For example, one may ask for a meaningful interpretation of
as an analogue of the functional square root for the differentiation operator, that is, 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 a linear operator
for every real number in such a way that, when takes an integer value , it coincides with the usual -fold differentiation if , and with the -th power of when .
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator is that the sets of operator powers defined in this way are continuoussemigroups with parameter , of which the original discrete semigroup of for integer is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional differential equations, also known as extraordinary differential equations, are a generalization of differential equations through the application of fractional calculus.
In applied mathematics and mathematical analysis, a fractional derivative is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to Guillaume de l'Hôpital by Gottfried Wilhelm Leibniz in 1695. Around the same time, Leibniz wrote to one of the Bernoulli brothers describing the similarity between the binomial theorem and the Leibniz rule for the fractional derivative of a product of two functions. Fractional calculus was introduced in one of Niels Henrik Abel's early papers where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized operation, and even the unified notation for differentiation and integration of arbitrary real order.
Independently, the foundations of the subject were laid by Liouville in a paper from 1832.
The autodidactOliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890. The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives and integrals.
Nature of the fractional derivative
The -th derivative of a function at a point is a local property only when is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of at depends on all values of , even those far away from . Therefore, it is expected that the fractional derivative operation involves some sort of boundary conditions, involving information on the function further out.
The fractional derivative of a function of order is nowadays often defined by means of the Fourier or Mellin integral transforms.
A fairly natural question to ask is whether there exists a linear operatorH, 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
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.
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 of fractional integration for periodic functions (therefore including the "boundary condition" of repeating after a period) is given by 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 zero). The Riemann–Liouville integral exists in two forms, upper and lower. Considering the interval [a,b], the integrals are defined as
Where the former is valid for t > a and the latter is valid for t < b.
Unlike classical Newtonian derivatives, fractional derivatives can be defined in a variety of different ways that often do not all lead to the same result even for smooth functions. Some of these are defined via a fractional integral. Because of the incompatibility of definitions, it is frequently necessary to be explicit about which definition is used.
Fractional derivatives of a Gaussian, interpolating continuously between the function and its first derivative.
Riemann–Liouville fractional derivative
The corresponding derivative is calculated using Lagrange's rule for differential operators. Computing nth order derivative over the integral of order (n − α), the α order derivative is obtained. It is important to remark that n is the smallest integer greater than α (that is, n = ⌈α⌉). Similar to the definitions for the Riemann–Liouville integral, the derivative has upper and lower variants.
Caputo fractional derivative
Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele 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, where again n = ⌈α⌉:
There is the Caputo fractional derivative defined as:
which has the advantage that is zero when f(t) is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative. Moreover, there is the Caputo fractional derivative of distributed order defined as
where φ(ν) is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.
Caputo–Fabrizio fractional derivative
In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function of given by:
In 2016, Atangana and Baleanu suggested differential operators based on the generalized Mittag-Leffler function. The aim was to introduce fractional differential operators with non-singular nonlocal kernel. Their fractional differential operators are given below in Riemann–Liouville sense and Caputo sense respectively. For a function of given by 
If the function is continuous, the Atangana–Baleanu derivative in Riemann–Liouville sense is given by:
The kernel used in Atangana–Baleanu fractional derivative has some properties of a cumulative distribution function. For example, for all , the function is increasing on the real line, converges to in , and . Therefore, we have that, the function is the cumulative distribution function of a probability measure on the positive real numbers. The distribution is therefore defined, and any of its multiples, is called a Mittag-Leffler distribution of order . It is also very well-known that, all these probability distributions are absolutely continuous. In particular, the function Mittag-Leffler has a particular case , which is the exponential function, the Mittag-Leffler distribution of order is therefore an exponential distribution. However, for , the Mittag-Leffler distributions are heavy-tailed. Their Laplace transform is given by:
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:
When studying the redox behavior of a substrate in solution, a voltage is applied at an electrode surface to force electron transfer between electrode and substrate. The resulting electron transfer is measured as a current. The current depends upon the concentration of substrate at the electrode surface. As substrate is consumed, fresh substrate diffuses to the electrode as described by Fick's laws of diffusion. Taking the Laplace transform of Fick's second law yields an ordinary second-order differential equation (here in dimensionless form):
whose solution C(x,s) contains a one-half power dependence on s. Taking the derivative of C(x,s) and then the inverse Laplace transform yields the following relationship:
which relates the concentration of substrate at the electrode surface to the current. This relationship is applied in electrochemical kinetics to elucidate mechanistic behavior. For example, it has been used to study the rate of dimerization of substrates upon electrochemical reduction.
