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분수 미적분 개념 소개

Title: Fractional calculus URL Source: https://en.wikipedia.org/wiki/Fractional_calculus Published Time: 2003-05-19T14:32:07Z Markdown Content: **Fractional calculus** is a branch of [mathematical analysis](https://en.wikipedia.org/wiki/Mathematical_analysis "Mathematical analysis") that studies the several different possibilities of defining [real number](https://en.wikipedia.org/wiki/Real_number "Real number") powers or [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number") powers of the [differentiation](https://en.wikipedia.org/wiki/Derivative "Derivative") [operator](https://en.wikipedia.org/wiki/Operator_(mathematics) "Operator (mathematics)") ![Image 1: {\displaystyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6) ![Image 2: {\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1c59da7454059970283a77a9a765100ce6a1f1) and of the [integration](https://en.wikipedia.org/wiki/Integral "Integral") operator ![Image 3: {\displaystyle J}](https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053) [\[Note 1\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-1) ![Image 4: {\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d39490673c02687cc2d3df07b164dcca9636018) and developing a [calculus](https://en.wikipedia.org/wiki/Calculus "Calculus") for such operators generalizing the classical one. In this context, the term _powers_ refers to iterative application of a [linear operator](https://en.wikipedia.org/wiki/Linear_operator "Linear operator") ![Image 5: {\displaystyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6) to a [function](https://en.wikipedia.org/wiki/Function_(mathematics) "Function (mathematics)") ![Image 6: {\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61), that is, repeatedly [composing](https://en.wikipedia.org/wiki/Function_composition "Function composition") ![Image 7: {\displaystyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6) with itself, as in ![Image 8: {\displaystyle {\begin{aligned}D^{n}(f)&=(\underbrace {D\circ D\circ D\circ \cdots \circ D} _{n})(f)\\&=\underbrace {D(D(D(\cdots D} _{n}(f)\cdots ))).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02e6ee53a72e5ef49af5ad57ec7ea2ad0453d184) For example, one may ask for a meaningful interpretation of ![Image 9: {\displaystyle {\sqrt {D}}=D^{\scriptstyle {\frac {1}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1448413c33ca0cbd5b40f11df0587cdd13b4320b) as an analogue of the [functional square root](https://en.wikipedia.org/wiki/Functional_square_root "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](https://en.wikipedia.org/wiki/Derivative "Derivative"). More generally, one can look at the question of defining a linear operator ![Image 10: {\displaystyle D^{a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11984b2968ff1326b343a3401db1c5af97cfe92) for every real number ![Image 11: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) in such a way that, when ![Image 12: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) takes an [integer](https://en.wikipedia.org/wiki/Integer "Integer") value ![Image 13: {\displaystyle n\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1cf6a513f2062531d95dbb198944936f312982), it coincides with the usual ![Image 14: {\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)\-fold differentiation ![Image 15: {\displaystyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6) if ![Image 16: {\displaystyle n>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96), and with the ![Image 17: {\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)\-th power of ![Image 18: {\displaystyle J}](https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053) when ![Image 19: {\displaystyle n<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9f68fb8426c4411972241aac6c359e7812eaba). One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator ![Image 20: {\displaystyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6) is that the [sets](https://en.wikipedia.org/wiki/Set_(mathematics) "Set (mathematics)") of operator powers ![Image 21: {\displaystyle \{D^{a}\mid a\in \mathbb {R} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82ddd07e3e3cebebecbe6fd8759a86c5b6e19573) defined in this way are _continuous_ [semigroups](https://en.wikipedia.org/wiki/Semigroup "Semigroup") with parameter ![Image 22: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc), of which the original _discrete_ semigroup of ![Image 23: {\displaystyle \{D^{n}\mid n\in \mathbb {Z} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7103633b3d6d0991222294fbdae4ab5ace36e85) for integer ![Image 24: {\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b) is a [denumerable](https://en.wikipedia.org/wiki/Denumerable_set "Denumerable set") subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics. Fractional [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation"), also known as extraordinary differential equations,[\[1\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Zwillinger2014-2) are a generalization of differential equations through the application of fractional calculus. In [applied mathematics](https://en.wikipedia.org/wiki/Applied_mathematics "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'Hpital](https://en.wikipedia.org/wiki/Guillaume_de_l%27H%C3%B4pital "Guillaume de l'Hpital") by [Gottfried Wilhelm Leibniz](https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz "Gottfried Wilhelm Leibniz") in 1695.[\[2\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Derivative-3) Around the same time, Leibniz wrote to [Johann Bernoulli](https://en.wikipedia.org/wiki/Johann_Bernoulli "Johann Bernoulli") about derivatives of "general order".[\[3\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-:1-4) In the correspondence between Leibniz and [John Wallis](https://en.wikipedia.org/wiki/John_Wallis "John Wallis") in 1697, Wallis's infinite product for ![Image 25: {\displaystyle {\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55) is discussed. Leibniz suggested using differential calculus to achieve this result. Leibniz further used the notation ![Image 26: {\displaystyle {d}^{1/2}{y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78c02c39608ab4adde1a58aa662b62494371aad8) to denote the derivative of order ![Image 27: {\displaystyle {\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55).[\[3\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-:1-4) Fractional calculus was introduced in one of [Niels Henrik Abel](https://en.wikipedia.org/wiki/Niels_Henrik_Abel "Niels Henrik Abel")'s early papers[\[4\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-5) 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.[\[5\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-6) Independently, the foundations of the subject were laid by [Liouville](https://en.wikipedia.org/wiki/Liouville "Liouville") in a paper from 1832.[\[6\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-7)[\[7\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-8)[\[8\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-9) [Oliver Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside") introduced the practical use of [fractional differential operators](https://en.wikipedia.org/wiki/Operational_calculus "Operational calculus") in electrical transmission line analysis circa 1890.[\[9\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-10) 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.