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Fractional calculus is a branch ofmathematical analysis that studies the several different possibilities of definingreal number powers orcomplex number powers of thedifferentiationoperator${\displaystyle D}$$${\displaystyle Df(x)=\frac{d}{dx}f(x),}$$

and of theintegration operator${\displaystyle J}$^{[Note 1]}$${\displaystyle Jf(x)={\int }_0^{x}f(s)ds,}$$

and developing acalculus for such operators generalizing the classical one.

In this context, the termpowers refers to iterative application of alinear operator${\displaystyle D}$ to afunction${\displaystyle f}$, that is, repeatedlycomposing${\displaystyle D}$ with itself, as in

For example, one may ask for a meaningful interpretation of$${\displaystyle \sqrt{D}=D^{\frac{1}{2}}}$$

as an analogue of thefunctional square root for the differentiation operator, that is, an expression for some linear operator that, when appliedtwice to any function, will have the same effect asdifferentiation. More generally, one can look at the question of defining a linear operator$${\displaystyle D^a}$$

for every real number${\displaystyle a}$ in such a way that, when${\displaystyle a}$ takes aninteger value${\displaystyle n\in \mathbb{Z}}$, it coincides with the usual${\displaystyle n}$-fold differentiation${\displaystyle D}$ if${\displaystyle n>0}$, and with the${\displaystyle n}$-th power of${\displaystyle J}$ when.

One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator${\displaystyle D}$ is that thesets of operator powers${\displaystyle \{D^a\mid a\in \mathbb{R}\}}$ defined in this way arecontinuoussemigroups with parameter${\displaystyle a}$, of which the originaldiscrete semigroup of${\displaystyle \{D^n\mid n\in \mathbb{Z}\}}$ for integer${\displaystyle n}$ is adenumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.

Fractionaldifferential equations, also known as extraordinary differential equations,^[1] are a generalization of differential equations through the application of fractional calculus.

Inapplied mathematics and mathematical analysis, afractional derivative is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written toGuillaume de l'Hôpital byGottfried Wilhelm Leibniz in 1695.^[2] Around the same time, Leibniz wrote toJohann Bernoulli about derivatives of "general order".^[3] In the correspondence between Leibniz andJohn Wallis in 1697, Wallis's infinite product for${\displaystyle \pi /2}$ is discussed. Leibniz suggested using differential calculus to achieve this result. Leibniz further used the notation${\displaystyle d^{1/2}y}$ to denote the derivative of order⁠1/2⁠.^[3]

Fractional calculus was introduced in one ofNiels Henrik Abel's early papers^[4] 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]Independently, the foundations of the subject were laid byLiouville in a paper from 1832.^[6][7][8]Oliver Heaviside introduced the practical use offractional differential operators in electrical transmission line analysis circa 1890.^[9] 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]

Let${\displaystyle f(x)}$ be a function defined for${\displaystyle x>0}$. Form the definite integral from 0 to${\displaystyle x}$. Call this$${\displaystyle (Jf)(x)={\int }_0^{x}f(t)dt.}$$

Repeating this process gives

and this can be extended arbitrarily.

TheCauchy formula for repeated integration, namely$${\displaystyle \left(J^{n}f\right)(x)=\frac{1}{(n-1)!}{\int }_0^x{\left(x-t\right)}^{n-1}f(t)dt,}$$leads in a straightforward way to a generalization for realn: using thegamma function to remove the discrete nature of the factorial function gives us a natural candidate for applications of the fractional integral operator as$${\displaystyle \left(J^{\alpha }f\right)(x)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^x{\left(x-t\right)}^{\alpha -1}f(t)dt.}$$

This is in fact a well-defined operator.

It is straightforward to show that theJ operator satisfies

This relationship is called the semigroup property of fractionaldifferintegral operators.

The classical form of fractional calculus is given by theRiemann–Liouville integral, which is essentially what has been described above. The theory of fractional integration forperiodic functions (therefore including the "boundary condition" of repeating after a period) is given by theWeyl integral. It is defined onFourier series, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on theunit 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 fort >a and the latter is valid fort <b.^[11]

It has been suggested^[12] that the integral on the positive real axis (i.e.${\displaystyle a=0}$) 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 theGrünwald–Letnikov derivative starts with the derivative instead of the integral.

TheHadamard fractional integral was introduced byJacques Hadamard^[13] and is given by the following formula,$${\displaystyle {\phantom{\mathbf{D}}}_a{\mathbf{D}}_t^{-\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_a^t{\left(\log \frac{t}{\tau }\right)}^{\alpha -1}f(\tau )\frac{d\tau }{\tau },t>a.}$$

The Atangana–Baleanu fractional integral of a continuous function is defined as:$${\displaystyle {\phantom{I}}_{\phantom{A}a}^{\mathrm{AB}}{I}_t^{\alpha }f(t)=\frac{1-\alpha }{\mathrm{AB}(\alpha )}f(t)+\frac{\alpha }{\mathrm{AB}(\alpha )\mathrm{\Gamma }(\alpha )}{\int }_a^t{\left(t-\tau \right)}^{\alpha -1}f(\tau )d\tau }$$

Unfortunately, the comparable process for the derivative operatorD is significantly more complex, but it can be shown thatD is neithercommutative noradditive in general.^[14]

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.

