# Fractional calculus

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*"Fractional derivative" redirects here.*

**Fractional calculus** is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator

and the integration operator *J*. (Usually *J* is used instead of *I* to avoid confusion with other *I*-like glyphs and identities.)

In this context, the term *powers* refers to iterative application of a linear operator acting on a function, in some analogy to function composition acting on a variable,
e.g., *f* ^{2}(*x*) = *f*(*f*(*x*)). For example, one may ask the question of meaningfully interpreting

as an analog of the functional square root for the differentiation operator (an operator half iterated), i.e., an expression for some linear operator that when applied *twice* to any function will have the same effect as differentiation.

More generally, one can look at the question of defining the linear functional

for real-number values of *a* in such a way that when *a* takes an integer value, *n*, the usual power of *n*-fold differentiation is recovered for *n* > 0, and the −*n*th power of *J* when *n* < 0.

The motivation behind this extension to the differential operator is that the semigroup of powers *D*^{a} will form a *continuous* semigroup with parameter *a*, inside which the original *discrete* semigroup of *D ^{n}* for integer

*n*can be recovered as a subgroup. Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that

*fraction*is then a misnomer for the exponent

*a*, since it need not be rational; the use of the term

*fractional calculus*is merely conventional.

**Fractional differential equations** (also known as **extraordinary differential equations**) are a generalization of differential equations through the application of fractional calculus.

## Nature of the fractional derivative

An important point is that the fractional derivative at a point *x* is a *local property* only when *a* is an integer; in non-integer cases we cannot say that the fractional derivative at *x* of a function *f* depends only on values 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.

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 or Mellin integral transforms.^{[1]}

## Heuristics

A fairly natural question to ask is whether there exists a linear operator *H*, or **half-derivative**, such that

It turns out that there is such an operator, and indeed for any *a* > 0, there exists an operator *P* such that

or to put it another way, the definition of *d ^{n}y*/

*dx*can be extended to all real values of

^{n}*n*.

Let *f*(*x*) be a function defined for *x* > 0. Form the definite integral from 0 to *x*. Call this

Repeating this process gives

and this can be extended arbitrarily.

The Cauchy formula for repeated integration, namely

leads in a straightforward way to a generalization for real *n*.

Using the gamma function to remove the discrete nature of the factorial function gives us a natural candidate for fractional applications of the integral operator.

This is in fact a well-defined operator.

It is straightforward to show that the *J* operator satisfies

Proof where in the last step we exchanged the order of integration and pulled out the

*f*(*s*) factor from the*t*integration. Changing variables to*r*defined by*t*=*s*+ (*x*−*s*)*r*,The inner integral is the beta function which satisfies the following property:

Substituting back into the equation

Interchanging

*α*and*β*shows that the order in which the*J*operator is applied is irrelevant and completes the proof.

This relationship is called the semigroup property of fractional differintegral operators. Unfortunately the comparable process for the derivative operator *D* is significantly more complex, but it can be shown that *D* is neither commutative nor additive in general.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## Fractional derivative of a basic power function

Let us assume that *f*(*x*) is a monomial of the form

The first derivative is as usual

Repeating this gives the more general result that

Which, after replacing the factorials with the gamma function, leads us to

For and , we obtain the half-derivative of the function as

Repeating this process yields

which is indeed the expected result of

For negative integer power k, the gamma function is undefined and we have to use the following relation:^{[2]}

This extension of the above differential operator need not be constrained only to real powers. For example, the (1 + *i*)th derivative of the (1 − *i*)th derivative yields the 2nd derivative. Also notice that setting negative values for *a* yields integrals.

For a general function *f*(*x*) and 0 < *α* < 1, the complete fractional derivative is

For arbitrary *α*, since the gamma function is undefined for arguments whose real part is a negative integer and whose imaginary part is zero, it is necessary to apply the fractional derivative after the integer derivative has been performed. For example,

## Laplace transform

We can also come at the question via the Laplace transform. Noting that

and

etc., we assert

For example

as expected. Indeed, given the convolution rule

and shorthanding *p*(*x*) = *x*^{α−1} for clarity, we find that

which is what Cauchy gave us above.