Groundwater flow problem
In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order. In these works, the classical Darcy law is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow.
Atangana and Kilicman extended the fractional advection dispersion equation to a variable order equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of a variational order derivative. The modified equation was numerically solved via the Crank–Nicolson method. The stability and convergence in numerical simulations showed that the modified equation is more reliable in predicting the movement of pollution in deformable aquifers than equations with constant fractional and integer derivatives
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 corresponds 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 the fractional derivative is the variable-order fractional derivative, α and β are changed into α(x, t) and β(x, t). Its applications in anomalous diffusion modeling can be found in the reference.
Generalizing PID controllers to use fractional orders can increase their degree of freedom. The new equation relating the control variableu(t) in terms of a measured error valuee(t) can be written as
where α and β are positive fractional orders and Kp, Ki, and Kd, all non-negative, denote the coefficients for the proportional, integral, and derivative terms, respectively (sometimes denoted P, I, and D).
Acoustic wave equations for complex media
The propagation of acoustical waves in complex media, such as in 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 Holm & Näsholm (2011) 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 Näsholm & Holm (2011b) and in the survey paper, as well as the Acoustic attenuation article. See Holm & Nasholm (2013) for a paper which compares fractional wave equations which model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.
Pandey and Holm gave a physical meaning to fractional differential equations by deriving them from physical principles and interpreting the fractional-order in terms of the parameters of the acoustical media, example in fluid-saturated granular unconsolidated marine sediments. Interestingly, Pandey and Holm derived Lomnitz's law in seismology and Nutting's law in non-Newtonian rheology using the framework of fractional calculus. Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.
Fractional Schrödinger equation in quantum theory
The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form:
Further, Δ = ∂2/∂r2 is the Laplace operator, and Dα is a scale constant with physical dimension[Dα] = J1 − α·mα·s−α = kg1 − α·m2 − α·sα − 2, (at α = 2, D2 = 1/2m for a particle of mass m), and the operator (−ħ2Δ)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.
Variable-order fractional Schrödinger equation
As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena:
where Δ = ∂2/∂r2 is the Laplace operator and the operator (−ħ2Δ)β(t)/2 is the variable-order fractional quantum Riesz derivative.
^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)
^Miller, Kenneth S. (1975). "The Weyl fractional calculus". In Ross, Bertram (ed.). Fractional Calculus and Its Applications. Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974. Lecture Notes in Mathematics. Vol. 457. Springer. pp. 80–89. doi:10.1007/bfb0067098. ISBN978-3-540-69975-0.
^Caputo, Michele; Fabrizio, Mauro (2016-01-01). "Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels". Progress in Fractional Differentiation and Applications. 2 (1): 1–11. doi:10.18576/pfda/020101. ISSN2356-9336.
^Erdélyi, Arthur (1950–1951). "On some functional transformations". Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino. 10: 217–234. MR0047818.
Articles regarding the history of fractional calculus
Ross, B. (1975). "A brief history and exposition of the fundamental theory of fractional calculus". Fractional Calculus and Its Applications. Fractional Calculus and Its Applications. Lecture Notes in Mathematics. Lecture Notes in Mathematics. Vol. 457. pp. 1–36. doi:10.1007/BFb0067096. ISBN978-3-540-07161-7.
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. ISBN978-0-12-525550-9.
Miller, Kenneth S.; Ross, Bertram, eds. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. ISBN978-0-471-58884-9.
Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN978-2-88124-864-1.
Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag Telos. ISBN978-3-211-82913-4.