The corresponding derivative is calculated using Lagrange's rule for differential operators. To find theαth order derivative, thenth order derivative of the integral of order(n −α) is computed, wheren is the smallest integer greater thanα (that is,n =⌈α⌉). 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.^[15][16] Similar to the definitions for the Riemann–Liouville integral, the derivative has upper and lower variants.^[17]

Another option for computing fractional derivatives is theCaputo fractional derivative. It was introduced byMichele Caputo in his 1967 paper.^[18] 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 againn = ⌈α⌉:$${\displaystyle {\phantom{D}}^C{D}_t^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_0^t\frac{f^{(n)}(\tau )}{{\left(t-\tau \right)}^{\alpha +1-n}}d\tau .}$$

There is the Caputo fractional derivative defined as:which has the advantage that it is zero whenf(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

whereϕ(ν) is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.

In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function${\displaystyle f(t)}$ of${\displaystyle C^1}$ given by:$${\displaystyle {\phantom{D}}_{\phantom{C}a}^{\text{CF}}{D}_t^{\alpha }f(t)=\frac{1}{1-\alpha }{\int }_a^t{f}^{\prime }(\tau )e^{\left(-\alpha \frac{t-\tau }{1-\alpha }\right)}d\tau ,}$$

where.^[19]

In 2016, Atangana and Baleanu suggested differential operators based on the generalizedMittag-Leffler function${\displaystyle E_{\alpha }}$. 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${\displaystyle f(t)}$ of${\displaystyle C^1}$ given by^[20][21]$${\displaystyle {\phantom{D}}_{\phantom{AB}a}^{\text{ABC}}{D}_t^{\alpha }f(t)=\frac{\mathrm{AB}(\alpha )}{1-\alpha }{\int }_a^t{f}^{\prime }(\tau )E_{\alpha }\left(-\alpha \frac{(t-\tau {)}^{\alpha }}{1-\alpha }\right)d\tau ,}$$

If the function is continuous, the Atangana–Baleanu derivative in Riemann–Liouville sense is given by:$${\displaystyle {\phantom{D}}_{\phantom{AB}a}^{\text{ABC}}{D}_t^{\alpha }f(t)=\frac{\mathrm{AB}(\alpha )}{1-\alpha }\frac{d}{dt}{\int }_a^{t}f(\tau )E_{\alpha }\left(-\alpha \frac{(t-\tau {)}^{\alpha }}{1-\alpha }\right)d\tau ,}$$

The kernel used in Atangana–Baleanu fractional derivative has some properties of a cumulative distribution function. For example, for all${\displaystyle \alpha \in (0,1]}$, the function${\displaystyle E_{\alpha }}$ is increasing on the real line, converges to${\displaystyle 0}$ in${\displaystyle -\mathrm{\infty }}$, and${\displaystyle E_{\alpha }(0)=1}$. Therefore, we have that, the function${\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })}$ 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 aMittag-Leffler distribution of order${\displaystyle \alpha }$. It is also well-known that all these probability distributions areabsolutely continuous. In particular, the Mittag-Leffler function has a particular case${\displaystyle E_1}$, which is the exponential function, the Mittag-Leffler distribution of order${\displaystyle 1}$ is therefore anexponential distribution. However, for${\displaystyle \alpha \in (0,1)}$, the Mittag-Leffler distributions areheavy-tailed. Their Laplace transform is given by:$${\displaystyle \mathbb{E}(e^{-\lambda X_{\alpha }})=\frac{1}{1+{\lambda }^{\alpha }},}$$

This directly implies that, for${\displaystyle \alpha \in (0,1)}$, the expectation is infinite. In addition, these distributions aregeometric stable distributions.