Laplace transforms "work" on relatively few functions, but they *are* often useful for solving fractional differential equations.

## Fractional integrals

### Riemann–Liouville fractional integral

The classical form of fractional calculus is given by the Riemann–Liouville integral, which is essentially what has been described above. The theory for periodic functions (therefore including the 'boundary condition' of repeating after a period) is the Weyl integral. It is defined on Fourier series, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the unit circle whose integrals evaluate to 0).

By contrast the Grünwald–Letnikov derivative starts with the derivative instead of the integral.

### Hadamard fractional integral

The *Hadamard fractional integral* is introduced by J. Hadamard ^{[3]} and is given by the following formula,

## Fractional derivatives

Not like classical Newtonian derivatives, a fractional derivative is defined via a fractional integral.

### Riemann–Liouville fractional derivative

The corresponding derivative is calculated using Lagrange's rule for differential operators. Computing *n*-th order derivative over the integral of order (*n* − *α*), the *α* order derivative is obtained. It is important to remark that *n* is the nearest integer bigger than *α*.

### Caputo fractional derivative

There is another option for computing fractional derivatives; the Caputo fractional derivative. It was introduced by M. Caputo in his 1967 paper.^{[4]} In contrast to the Riemann Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Caputo's definition is illustrated as follows.

## Generalizations

### Erdélyi–Kober operator

The **Erdélyi–Kober operator** is an integral operator introduced by Arthur Erdélyi (1940).^{[5]} and Hermann Kober (1940)^{[6]} and is given by

which generalizes the Riemann fractional integral and the Weyl integral. A recent generalization is the following, which generalizes the *Riemann-Liouville fractional integral* and the *Hadamard fractional integral*. It is given by,^{[7]}

## Functional calculus

In the context of functional analysis, functions *f(D)* more general than powers are studied in the functional calculus of spectral theory. The theory of pseudo-differential operators also allows one to consider powers of *D*. The operators arising are examples of singular integral operators; and the generalisation of the classical theory to higher dimensions is called the theory of Riesz potentials. So there are a number of contemporary theories available, within which *fractional calculus* can be discussed. See also Erdélyi–Kober operator, important in special function theory Template:Harv, Template:Harv.

## Applications

### Fractional conservation of mass

As described by Wheatcraft and Meerschaert (2008),^{[8]} a fractional conservation of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is:

### Fractional advection dispersion equation

This equation has been shown useful for modeling contaminant flow in heterogenous porous media.^{[9]}^{[10]}^{[11]}

### Time-space fractional diffusion equation models

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.^{[12]}^{[13]} The time derivative term is corresponding to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation can be written as

A simple extension of fractional derivative is the variable-order fractional derivative, the *α*, *β* are changed into *α*(*x*, *t*), *β*(*x*, *t*). Its applications in anomalous diffusion modeling can be found in reference.^{[14]}

### Structural damping models

Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.^{[15]}

### Acoustical wave equations for complex media

The propagation of acoustical waves in complex media, e.g. biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:

See also ^{[16]} and the references therein. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This link is further described in ^{[17]} and in the survey paper,^{[18]} as well as the acoustic attenuation article. See ^{[19]} for a recent paper which compares fractional wave equations which model power-law attenuation.

### Fractional Schrödinger equation in quantum theory

The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics discovered by Nick Laskin,^{[20]} has the following form:^{[21]}

where the solution of the equation is the wavefunction *ψ*(**r**, *t*) - the quantum mechanical probability amplitude for the particle to have a given position vector **r** at any given time *t*, and *ħ* is the reduced Planck constant. The potential energy function *V*(**r**, *t*) depends on the system.

Further, Δ = {{ safesubst:#invoke:Unsubst||$B=∂^{2}/∂**r**^{2}}} is the Laplace operator, and *D _{α}* is a scale constant with physical dimension [

*D*] = erg

_{α}^{1 − α}·cm

^{α}·sec

^{−α}, (at

*α*= 2,

*D*

_{2}= 1/2

*m*for a particle of mass

*m*), and the operator (−

*ħ*

^{2}Δ)

^{α/2}is the 3-dimensional fractional quantum Riesz derivative defined by

The index *α* in the fractional Schrödinger equation is the Lévy index, 1 < *α* ≤ 2.