[\[10\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-11) Computing the fractional integral --------------------------------- \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=2 "Edit section: Computing the fractional integral")\] Let _f_(_x_) be a function defined for _x_ \> 0. Form the definite integral from 0 to x. Call this ![Image 28: {\displaystyle (Jf)(x)=\int _{0}^{x}f(t)\,dt\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a39a447696b8ce70c8cd8600133f878116656b9f) Repeating this process gives ![Image 29: {\displaystyle {\begin{aligned}\left(J^{2}f\right)(x)&=\int _{0}^{x}(Jf)(t)\,dt\\&=\int _{0}^{x}\left(\int _{0}^{t}f(s)\,ds\right)dt\,,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a28fbaeefa4ad370d51343035834f136c43f7ae) and this can be extended arbitrarily. The [Cauchy formula for repeated integration](https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration "Cauchy formula for repeated integration"), namely ![Image 30: {\displaystyle \left(J^{n}f\right)(x)={\frac {1}{(n-1)!}}\int _{0}^{x}\left(x-t\right)^{n-1}f(t)\,dt\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1fc6e89735bb0e11e01f2255bb03522bd5a5c4) leads in a straightforward way to a generalization for real n: using the [gamma function](https://en.wikipedia.org/wiki/Gamma_function "Gamma function") to remove the discrete nature of the factorial function gives us a natural candidate for applications of the fractional integral operator as ![Image 31: {\displaystyle \left(J^{\alpha }f\right)(x)={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}\left(x-t\right)^{\alpha -1}f(t)\,dt\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b50d2a48f75b14dda93d1d832b64169eedc4197) This is in fact a well-defined operator. It is straightforward to show that the J operator satisfies ![Image 32: {\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&=\left(J^{\beta }\right)\left(J^{\alpha }f\right)(x)\\&=\left(J^{\alpha +\beta }f\right)(x)\\&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-t\right)^{\alpha +\beta -1}f(t)\,dt\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5b3a3987bdf4997bb9ff0e5ed77be1725f0223) | Proof of this identity | | --- | | ![Image 33: {\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}(x-t)^{\alpha -1}\left(J^{\beta }f\right)(t)\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\int _{0}^{t}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}f(s)\,ds\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}f(s)\left(\int _{s}^{x}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}\,dt\right)\,ds\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d341e8a5e8e65cc98fb84efbdb6bbeba32235fd) 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_, ![Image 34: {\displaystyle \left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\left(\int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr\right)\,ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4e762156b4241d765e6042946c170fb8dc34b4) The inner integral is the [beta function](https://en.wikipedia.org/wiki/Beta_function "Beta function") which satisfies the following property: ![Image 35: {\displaystyle \int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr=B(\alpha ,\beta )={\frac {\Gamma (\alpha )\,\Gamma (\beta )}{\Gamma (\alpha +\beta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d7ce1da3589b058ffc6876da2ef32089fdc775) Substituting back into the equation: ![Image 36: {\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\,ds\\&=\left(J^{\alpha +\beta }f\right)(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/900cdbb1b27cddc46aefff3fe284d412fd0321ff) Interchanging and 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](https://en.wikipedia.org/wiki/Differintegral "Differintegral") operators. ### RiemannLiouville fractional integral \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=3 "Edit section: RiemannLiouville fractional integral")\] The classical form of fractional calculus is given by the [RiemannLiouville integral](https://en.wikipedia.org/wiki/Riemann%E2%80%93Liouville_integral "RiemannLiouville integral"), which is essentially what has been described above. The theory of fractional integration for [periodic functions](https://en.wikipedia.org/wiki/Periodic_function "Periodic function") (therefore including the "boundary condition" of repeating after a period) is given by the [Weyl integral](https://en.wikipedia.org/wiki/Weyl_integral "Weyl integral"). It is defined on [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series"), and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle") whose integrals evaluate to zero). The RiemannLiouville integral exists in two forms, upper and lower. Considering the interval \[_a_,_b_\], the integrals are defined as ![Image 37: {\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{-\alpha }}Df(t)&=\sideset {_{a}}{_{t}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau \\\sideset {_{t}}{_{b}^{-\alpha }}Df(t)&=\sideset {_{t}}{_{b}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{t}^{b}\left(\tau -t\right)^{\alpha -1}f(\tau )\,d\tau \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb153742f9f7427df87cc3696cf830833982eb2) Where the former is valid for _t_ \> _a_ and the latter is valid for _t_ < _b_.[\[11\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-12) It has been suggested[\[12\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Mainardi-13) that the integral on the positive real axis (i.e. ![Image 38: {\displaystyle a=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8)) would be more appropriately named the AbelRiemann integral, on the basis of history of discovery and use, and in the same vein the integral over the entire real line be named LiouvilleWeyl integral. By contrast the [GrnwaldLetnikov derivative](https://en.wikipedia.org/wiki/Gr%C3%BCnwald%E2%80%93Letnikov_derivative "GrnwaldLetnikov derivative") starts with the derivative instead of the integral. ### Hadamard fractional integral \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=4 "Edit section: Hadamard fractional integral")\] The _Hadamard fractional integral_ was introduced by [Jacques Hadamard](https://en.wikipedia.org/wiki/Jacques_Hadamard "Jacques Hadamard")[\[13\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-14) and is given by the following formula, ![Image 39: {\displaystyle \sideset {_{a}}{_{t}^{-\alpha }}{\mathbf {D} }f(t)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(\log {\frac {t}{\tau }}\right)^{\alpha -1}f(\tau ){\frac {d\tau }{\tau }},\qquad t>a\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/022f1350040ca31ff704e2809eeac34a1c216ea8) ### AtanganaBaleanu fractional integral (AB fractional integral) \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=5 "Edit section: AtanganaBaleanu fractional integral (AB fractional integral)")\] The AtanganaBaleanu fractional integral of a continuous function is defined as: ![Image 40: {\displaystyle \sideset {_{{\hphantom {A}}a}^{\operatorname {AB} }}{_{t}^{\alpha }}If(t)={\frac {1-\alpha }{\operatorname {AB} (\alpha )}}f(t)+{\frac {\alpha }{\operatorname {AB} (\alpha )\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b27768c8874c4fc99fe58bc7d51741a556598594) Fractional derivatives ---------------------- \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=6 "Edit section: Fractional derivatives")\] Unfortunately, the comparable process for the derivative operator D is significantly more complex, but it can be shown that D is neither [commutative](https://en.wikipedia.org/wiki/Commutative "Commutative") nor [additive](https://en.wikipedia.org/wiki/Additive_map "Additive map") in general.