The Riesz derivative is defined as$${\displaystyle \mathcal{F}\left\{\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{\left|x\right|}^{\alpha }}\right\}(k)=-{\left|k\right|}^{\alpha }\mathcal{F}\{u\}(k),}$$

where${\displaystyle \mathcal{F}}$ denotes theFourier transform.^[22][23]

The conformable fractional derivative of a function${\displaystyle f}$ of order${\displaystyle \alpha }$ is given by$${\displaystyle T_{\alpha }(f)(t)=\underset{ϵ\to 0}{lim}\frac{f\left(t+ϵt^{1-\alpha }\right)-f(t)}{ϵ}}$$Unlike other definitions of the fractional derivative, the conformable fractional derivative obeys theproduct andquotient rule has analogs toRolle's theorem and themean value theorem.^[24][25] 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]$${\displaystyle {}_a^C{\tilde{T}}_{\alpha }(f)(t)=\underset{ϵ\to 0}{lim}\left[(1-\alpha )(f(t)-f(a))+\alpha \frac{f\left(t+ϵ(t-a{)}^{1-\alpha }\right)-f(t)}{ϵ}\right]}$$where${\displaystyle a}$ and${\displaystyle t}$ arereal numbers and. They also defined the improved Riemann-Liouville-type conformable fractional derivative to similarly approximate the Riemann-Liouville fractional derivative:^[25]

$${\displaystyle {}_a^{RL}{\tilde{T}}_{\alpha }(f)(t)=\underset{ϵ\to 0}{lim}\left[(1-\alpha )f(t)+\alpha \frac{f\left(t+ϵ(t-a{)}^{1-\alpha }\right)-f(t)}{ϵ}\right]}$$where${\displaystyle a}$ and${\displaystyle t}$ arereal numbers and. Both improved conformable fractional derivatives have analogs to Rolle's theorem and theinterior extremum theorem.^[26]

Classical fractional derivatives include:

• Grünwald–Letnikov derivative^[27][28]
• Sonin–Letnikov derivative^[28]
• Liouville derivative^[27]
• Caputo derivative^[27]
• Hadamard derivative^[27][29]
• Marchaud derivative^[27]
• Riesz derivative^[28]
• Miller–Ross derivative^[27]
• Weyl derivative^[30][31][27]
• Erdélyi–Kober derivative^[27]
• ${\displaystyle F^{\alpha }}$-derivative^[32]

New fractional derivatives include:

• Coimbra derivative^[27]
• Katugampola derivative^[33]
• Hilfer derivative^[27]
• Davidson derivative^[27]
• Chen derivative^[27]
• Caputo-Fabrizio derivative^[20][34]
• Atangana–Baleanu derivative^[20][21]

Novel fractional derivatives with nonsingular kernels, namely the Caputo-Fabrizio and Atangana–Baleanu derivatives, are subject to controversy in applied mathematics literature.^[35][36] 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.^[37][38][39]

TheCoimbra derivative is used for physical modeling:^[40] A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,^{[41][42][43][44][45][46][47]} as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors^{[48][49][50][51]}

For$${\displaystyle \begin{array}{r}{}_a^{\mathbb{C}}{\mathbb{D}}^{q(t)}f(t)=\frac{1}{\mathrm{\Gamma }[1-q(t)]}{\int }_{0^{+}}^t(t-\tau {)}^{-q(t)}\frac{df(\tau )}{d\tau }d\tau +\frac{(f(0^{+})-f(0^{-}))t^{-q(t)}}{\mathrm{\Gamma }(1-q(t))},\end{array}}$$where the lower limit${\displaystyle a}$ can be taken as either${\displaystyle 0^{-}}$ or${\displaystyle -\mathrm{\infty }}$ as long as${\displaystyle f(t)}$ is identically zero from or${\displaystyle -\mathrm{\infty }}$ to${\displaystyle 0^{-}}$. Note that this operator returns the correct fractional derivatives for all values of${\displaystyle t}$ and can be applied to either the dependent function itself${\displaystyle f(t)}$ with a variable order of the form${\displaystyle q(f(t))}$ or to the independent variable with a variable order of the form${\displaystyle q(t)}$.${\displaystyle {}^{[1]}}$

The Coimbra derivative can be generalized to any order,^[52] leading to the Coimbra Generalized Order Differintegration Operator (GODO)^[53]For$${\displaystyle \begin{array}{r}{}_{-\mathrm{\infty }}^{\mathbb{C}}{\mathbb{D}}^{q(t)}f(t)=\frac{1}{\mathrm{\Gamma }[m-q(t)]}{\int }_{0^{+}}^t(t-\tau {)}^{m-1-q(t)}\frac{d^{m}f(\tau )}{d{\tau }^m}d\tau +\sum _{n=0}^{m-1}\frac{(\frac{d^{n}f(t)}{d{t}^n}{|}_{0^{+}}-\frac{d^{n}f(t)}{d{t}^n}{|}_{0^{-}})t^{n-q(t)}}{\mathrm{\Gamma }[n+1-q(t)]},\end{array}}$$where${\displaystyle m}$ is an integer larger than the larger value of${\displaystyle q(t)}$ for all values of${\displaystyle t}$. Note that the second (summation) term on the right side of the definition above can be expressed as

$${\displaystyle \begin{array}{r}\frac{1}{\mathrm{\Gamma }[m-q(t)]}\sum _{n=0}^{m-1}\{[\frac{d^{n}f(t)}{d{t}^n}{|}_{0^{+}}-\frac{d^{n}f(t)}{d{t}^n}{|}_{0^{-}}]t^{n-q(t)}\prod _{j=n+1}^{m-1}[j-q(t)]\}\end{array}}$$so to keep the denominator on the positive branch of the Gamma (${\displaystyle \mathrm{\Gamma }}$) function and for ease of numerical calculation.