## See also

- Acoustic attenuation
- Differintegral
- Differential equation
- Fractional dynamics
- Fractional Fourier transform
- Neopolarogram
- Fractional Schrödinger equation
- Autoregressive fractionally integrated moving average

## Notes

- ↑ For the history of the subject, see the thesis (in French): Stéphane Dugowson,
*Les différentielles métaphysiques*(*histoire et philosophie de la généralisation de l'ordre de dérivation*), Thèse, Université Paris Nord (1994) - ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ Hadamard, J.,
*Essai sur l'étude des fonctions données par leur développement de Taylor*, Journal of pure and applied mathematics, vol. 4, no. 8, pp. 101–186, 1892. - ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Katugampola, U.N., New Approach To A Generalized Fractional Integral, Appl. Math. Comput. Vol 218, Issue 3, 1 October 2011, pages 860–865
- ↑ Wheatcraft, S., Meerschaert, M., (2008). "Fractional Conservation of Mass." Advances in Water Resources 31, 1377–1381.
- ↑ Benson, D., Wheatcraft, S., Meerschaert, M., (2000). "Application of a fractional advection-dispersion equation." Water Resources Res 36, 1403–1412.
- ↑ Benson, D., Wheatcraft, S., Meerschaert, M., (2000). "The fractional-order governing equation of Lévy motion." Water Resources Res 36, 1413–1423.
- ↑ Benson, D., Schumer, R., Wheatcraft, S., Meerschaert, M., (2001). "Fractional dispersion, Lévy motion, and the MADE tracer tests." Transport Porous Media 42, 211–240.
- ↑ Metzler, R., Klafter, J., (2000). "The random walk's guide to anomalous diffusion: a fractional dynamics approach." Phys. Rep., 339, 1-77.
- ↑ Chen, W., Sun, H.G., Zhang, X., Korosak, D., (2010). "Anomalous diffusion modeling by fractal and fractional derivatives." Computers and Mathematics with Applications, 59(5), 1754-1758. [1]
- ↑ Sun, H.G., Chen, W., Chen, Y.Q., (2009). "Variable-order fractional differential operators in anomalous diffusion modeling." Physica A, 2009, 388: 4586-4592.[2]
- ↑ Nolte, Kempfle and Schäfer (2003). "Does a Real Material Behave Fractionally? Applications of Fractional Differential Operators to the Damped Structure Borne Sound in Viscoelastic Solids", Journal of Computational Acoustics (JCA), Volume 11, Issue 3.
- ↑ S. Holm and S. P. Näsholm, "A causal and fractional all-frequency wave equation for lossy media," Journal of the Acoustical Society of America, Volume 130, Issue 4, pp. 2195–2201 (October 2011)
- ↑ S. P. Näsholm and S. Holm, "Linking multiple relaxation, power-law attenuation, and fractional wave equations," Journal of the Acoustical Society of America, Volume 130, Issue 5, pp. 3038-3045 (November 2011).
- ↑ S. P. Näsholm and S. Holm, "On a Fractional Zener Elastic Wave Equation," Fract. Calc. Appl. Anal. Vol. 16, No 1 (2013), pp. 26-50, DOI: 10.2478/s13540-013--0003-1 Link to e-print
- ↑ Holm S., Näsholm, S. P., "Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography," Ultrasound Med. Biol., 40(4), pp. 695-703, DOI: 10.1016/j.ultrasmedbio.2013.09.033 Link to e-print
- ↑ N. Laskin, (2000), Fractional Quantum Mechanics and Lévy Path Integrals.
*Physics Letters*268A, 298-304. - ↑ N. Laskin, (2002), Fractional Schrödinger equation,
*Physical Review*E66, 056108 7 pages.*(also available online: http://arxiv.org/abs/quant-ph/0206098)*