[\[14\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-15) 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. [![Image 41](https://upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Fractionalderivative.gif/220px-Fractionalderivative.gif)](https://en.wikipedia.org/wiki/File:Fractionalderivative.gif) Fractional derivatives of a Gaussian, interpolating continuously between the function and its first derivative ### RiemannLiouville fractional derivative \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=7 "Edit section: RiemannLiouville fractional derivative")\] The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the th order derivative, the nth order derivative of the integral of order (_n_ __) is computed, where n is the smallest integer greater than (that is, _n_ = __). The RiemannLiouville 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.[\[15\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Mostafanejad-16)[\[16\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Al-Raeei-17) Similar to the definitions for the RiemannLiouville integral, the derivative has upper and lower variants.[\[17\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-18) ![Image 42: {\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{n-\alpha }}If(t)\\\sideset {_{t}}{_{b}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{n-\alpha }}If(t)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45ac7dd211f711572b5a519ac76a5e5ea816cef8) ### Caputo fractional derivative \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=8 "Edit section: Caputo fractional derivative")\] Another option for computing fractional derivatives is the [Caputo fractional derivative](https://en.wikipedia.org/wiki/Caputo_fractional_derivative "Caputo fractional derivative"). It was introduced by [Michele Caputo](https://en.wikipedia.org/w/index.php?title=Michele_Caputo&action=edit&redlink=1 "Michele Caputo (page does not exist)") in his 1967 paper.[\[18\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-19) In contrast to the RiemannLiouville 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_ = __: ![Image 43: {\displaystyle \sideset {^{C}}{_{t}^{\alpha }}Df(t)={\frac {1}{\Gamma (n-\alpha )}}\int _{0}^{t}{\frac {f^{(n)}(\tau )}{\left(t-\tau \right)^{\alpha +1-n}}}\,d\tau .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1d0e67e4046b630424896df8d353ca8031bd1c) There is the Caputo fractional derivative defined as: ![Image 44: {\displaystyle D^{\nu }f(t)={\frac {1}{\Gamma (n-\nu )}}\int _{0}^{t}(t-u)^{(n-\nu -1)}f^{(n)}(u)\,du\qquad (n-1)<\nu <n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46b1a78fb1f0fb431ce8204decf418f3b369f595) which has the advantage that it 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 ![Image 45: {\displaystyle {\begin{aligned}\sideset {_{a}^{b}}{^{n}u}Df(t)&=\int _{a}^{b}\phi (\nu )\left[D^{(\nu )}f(t)\right]\,d\nu \\&=\int _{a}^{b}\left[{\frac {\phi (\nu )}{\Gamma (1-\nu )}}\int _{0}^{t}\left(t-u\right)^{-\nu }f'(u)\,du\right]\,d\nu \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50d42263d28b3965e20b9d49daadc8625ce5432e) where __(__) is a weight function and which is used to represent mathematically the presence of multiple memory formalisms. ### CaputoFabrizio fractional derivative \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=9 "Edit section: CaputoFabrizio fractional derivative")\] In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function ![Image 46: {\displaystyle f(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6) of ![Image 47: {\displaystyle C^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91) given by: ![Image 48: {\displaystyle \sideset {_{{\hphantom {C}}a}^{\text{CF}}}{_{t}^{\alpha }}Df(t)={\frac {1}{1-\alpha }}\int _{a}^{t}f'(\tau )\ e^{\left(-\alpha {\frac {t-\tau }{1-\alpha }}\right)}\ d\tau ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31c88582ef182d0b6f0961ce2ff39638a1537421) where ![Image 49: {\displaystyle a<0,\alpha \in (0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/191e796c1152367c593e26bced3b6bf24171e6b9).[\[19\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-20) ### AtanganaBaleanu fractional derivative \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=10 "Edit section: AtanganaBaleanu fractional derivative")\] In 2016, Atangana and Baleanu suggested differential operators based on the generalized [Mittag-Leffler function](https://en.wikipedia.org/wiki/Mittag-Leffler_function "Mittag-Leffler function") ![Image 50: {\displaystyle E_{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/236b689dce03c17d59033e746b4e17860242cfe4). The aim was to introduce fractional differential operators with non-singular nonlocal kernel. Their fractional differential operators are given below in RiemannLiouville sense and Caputo sense respectively. For a function ![Image 51: {\displaystyle f(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6) of ![Image 52: {\displaystyle C^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91) given by [\[20\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Algahtani2016-21)[\[21\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-doiserbia.nb.rs-22) ![Image 53: {\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(t)={\frac {\operatorname {AB} (\alpha )}{1-\alpha }}\int _{a}^{t}f'(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2238bc8633fea409c150e04ff7c08e4e4fe815b1) If the function is continuous, the AtanganaBaleanu derivative in RiemannLiouville sense is given by: ![Image 54: {\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(t)={\frac {\operatorname {AB} (\alpha )}{1-\alpha }}{\frac {d}{dt}}\int _{a}^{t}f(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/014c2d10b482e4b3265c9896dddd9b79ca20fd38) The kernel used in AtanganaBaleanu fractional derivative has some properties of a cumulative distribution function. For example, for all ![Image 55: {\displaystyle \alpha \in (0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea174265d33ba6eed47a00e6ebffedda57cb03e), the function ![Image 56: {\displaystyle E_{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/236b689dce03c17d59033e746b4e17860242cfe4) is increasing on the real line, converges to ![Image 57: {\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) in ![Image 58: {\displaystyle -\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1), and ![Image 59: {\displaystyle E_{\alpha }(0)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/292e416fd870311ff5a6c8f5ac4cc1c65ac5838a). Therefore, we have that, the function ![Image 60: {\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76faf9a5c1222228b6b2f50367598f42711d6ec8) 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](https://en.wikipedia.org/wiki/Mittag-Leffler_distribution "Mittag-Leffler distribution") of order ![Image 61: {\displaystyle \alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3). It is also very well-known that, all these probability distributions are [absolutely continuous](https://en.wikipedia.org/wiki/Absolute_continuity "Absolute continuity"). In particular, the function Mittag-Leffler has a particular case ![Image 62: {\displaystyle E_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26), which is the exponential function, the Mittag-Leffler distribution of order ![Image 63: {\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) is therefore an [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution "Exponential distribution"). However, for ![Image 64: {\displaystyle \alpha \in (0,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5df576f7940384416d1553ab063704d37bf99420), the Mittag-Leffler distributions are [heavy-tailed](https://en.wikipedia.org/wiki/Heavy-tailed_distribution "Heavy-tailed distribution"). Their Laplace transform is given by: ![Image 65: {\displaystyle \mathbb {E} (e^{-\lambda X_{\alpha }})={\frac {1}{1+\lambda ^{\alpha }}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0afa259794949bf4db5ea3d9023c0250df7a359) This directly implies that, for ![Image 66: {\displaystyle \alpha \in (0,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5df576f7940384416d1553ab063704d37bf99420), the expectation is infinite. In addition, these distributions are [geometric stable distributions](https://en.wikipedia.org/wiki/Geometric_stable_distribution "Geometric stable distribution"). The Riesz derivative is defined as ![Image 67: {\displaystyle {\mathcal {F}}\left\{{\frac {\partial ^{\alpha }u}{\partial \left|x\right|^{\alpha }}}\right\}(k)=-\left|k\right|^{\alpha }{\mathcal {F}}\{u\}(k),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f35d44d8d559713d031a60a8898412dd8cd33f10) where ![Image 68: {\displaystyle {\mathcal {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676) denotes the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform").