Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator
and of the integration operator The symbol is commonly used instead of the intuitive in order to avoid confusion with other concepts identified by similar glyphs, such as identities.
and developing a calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator to a function that is, repeatedly composing with itself, as in
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 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 differentiation if and with the 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 continuous semigroups 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.
Historical notes 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 Johann Bernoulli about derivatives of "general order". In the correspondence between Leibniz and John Wallis in 1697, Wallis's infinite product for is discussed. Leibniz suggested using differential calculus to achieve this result. Leibniz further used the notation to denote the derivative of order .
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 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...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) Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890.For a historical review of the subject up to the beginning of the 20th century, see: 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.
Computing the fractional integral Let be a function defined for . Form the definite integral from 0 to . 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 : using the gamma function to remove the discrete nature of the factorial function gives us a natural candidate for applications of the fractional integral operator as
This is in fact a well-defined operator.
It is straightforward to show that the operator satisfies
This relationship is called the semigroup property of fractional differintegral operators.
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 , the integrals are defined as
Where the former is valid for and the latter is valid for .
It has been suggested that the integral on the positive real axis (i.e. ) would be more appropriately named the Abel–Riemann integral, on the basis of history of discovery and use, and in the same vein the integral over the entire real line be named Liouville–Weyl integral.
By contrast the Grünwald–Letnikov derivative starts with the derivative instead of the integral.
Hadamard fractional integral The Hadamard fractional integral was introduced by Jacques Hadamard and is given by the following formula,
Atangana–Baleanu fractional integral (AB fractional integral) The Atangana–Baleanu fractional integral of a continuous function is defined as:
Fractional derivatives
Unfortunately, the comparable process for the derivative operator is significantly more complicated, but it can be shown that is neither commutative nor additive in general.
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.
Riemann–Liouville fractional derivative The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the th order derivative, the th order derivative of the integral of order is computed, where is the smallest integer greater than (that is, ). The Riemann–Liouville fractional derivative and integral has multiple applications, such as in case of solutions to the equation in the case of multiple systems such as the tokamak systems, and variable order fractional parameter. 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 :
There is the Caputo fractional derivative defined as:
which has the advantage that it is zero when 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 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:
where
Atangana–Baleanu fractional derivative 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 well-known that all these probability distributions are absolutely continuous. In particular, the Mittag-Leffler function 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:
This directly implies that, for the expectation is infinite. In addition, these distributions are geometric stable distributions.
Riesz derivative The Riesz derivative is defined as
where denotes the Fourier transform.
Conformable fractional derivative The conformable fractional derivative of a function of order is given by Unlike other definitions of the fractional derivative, the conformable fractional derivative obeys the product and quotient rule and has analogs to Rolle's theorem and the mean value theorem. However, this fractional derivative produces significantly different results compared to the Riemann-Liouville and Caputo fractional derivative. In 2020, Feng Gao and Chunmei Chi defined the improved Caputo-type conformable fractional derivative, which more closely approximates the behavior of the Caputo fractional derivative:where and are real numbers and . They also defined the improved Riemann-Liouville-type conformable fractional derivative to similarly approximate the Riemann-Liouville fractional derivative:
where and are real numbers and . Both improved conformable fractional derivatives have analogs to Rolle's theorem and the interior extremum theorem.
Other types Classical fractional derivatives include: Grünwald–Letnikov derivative Sonin–Letnikov derivative Liouville derivative Caputo derivative Hadamard derivative Marchaud derivative Riesz derivative Miller–Ross derivative Weyl derivative Erdélyi–Kober derivative -derivative New fractional derivatives include: Coimbra derivative Katugampola derivative Hilfer derivative Davidson derivative Chen derivative Caputo-Fabrizio derivative Atangana–Baleanu derivative
Novel fractional derivatives with nonsingular kernels, namely the Caputo-Fabrizio and Atangana–Baleanu derivatives, are subject to controversy in applied mathematics literature. Criticism is leveled against the validity and applications of these derivatives, with critics arguing these derivatives to be simple realizations of either integer derivatives or standard Caputo fractional derivatives, or incompatible with the fundamental theorem of fractional calculus.
On the other hand, subsequent work has argued that the claim that fractional derivatives with continuous (non-singular) kernels are too restrictive is a consequence of initializing fractional differential equations using single-time values instead of past histories. Since memory is an intrinsic property of fractional models, such initialization is physically and mathematically inconsistent. When the model past is treated correctly, the alleged incompatibilities of nonsingular kernels no longer arise. The paper concludes that the limitations attributed to operators such as the Caputo–Fabrizio and Atangana–Baleanu derivatives stem from modeling assumptions, not from the operators.