## Further reading

*Fractional Integrals and Derivatives: Theory and Applications*, by Samko, S.; Kilbas, A.A.; and Marichev, O. Hardcover: 1006 pages. Publisher: Taylor & Francis Books. ISBN 2-88124-864-0*Theory and Applications of Fractional Differential Equations*, by Kilbas, A. A.; Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Elsevier, February 2006. ISBN 0-444-51832-0 (http://www.elsevier.com/wps/find/bookdescription.cws_home/707212/description#description)*An Introduction to the Fractional Calculus and Fractional Differential Equations*, by Kenneth S. Miller, Bertram Ross (Editor). Hardcover: 384 pages. Publisher: John Wiley & Sons; 1 edition (May 19, 1993). ISBN 0-471-58884-9*The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V)*, by Keith B. Oldham, Jerome Spanier. Hardcover. Publisher: Academic Press; (November 1974). ISBN 0-12-525550-0*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), by Igor Podlubny. Hardcover. Publisher: Academic Press; (October 1998) ISBN 0-12-558840-2- Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. by F. Mainardi, Imperial College Press, 2010. 368 pages.
- Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. by V.E. Tarasov, Springer, 2010. 450 pages.
- Fractional Derivatives for Physicists and Engineers by V.V. Uchaikin, Springer, Higher Education Press, 2012, 385 pages.
- Fractional Calculus - An Introduction for Physicists by R. Herrmann, World Scientific, Singapore 2014. 500 pages.
*Fractals and Fractional Calculus in Continuum Mechanics*, by A. Carpinteri (Editor), F. Mainardi (Editor). Paperback: 348 pages. Publisher: Springer-Verlag Telos; (January 1998). ISBN 3-211-82913-X*Physics of Fractal Operators*, by Bruce J. West, Mauro Bologna, Paolo Grigolini. Hardcover: 368 pages. Publisher: Springer Verlag; (January 14, 2003). ISBN 0-387-95554-2- Special Functions in Fractional Calculus and Related Fractional Differintegral Equations. y Hari M Srivastava, World Scientific, Singapore, 2014. 300 pages
- Fractional Calculus: Theory and Applications. by Varsha Daftardar-gejji, Narosa Publishing House, 2013. 222 pages
- Basic Theory of Fractional Differential Equations. by Y. Zhou, World Scientific, Singapore, 201).

## History of fractional calculus

- B. Ross, "A brief history and exposition of the fundamental theory of fractional calculus", in Fractional Calculus and Its Applications. Lecture Notes in Mathematics. Vol.457. (1975) 1-36.
- J. Tenreiro Machado, V. Kiryakova, F. Mainardi, "Recent history of fractional calculus", Communications in Nonlinear Science and Numerical Simulation. Vol.16. No.3. (2011) 1140–1153.
- L. Debnath, "A brief historical introduction to fractional calculus", International Journal of Mathematical Education in Science and Technology. Vol.35. No.4. (2004) 487-501.
- J.A. Tenreiro Machado, A.M.S.F. Galhano, J.J. Trujillo, "On development of fractional calculus during the last fifty years", Scientometrics. Vol.98. No.1. (2014) 577-582.
- J.A. Tenreiro Machado, A.M. Galhano, J.J. Trujillo, "Science metrics on fractional calculus development since 1966", Fractional Calculus and Applied Analysis. Vol.16. No.2. (2013) 479-500.

## External links

- Eric W. Weisstein. "Fractional Differential Equation." From MathWorld — A Wolfram Web Resource.
- MathWorld - Fractional calculus
- MathWorld - Fractional derivative
- Fractional Calculus at MathPages
- Specialized journal: Fractional Calculus and Applied Analysis
- Specialized journal: Fractional Differential Equations (FDE)
- Specialized journal: Communications in Fractional Calculus (ISSN 2218-3892)
- www.nasatech.com
- unr.edu (Broken Link)
- Igor Podlubny's collection of related books, articles, links, software, etc.
- GigaHedron - Richard Herrmann's collection of books, articles, preprints, etc.
- s.dugowson.free.fr
- History, Definitions, and Applications for the Engineer (PDF), by Adam Loverro, University of Notre Dame
- Fractional Calculus Modelling
- Introductory Notes on Fractional Calculus
- Pseudodifferential operators and diffusive representation in modeling, control and signal
- Power Law & Fractional Dynamics
- The CRONE (R) Toolbox, a Matlab and Simulink Toolbox dedicated to fractional calculus, which is freely downloadable