- Specialized journal: Journal of Fractional Calculus and Applications (JFCA)
- Lorenzo, Carl F.; Hartley, Tom T. (2002). "Initialized Fractional Calculus". Information Technology. Tech Briefs Media Group.
- 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.
- 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.