[\[22\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-23)[\[23\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-24) ### Conformable fractional derivative \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=12 "Edit section: Conformable fractional derivative")\] The conformable fractional derivative of a function ![Image 69: {\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61) of order ![Image 70: {\displaystyle \alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3) is given by![Image 71: {\displaystyle T_{a}(f)(t)=\lim _{\epsilon \rightarrow 0}{\frac {f\left(t+\epsilon t^{1-\alpha }\right)-f(t)}{\epsilon }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d295c67461b2f182897df3db3cc99d2ec2886bf)Unlike other definitions of the fractional derivative, the conformable fractional derivative obeys the [product](https://en.wikipedia.org/wiki/Product_rule "Product rule") and [quotient rule](https://en.wikipedia.org/wiki/Quotient_rule "Quotient rule") has analogs to [Rolle's theorem](https://en.wikipedia.org/wiki/Rolle%27s_theorem "Rolle's theorem") and the [mean value theorem](https://en.wikipedia.org/wiki/Mean_value_theorem "Mean value theorem").[\[24\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-25)[\[25\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-:0-26) 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:[\[25\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-:0-26)![Image 72: {\displaystyle _{a}^{C}{\widetilde {T}}_{a}(f)(t)=\lim _{\epsilon \rightarrow 0}\left[(1-\alpha )(f(t)-f(a))+\alpha {\frac {f\left(t+\epsilon (t-a)^{1-\alpha }\right)-f(t)}{\epsilon }}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b734b1eff3248611a8427113297b5d399c716009)where ![Image 73: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) and ![Image 74: {\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560) are [real numbers](https://en.wikipedia.org/wiki/Real_numbers "Real numbers") and ![Image 75: {\displaystyle a<t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22cab004dff5b953f06658cda97e91bf56e55240). They also defined the improved Riemann-Liouville-type conformable fractional derivative to similarly approximate the Riemann-Liouville fractional derivative:[\[25\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-:0-26) ![Image 76: {\displaystyle _{a}^{RL}{\widetilde {T}}_{a}(f)(t)=\lim _{\epsilon \rightarrow 0}\left[(1-\alpha )f(t)+\alpha {\frac {f\left(t+\epsilon (t-a)^{1-\alpha }\right)-f(t)}{\epsilon }}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a7ac35d579ba046a3d5066622fe155b09d7af8)where ![Image 77: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) and ![Image 78: {\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560) are [real numbers](https://en.wikipedia.org/wiki/Real_numbers "Real numbers") and ![Image 79: {\displaystyle a<t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22cab004dff5b953f06658cda97e91bf56e55240). Both improved conformable fractional derivatives have analogs to Rolle's theorem and the [interior extremum theorem](https://en.wikipedia.org/wiki/Interior_extremum_theorem "Interior extremum theorem").[\[26\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-27) Classical fractional derivatives include: * [GrnwaldLetnikov derivative](https://en.wikipedia.org/wiki/Gr%C3%BCnwald%E2%80%93Letnikov_derivative "GrnwaldLetnikov derivative")[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28)[\[28\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Aslan2015-29) * SoninLetnikov derivative[\[28\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Aslan2015-29) * Liouville derivative[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * [Caputo derivative](https://en.wikipedia.org/wiki/Differintegral "Differintegral")[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * Hadamard derivative[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28)[\[29\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-30) * Marchaud derivative[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * Riesz derivative[\[28\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Aslan2015-29) * MillerRoss derivative[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * [Weyl derivative](https://en.wikipedia.org/wiki/Weyl_integral "Weyl integral")[\[30\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-31)[\[31\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-32)[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * [ErdlyiKober derivative](https://en.wikipedia.org/wiki/Erdelyi%E2%80%93Kober_operator "ErdelyiKober operator")[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * [![Image 80: {\displaystyle F^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81fb943af52f056cca6657ad9702de8fc4603f02)\-derivative](https://en.wikipedia.org/w/index.php?title=Fractal_calculus&action=edit&redlink=1 "Fractal calculus (page does not exist)")[\[32\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Ali-33) New fractional derivatives include: * Coimbra derivative[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * [Katugampola derivative](https://en.wikipedia.org/wiki/Katugampola_fractional_operators "Katugampola fractional operators")[\[33\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-34) * Hilfer derivative[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * Davidson derivative[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * Chen derivative[\[27\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-deOliveira2014-28) * [Caputo Fabrizio derivative](https://en.wikipedia.org/w/index.php?title=Caputo_Fabrizio_derivative&action=edit&redlink=1 "Caputo Fabrizio derivative (page does not exist)")[\[20\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Algahtani2016-21)[\[34\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-35) * AtanganaBaleanu derivative[\[20\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Algahtani2016-21)[\[21\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-doiserbia.nb.rs-22) The **Coimbra derivative** is used for physical modeling:[\[35\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-36) A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,[\[36\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-37)[\[37\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-38)[\[38\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-39)[\[39\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-40)[\[40\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-41)[\[41\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-42)[\[42\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-43) as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors[\[43\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-44)[\[44\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-45)[\[45\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-46)[\[46\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-47) For ![Image 81: {\displaystyle q(t)<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bbf2894b0a8cbcfcb8deae1d39475604208bdec) ![Image 82: {\displaystyle {\begin{aligned}^{\mathbb {C} }_{a}\mathbb {D} ^{q(t)}f(t)={\frac {1}{\Gamma [1-q(t)]}}\int _{0^{+}}^{t}(t-\tau )^{-q(t)}{\frac {d\,f(\tau )}{d\tau }}d\tau \,+\,{\frac {(f(0^{+})-f(0^{-}))\,t^{-q(t)}}{\Gamma (1-q(t))}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b86ef386ed041f1f311995e8445ae3966aae5c81) where the lower limit ![Image 83: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) can be taken as either ![Image 84: {\displaystyle 0^{-}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb153c3e2b753e4b1627d97a00817d6ecab67eca) or ![Image 85: {\displaystyle -\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1) as long as ![Image 86: {\displaystyle f(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6) is identically zero from or ![Image 87: {\displaystyle -\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1) to ![Image 88: {\displaystyle 0^{-}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb153c3e2b753e4b1627d97a00817d6ecab67eca). Note that this operator returns the correct fractional derivatives for all values of ![Image 89: {\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560) and can be applied to either the dependent function itself ![Image 90: {\displaystyle f(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6) with a variable order of the form ![Image 91: {\displaystyle q(f(t))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdb625dbe9e4a39380dcb82e6860769ff44ad9e) or to the independent variable with a variable order of the form ![Image 92: {\displaystyle q(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1f8079f76d0e8a89cf19db8fc43f34ec569d25).![Image 93: {\displaystyle ^{[1]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c97b3809485cd490e1a61475329776d6b0efee76) The Coimbra derivative can be generalized to any order,[\[47\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-48) leading to the Coimbra Generalized Order Differintegration Operator (GODO)[\[48\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-49) For ![Image 94: {\displaystyle q(t)<m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dac39f3efa2c564ef574faf91d7dfebbe07f7424) ![Image 95: {\displaystyle {\begin{aligned}^{\mathbb {\quad C} }_{\,\,-\infty }\mathbb {D} ^{q(t)}f(t)={\frac {1}{\Gamma [m-q(t)]}}\int _{0^{+}}^{t}(t-\tau )^{m-1-q(t)}{\frac {d^{m}f(\tau )}{d\tau ^{m}}}d\tau \,+\,\sum _{n=0}^{m-1}{\frac {({\frac {d^{n}f(t)}{dt^{n}}}|_{0^{+}}-{\frac {d^{n}f(t)}{dt^{n}}}|_{0^{-}})\,t^{n-q(t)}}{\Gamma [n+1-q(t)]}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c269b204f85974c9be81014f037ea5ef374a62c) where ![Image 96: {\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) is an integer larger than the larger value of ![Image 97: {\displaystyle q(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1f8079f76d0e8a89cf19db8fc43f34ec569d25) for all values of ![Image 98: {\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560). Note that the second (summation) term on the right side of the definition above can be expressed as ![Image 99: {\displaystyle {\begin{aligned}{\frac {1}{\Gamma [m-q(t)]}}\sum _{n=0}^{m-1}\{[{\frac {d^{n}\!f(t)}{dt^{n}}}|_{0^{+}}-{\frac {d^{n}\!f(t)}{dt^{n}}}|_{0^{-}}]\,t^{n-q(t)}\prod _{j=n+1}^{m-1}[j-q(t)]\}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ed12d6afff071f04afe1bc677ef43225ac4b82) so to keep the denominator on the positive branch of the Gamma (![Image 100: {\displaystyle \Gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)) function and for ease of numerical calculation. ### Nature of the fractional derivative \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=15 "Edit section: Nature of the fractional derivative")\] The ![Image 101: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)\-th derivative of a function ![Image 102: {\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61) at a point ![Image 103: {\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) is a _local property_ only when ![Image 104: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of ![Image 105: {\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61) at ![Image 106: {\displaystyle x=c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3642398a43ac55e2d858529b48f8e113682801d6) depends on all values of ![Image 107: {\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61), even those far away from ![Image 108: {\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455). Therefore, it is expected that the fractional derivative operation involves some sort of [boundary conditions](https://en.wikipedia.org/wiki/Boundary_condition "Boundary condition"), involving information on the function further out.[\[49\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-50) The fractional derivative of a function of order ![Image 109: {\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) is nowadays often defined by means of the [Fourier](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") or [Mellin](https://en.wikipedia.org/wiki/Mellin_transform "Mellin transform") integral transforms.\[_[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")_\] ### ErdlyiKober operator \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=17 "Edit section: ErdlyiKober operator")\] The [ErdlyiKober operator](https://en.wikipedia.org/wiki/Erd%C3%A9lyi%E2%80%93Kober_operator "ErdlyiKober operator") is an integral operator introduced by [Arthur Erdlyi](https://en.wikipedia.org/wiki/Arthur_Erd%C3%A9lyi "Arthur Erdlyi") (1940).[\[50\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-51) and [Hermann Kober](https://en.wikipedia.org/wiki/Hermann_Kober "Hermann Kober") (1940)[\[51\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-52) and is given by ![Image 110: {\displaystyle {\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0}^{x}\left(t-x\right)^{\alpha -1}t^{-\alpha -\nu }f(t)\,dt\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/672aacf66c86143469c4a65e4dad0f58810d20f5) which generalizes the [RiemannLiouville fractional integral](https://en.wikipedia.org/wiki/Fractional_calculus#Fractional_integrals) and the Weyl integral. Functional calculus ------------------- \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=18 "Edit section: Functional calculus")\] In the context of [functional analysis](https://en.wikipedia.org/wiki/Functional_analysis "Functional analysis"), functions _f_(_D_) more general than powers are studied in the [functional calculus](https://en.wikipedia.org/wiki/Functional_calculus "Functional calculus") of [spectral theory](https://en.wikipedia.org/wiki/Spectral_theorem "Spectral theorem"). The theory of [pseudo-differential operators](https://en.wikipedia.org/wiki/Pseudo-differential_operator "Pseudo-differential operator") also allows one to consider powers of D. The operators arising are examples of [singular integral operators](https://en.wikipedia.org/wiki/Singular_integral_operator "Singular integral operator"); and the generalisation of the classical theory to higher dimensions is called the theory of [Riesz potentials](https://en.wikipedia.org/wiki/Riesz_potential "Riesz potential"). So there are a number of contemporary theories available, within which _fractional