- "Fractional derivative" redirects to here.
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 or composition, in the same sense that f2(x) = f(f(x)). For example, one may ask the question of meaningfully interpreting
as a square root of the differentiation operator (an operator half iterate), i.e., an expression for 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
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 −nth power of J when n < 0.
The motivation behind this extension to the differential operator is that the semigroup of powers Da will form a continuous semigroup with parameter a, inside which the original discrete semigroup of Dn 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 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.
A fairly natural question to ask is whether there exists an 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 dny/dxn can be extended to all real values of 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
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.
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
For and , we obtain the half-derivative of the function as
Repeating this process yields
which is indeed the expected result of
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, 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
etc., we assert
as expected. Indeed, given the convolution rule
and shorthanding p(x) = xα − 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, 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 (so, applies to functions on the unit circle integrating 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  and is given by the following formula,
for t > a.
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. 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.
Erdélyi–Kober operator 
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 inegral. It is given by,
for x > a.
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 (Kober 1940), (Erdélyi 1950–51).
Fractional conservation of mass 
As described by Wheatcraft and Meerschaert (2008), 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 
Time-space fractional diffusion equation models 
Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models. 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.
Structural damping models 
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  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  and in the survey paper, as well as the acoustic attenuation article.
Fractional Schrödinger equation in quantum theory 
Here r is a 3-dimensional vector, ħ is the Planck constant, ψ(r, t) is the wavefunction, which is the quantum mechanical probability amplitude for the particle to have a given position r at any given time t, V(r, t) is a potential energy, and Δ = ∂2/∂r2 is the Laplace operator. Further, Dα is a scale constant with physical dimension [Dα] = erg1 − α·cmα·sec−α, (at α = 2, D2 = 1/2m, where m is a particle mass), and the operator (−ħ2Δ)α/2 is the 3-dimensional fractional quantum Riesz derivative defined by
Here the wave functions in the space ψ(r, t) and momentum φ(p, t) representations are related each other by the 3-dimensional Fourier transforms
The index α in the fractional Schrödinger equation is the Lévy index, 1 < α ≤ 2.
See also 
- Acoustic attenuation
- Differential equation
- Fractional dynamics
- Fractional Fourier transform
- Fractional Schrödinger equation
- Autoregressive fractionally integrated moving average
- 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)
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- 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
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- 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)
- 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.
- 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
- Introduction to fractional derivatives