calculus_ can be discussed. See also [ErdlyiKober operator](https://en.wikipedia.org/wiki/Erd%C3%A9lyi%E2%80%93Kober_operator "ErdlyiKober operator"), important in [special function](https://en.wikipedia.org/wiki/Special_function "Special function") theory ([Kober 1940](https://en.wikipedia.org/wiki/Fractional_calculus#CITEREFKober1940)), ([Erdlyi 19501951](https://en.wikipedia.org/wiki/Fractional_calculus#CITEREFErd%C3%A9lyi1950%E2%80%931951)). ### Fractional conservation of mass \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=20 "Edit section: Fractional conservation of mass")\] As described by Wheatcraft and Meerschaert (2008),[\[52\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-53) a fractional conservation of mass equation is needed to model fluid flow when the [control volume](https://en.wikipedia.org/wiki/Control_volume "Control volume") is not large enough compared to the scale of [heterogeneity](https://en.wikipedia.org/wiki/Heterogeneity "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: ![Image 111: {\displaystyle -\rho \left(\nabla ^{\alpha }\cdot {\vec {u}}\right)=\Gamma (\alpha +1)\Delta x^{1-\alpha }\rho \left(\beta _{s}+\phi \beta _{w}\right){\frac {\partial p}{\partial t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f29bebbff635e7c00e0a7ece9bdae1a487aca38) ### Electrochemical analysis \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=21 "Edit section: Electrochemical analysis")\] 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](https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion "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): ![Image 112: {\displaystyle {\frac {d^{2}}{dx^{2}}}C(x,s)=sC(x,s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5d8a846d25447fdca17a284b2f545c1c667fc9) 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: ![Image 113: {\displaystyle {\frac {d}{dx}}C(x,t)={\frac {d^{\scriptstyle {\frac {1}{2}}}}{dt^{\scriptstyle {\frac {1}{2}}}}}C(x,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2940ba8a22d51bded0831c72b35278cda255395b) which relates the concentration of substrate at the electrode surface to the current.[\[53\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-54) 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.[\[54\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-55) ### Groundwater flow problem \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=22 "Edit section: Groundwater flow problem")\] In 20132014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order.[\[55\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-56)[\[56\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-57) In these works, the classical [Darcy law](https://en.wikipedia.org/wiki/Darcy_law "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. ### Fractional advection dispersion equation \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=23 "Edit section: Fractional advection dispersion equation")\] This equation\[_[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")_\] has been shown useful for modeling contaminant flow in heterogenous porous media.[\[57\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-58)[\[58\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-59)[\[59\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-60) 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](https://en.wikipedia.org/w/index.php?title=Variational_order_derivative&action=edit&redlink=1 "Variational order derivative (page does not exist)"). The modified equation was numerically solved via the [CrankNicolson method](https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method "CrankNicolson 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[\[60\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Atangana2014a-61) ### Time-space fractional diffusion equation models \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=24 "Edit section: Time-space fractional diffusion equation models")\] Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.[\[61\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-62)[\[62\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-63) 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 ![Image 114: {\displaystyle {\frac {\partial ^{\alpha }u}{\partial t^{\alpha }}}=-K(-\Delta )^{\beta }u.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6adbc7788741db4e7e17b847400d2adaefe6b652) A simple extension of the fractional derivative is the variable-order fractional derivative, and are changed into __(_x_, _t_) and __(_x_, _t_). Its applications in anomalous diffusion modeling can be found in the reference.[\[60\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Atangana2014a-61)[\[63\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-64)[\[64\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-65) ### Structural damping models \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=25 "Edit section: Structural damping models")\] Fractional derivatives are used to model [viscoelastic](https://en.wikipedia.org/wiki/Viscoelastic "Viscoelastic") [damping](https://en.wikipedia.org/wiki/Damping "Damping") in certain types of materials like polymers.[\[12\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Mainardi-13) Generalizing [PID controllers](https://en.wikipedia.org/wiki/PID_controller "PID controller") to use fractional orders can increase their degree of freedom. The new equation relating the _control variable_ _u_(_t_) in terms of a measured _error value_ _e_(_t_) can be written as ![Image 115: {\displaystyle u(t)=K_{\mathrm {p} }e(t)+K_{\mathrm {i} }D_{t}^{-\alpha }e(t)+K_{\mathrm {d} }D_{t}^{\beta }e(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd7d26c2245e2d6a8227b48ee7ffcc9e770fa90) where and are positive fractional orders and _K_p, _K_i, and _K_d, all non-negative, denote the coefficients for the [proportional](https://en.wikipedia.org/wiki/Proportional_control "Proportional control"), [integral](https://en.wikipedia.org/wiki/Integral "Integral"), and [derivative](https://en.wikipedia.org/wiki/Derivative "Derivative") terms, respectively (sometimes denoted P, I, and D).[\[65\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-66) ### Acoustic wave equations for complex media \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=27 "Edit section: Acoustic wave equations for complex media")\] 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: ![Image 116: {\displaystyle \nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial t^{\alpha }}}\nabla ^{2}u-{\dfrac {\tau _{\epsilon }^{\beta }}{c_{0}^{2}}}{\dfrac {\partial ^{\beta +2}u}{\partial t^{\beta +2}}}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89aba7f1c6b3d5a29461e324dd057fb57db722eb) See also Holm & Nsholm (2011)[\[66\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-67) 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 Nsholm & Holm (2011b)[\[67\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-68) and in the survey paper,[\[68\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Nasholm2-69) as well as the _[Acoustic attenuation](https://en.wikipedia.org/wiki/Acoustic_attenuation "Acoustic attenuation")_ article. See Holm & Nasholm (2013)[\[69\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-HolmNasholm2014-70) 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.[\[70\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Holm2019-71) 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.[\[71\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Pandey2016-72) Interestingly, Pandey and Holm derived [Lomnitz's law](https://en.wikipedia.org/wiki/Cinna_Lomnitz "Cinna Lomnitz") in [seismology](https://en.wikipedia.org/wiki/Seismology "Seismology") and Nutting's law in [non-Newtonian rheology](https://en.wikipedia.org/wiki/Non-Newtonian_fluid "Non-Newtonian fluid") using the framework of fractional calculus.[\[72\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-73) Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.[\[71\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-Pandey2016-72) ### Fractional Schrdinger equation in quantum theory \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=28 "Edit section: Fractional Schrdinger equation in quantum theory")\] The [fractional Schrdinger equation](https://en.wikipedia.org/w/index.php?title=Fractional_Schr%C3%B6dinger_equation&action=edit&redlink=1 "Fractional Schrdinger equation (page does not exist)"), a fundamental equation of [fractional quantum mechanics](https://en.wikipedia.org/w/index.php?title=Fractional_quantum_mechanics&action=edit&redlink=1 "Fractional quantum mechanics (page does not exist)"), has the following form:[\[73\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-74)[\[74\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-75) ![Image 117: {\displaystyle i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}=D_{\alpha }\left(-\hbar ^{2}\Delta \right)^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9306359d8ae6c87ae1144f721d44ea1678940f38) where the solution of the equation is the [wavefunction](https://en.wikipedia.org/wiki/Wavefunction "Wavefunction") __(**r**, _t_) the quantum mechanical [probability amplitude](https://en.wikipedia.org/wiki/Probability_amplitude "Probability amplitude") for the particle to have a given [position vector](https://en.wikipedia.org/wiki/Position_vector "Position vector") **r** at any given time t, and is the [reduced Planck constant](https://en.wikipedia.org/wiki/Reduced_Planck_constant "Reduced Planck constant"). The [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy") function _V_(**r**, _t_) depends on the system. Further, ![Image 118: {\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7f5138716ff9f4e227d3174a3dee47a859c7b3) is the [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator"), and D is a scale constant with physical [dimension](https://en.wikipedia.org/wiki/Dimensional_analysis "Dimensional analysis") \[_D_\] = J1 __m__s__ = kg1 __m2 __s__ 2, (at __ = 2, ![Image 119: {\textstyle D_{2}={\frac {1}{2m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14fe9bd3f2a4a239a67b2951311f0572bf71d47c) for a particle of mass m), and the operator (__2)__/2 is the 3-dimensional fractional quantum Riesz derivative defined by ![Image 120: {\displaystyle (-\hbar ^{2}\Delta )^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)={\frac {1}{(2\pi \hbar )^{3}}}\int d^{3}pe^{{\frac {i}{\hbar }}\mathbf {p} \cdot \mathbf {r} }|\mathbf {p} |^{\alpha }\varphi (\mathbf {p} ,t)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad70321f5d5c8212f31b81e45bff4184bd76dfd) The index in the fractional Schrdinger equation is the Lvy index, 1 < __ 2. #### Variable-order fractional Schrdinger equation \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=29 "Edit section: Variable-order fractional Schrdinger equation")\] As a natural generalization of the [fractional Schrdinger equation](https://en.wikipedia.org/w/index.php?title=Fractional_Schr%C3%B6dinger_equation&action=edit&redlink=1 "Fractional Schrdinger equation (page does not exist)"), the variable-order fractional Schrdinger equation has been exploited to study fractional quantum phenomena:[\[75\]](https://en.wikipedia.org/wiki/Fractional_calculus#cite_note-76) ![Image 121: {\displaystyle i\hbar {\frac {\partial \psi ^{\alpha (\mathbf {r} )}(\mathbf {r} ,t)}{\partial t^{\alpha (\mathbf {r} )}}}=\left(-\hbar ^{2}\Delta \right)^{\frac {\beta (t)}{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57f444c1da75e4b2ab19dad1ee46621eea7773a2) where ![Image 122: {\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7f5138716ff9f4e227d3174a3dee47a859c7b3) is the [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator") and the operator (__2)__(_t_)/2 is the variable-order fractional quantum Riesz derivative. * [Acoustic attenuation](https://en.wikipedia.org/wiki/Acoustic_attenuation "Acoustic attenuation") * [Autoregressive fractionally integrated moving average](https://en.wikipedia.org/wiki/Autoregressive_fractionally_integrated_moving_average "Autoregressive fractionally integrated moving average") * [Initialized fractional calculus](https://en.wikipedia.org/wiki/Initialized_fractional_calculus "Initialized fractional calculus") * [Nonlocal operator](https://en.wikipedia.org/wiki/Nonlocal_operator "Nonlocal operator") ### Other fractional theories \[[edit](https://en.wikipedia.org/w/index.php?title=Fractional_calculus&action=edit&section=31 "Edit section: Other fractional theories")\] * [Fractional-order system](https://en.wikipedia.org/wiki/Fractional-order_system "Fractional-order system") * [Fractional Fourier transform](https://en.wikipedia.org/wiki/Fractional_Fourier_transform "Fractional Fourier transform") * [Prabhakar function](https://en.wikipedia.org/wiki/Prabhakar_function "Prabhakar function") 1. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-Zwillinger2014_2-0)** Daniel Zwillinger (12 May 2014). [_Handbook of Differential Equations_](https://books.google.com/books?id=9QLjBQAAQBAJ). Elsevier Science. [ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-1-4832-2096-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4832-2096-3 "Special:BookSources/978-1-4832-2096-3"). 2. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-Derivative_3-0)** Katugampola, Udita N. (15 October 2014). ["A New Approach To Generalized Fractional Derivatives"](https://www.emis.de/journals/BMAA/repository/docs/BMAA6-4-1.pdf) (PDF). _Bulletin of Mathematical Analysis and Applications_. **6** (4): 115\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_(identifier) "ArXiv (identifier)"):[1106.0965](https://arxiv.org/abs/1106.0965). 3. ^ [_**a**_](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-:1_4-0) [_**b**_](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-:1_4-1) Miller, Kenneth S.; Ross, Bertram (1993). _An Introduction to the Fractional Calculus and Fractional Differential Equations_. New York: Wiley. pp.12\. [ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-0-471-58884-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-58884-9 "Special:BookSources/978-0-471-58884-9"). 4. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-5)** Niels Henrik Abel (1823). ["Oplsning af et Par Opgaver ved Hjelp af bestemte Integraler (Solution de quelques problmes l'aide d'intgrales dfinies, Solution of a couple of problems by means of definite integrals)"](https://abelprize.no/sites/default/files/2021-04/Magazin_for_Naturvidenskaberne_oplosning_av_et_par1_opt.pdf) (PDF). _Magazin for Naturvidenskaberne_. Kristiania (Oslo): 5568. 5. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-6)** Podlubny, Igor; Magin, Richard L.; Trymorush, Irina (2017). "Niels Henrik Abel and the birth of fractional calculus". _Fractional Calculus and Applied Analysis_. **20** (5): 10681075\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_(identifier) "ArXiv (identifier)"):[1802.05441](https://arxiv.org/abs/1802.05441). [doi](https://en.wikipedia.org/wiki/Doi_(identifier) "Doi (identifier)"):[10.1515/fca-2017-0057](https://doi.org/10.1515%2Ffca-2017-0057). [S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier) "S2CID (identifier)")[119664694](https://api.semanticscholar.org/CorpusID:119664694). 6. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-7)** [Liouville, Joseph](https://en.wikipedia.org/wiki/Joseph_Liouville "Joseph Liouville") (1832), ["Mmoire sur quelques questions de gomtrie et de mcanique, et sur un nouveau genre de calcul pour rsoudre ces questions"](https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f2.item.r=Joseph%20Liouville), _Journal de l'cole Polytechnique_, **13**, Paris: 169. 7. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-8)** [Liouville, Joseph](https://en.wikipedia.org/wiki/Joseph_Liouville "Joseph Liouville") (1832), ["Mmoire sur le calcul des diffrentielles indices quelconques"](https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f72.image), _Journal de l'cole Polytechnique_, **13**, Paris: 71162. 8. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-9)** For the history of the subject, see the thesis (in French): Stphane Dugowson, [_Les diffrentielles mtaphysiques_](http://s.dugowson.free.fr/recherche/dones/index.html) (_histoire et philosophie de la gnralisation de l'ordre de drivation_), Thse, Universit Paris Nord (1994) 9. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-10)** For a historical review of the subject up to the beginning of the 20th century, see: Bertram Ross (1977). "The development of fractional calculus 16951900". _Historia Mathematica_. **4**: 7589\. [doi](https://en.wikipedia.org/wiki/Doi_(identifier) "Doi (identifier)"):[10.1016/0315-0860(77)90039-8](https://doi.org/10.1016%2F0315-0860%2877%2990039-8). [S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier) "S2CID (identifier)")[122146887](https://api.semanticscholar.org/CorpusID:122146887). 10. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-11)** Valrio, Duarte; Machado, Jos; [Kiryakova, Virginia](https://en.wikipedia.org/wiki/Virginia_Kiryakova "Virginia Kiryakova") (2014-01-01). "Some pioneers of the applications of fractional calculus". _Fractional Calculus and Applied Analysis_. **17** (2): 552578\. [doi](https://en.wikipedia.org/wiki/Doi_(identifier) "Doi (identifier)"):[10.2478/s13540-014-0185-1](https://doi.org/10.2478%2Fs13540-014-0185-1). [hdl](https://en.wikipedia.org/wiki/Hdl_(identifier) "Hdl (identifier)"):[10400.22/5491](https://hdl.handle.net/10400.22%2F5491). [ISSN](https://en.wikipedia.org/wiki/ISSN_(identifier) "ISSN (identifier)")[1314-2224](https://search.worldcat.org/issn/1314-2224). [S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier) "S2CID (identifier)")[121482200](https://api.semanticscholar.org/CorpusID:121482200). 11. **[^](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-12)** Hermann, Richard (2014). _Fractional Calculus: An Introduction for Physicists_ (2nded.). New Jersey: World Scientific Publishing. p.46\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_(identifier) "Bibcode (identifier)"):[2014fcip.book.....H](https://ui.adsabs.harvard.edu/abs/2014fcip.book.....H). [doi](https://en.wikipedia.org/wiki/Doi_(identifier) "Doi (identifier)"):[10.1142/8934](https://doi.org/10.1142%2F8934). 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[ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-3-030-14926-0](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-14926-0 "Special:BookSources/978-3-030-14926-0"). [S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier) "S2CID (identifier)")[145880744](https://api.semanticscholar.org/CorpusID:145880744). 71. ^ [_**a**_](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-Pandey2016_72-0) [_**b**_](https://en.wikipedia.org/wiki/Fractional_calculus#cite_ref-Pandey2016_72-1) Pandey, Vikash; Holm, Sverre (2016-12-01). "Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations". _The Journal of the Acoustical Society of America_. **140** (6): 42254236\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_(identifier) "ArXiv (identifier)"):[1612.05557](https://arxiv.org/abs/1612.05557). 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[S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier) "S2CID (identifier)")[122198977](https://api.semanticscholar.org/CorpusID:122198977). * Miller, Kenneth S.; Ross, Bertram, eds. (1993). _An Introduction to the Fractional Calculus and Fractional Differential Equations_. John Wiley & Sons. [ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-0-471-58884-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-58884-9 "Special:BookSources/978-0-471-58884-9"). * Samko, S.; Kilbas, A.A.; Marichev, O. (1993). _Fractional Integrals and Derivatives: Theory and Applications_. Taylor & Francis Books. [ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-2-88124-864-1](https://en.wikipedia.org/wiki/Special:BookSources/978-2-88124-864-1 "Special:BookSources/978-2-88124-864-1"). * Carpinteri, A.; Mainardi, F., eds. (1998). _Fractals and Fractional Calculus in Continuum Mechanics_. Springer-Verlag Telos. [ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-3-211-82913-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-211-82913-4 "Special:BookSources/978-3-211-82913-4"). * Igor Podlubny (27 October 1998). [_Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications_](https://books.google.com/books?id=K5FdXohLto0C). Elsevier. [ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-0-08-053198-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-053198-4 "Special:BookSources/978-0-08-053198-4"). * Tarasov, V.E. (2010). [_Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media_](https://link.springer.com/book/10.1007/978-3-642-14003-7). Nonlinear Physical Science. Springer. [doi](https://en.wikipedia.org/wiki/Doi_(identifier) "Doi (identifier)"):[10.1007/978-3-642-14003-7](https://doi.org/10.1007%2F978-3-642-14003-7). [ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-3-642-14003-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-14003-7 "Special:BookSources/978-3-642-14003-7"). * Li, Changpin; Cai, Min (2019). [_Theory and Numerical Approximations of Fractional Integrals and Derivatives_](https://doi.org/10.1137/1.9781611975888). SIAM. [doi](https://en.wikipedia.org/wiki/Doi_(identifier) "Doi (identifier)"):[10.1137/1.9781611975888](https://doi.org/10.1137%2F1.9781611975888). [ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier) "ISBN (identifier)")[978-1-61197-587-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-61197-587-1 "Special:BookSources/978-1-61197-587-1"). * [Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Fractional calculus"](https://mathworld.wolfram.com/FractionalCalculus.html). _[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")_. * ["Fractional Calculus"](http://www.mathpages.com/home/kmath616/kmath616.htm.htm). _MathPages.com_. * [Journal of Fractional Calculus and Applied Analysis](https://www.degruyter.com/journal/key/fca/html) [ISSN](https://en.wikipedia.org/wiki/ISSN_(identifier) "ISSN (identifier)")[1314-2224](https://www.worldcat.org/search?fq=x0:jrnl&q=n2:1314-2224) 2015 * Lorenzo, Carl F.; Hartley, Tom T. (2002). ["Initialized Fractional Calculus"](https://www.techbriefs.com/component/content/article/tb/pub/briefs/information-sciences/2264). _Tech Briefs_. NASA John H. Glenn Research Center. * Herrmann, Richard (2018). ["GigaHedron"](https://fractionalcalculus.org/). collection of books, articles, preprints, etc. * Loverro, Adam (2005). ["History, Definitions, and Applications for the Engineer"](https://web.archive.org/web/20051029113800/http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf) (PDF). [University of Notre Dame](https://en.wikipedia.org/wiki/University_of_Notre_Dame "University of Notre Dame"). Archived from [the original](http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf) (PDF) on 2005-10-29.