The${\displaystyle a}$-th derivative of a function${\displaystyle f}$ at a point${\displaystyle x}$ is alocal property only when${\displaystyle a}$ is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of${\displaystyle f}$ at${\displaystyle x=c}$ depends on all values of${\displaystyle f}$, even those far away from${\displaystyle c}$. Therefore, it is expected that the fractional derivative operation involves some sort ofboundary conditions, involving information on the function further out.^[54]

The fractional derivative of a function of order${\displaystyle a}$ is nowadays often defined by means of theFourier orMellin integral transforms.^{[55][citation needed]}

TheErdélyi–Kober operator is an integral operator introduced byArthur Erdélyi (1940).^[56] andHermann Kober (1940)^[57] and is given by$${\displaystyle \frac{x^{-\nu -\alpha +1}}{\mathrm{\Gamma }(\alpha )}{\int }_0^x{\left(t-x\right)}^{\alpha -1}{t}^{-\alpha -\nu }f(t)dt,}$$

which generalizes theRiemann–Liouville fractional integral and the Weyl integral.

In the context offunctional analysis, functionsf(D) more general than powers are studied in thefunctional calculus ofspectral theory. The theory ofpseudo-differential operators also allows one to consider powers ofD. The operators arising are examples ofsingular integral operators; and the generalisation of the classical theory to higher dimensions is called the theory ofRiesz potentials. So there are a number of contemporary theories available, within whichfractional calculus can be discussed.(See alsoErdélyi–Kober operator, important inspecial function theory.)^[58]

As described by Wheatcraft and Meerschaert (2008),^[59] a fractional conservation of mass equation is needed to model fluid flow when thecontrol volume is not large enough compared to the scale ofheterogeneity 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:$${\displaystyle -\rho \left({\mathrm{\nabla }}^{\alpha }\cdot \overrightarrow{u}\right)=\mathrm{\Gamma }(\alpha +1)\mathrm{\Delta }{x}^{1-\alpha }\rho \left({\beta }_s+\varphi {\beta }_w\right)\frac{\mathrm{\partial }p}{\mathrm{\partial }t}}$$

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 byFick's laws of diffusion. Taking the Laplace transform of Fick's second law yields an ordinary second-order differential equation (here in dimensionless form):$${\displaystyle \frac{d^2}{d{x}^2}C(x,s)=sC(x,s)}$$

whose solutionC(x,s) contains a one-half power dependence ons. Taking the derivative ofC(x,s) and then the inverse Laplace transform yields the following relationship:$${\displaystyle \frac{d}{dx}C(x,t)=\frac{d^{\frac{1}{2}}}{d{t}^{\frac{1}{2}}}C(x,t)}$$

which relates the concentration of substrate at the electrode surface to the current.^[60] 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.^[61]

In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order.^[62][63] In these works, the classicalDarcy 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.

This equation^{[clarification needed]} has been shown useful for modeling contaminant flow in heterogenous porous media.^[64][65][66]

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 avariational order derivative. The modified equation was numerically solved via theCrank–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^[67]

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.^[68][69] 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$${\displaystyle \frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }}=-K(-\mathrm{\Delta }{)}^{\beta }u.}$$

A simple extension of the fractional derivative is the variable-order fractional derivative,α andβ are changed intoα(x,t) andβ(x,t). Its applications in anomalous diffusion modeling can be found in the reference.^[67][70][71]

Fractional derivatives are used to modelviscoelasticdamping in certain types of materials like polymers.^[12]

GeneralizingPID controllers to use fractional orders can increase their degree of freedom. The new equation relating thecontrol variableu(t) in terms of a measurederror valuee(t) can be written as$${\displaystyle u(t)=K_{\mathrm{p}}e(t)+K_{\mathrm{i}}{D}_t^{-\alpha }e(t)+K_{\mathrm{d}}{D}_t^{\beta }e(t)}$$

whereα andβ are positive fractional orders andK_p,K_i, andK_d, all non-negative, denote the coefficients for theproportional,integral, andderivative terms, respectively (sometimes denotedP,I, andD).^[72]

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:$${\displaystyle {\mathrm{\nabla }}^{2}u-{\displaystyle \frac{1}{c_0^2}}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}+{\tau }_{\sigma }^{\alpha }{\displaystyle \frac{{\mathrm{\partial }}^{\alpha }}{\mathrm{\partial }{t}^{\alpha }}}{\mathrm{\nabla }}^{2}u-{\displaystyle \frac{{\tau }_{ϵ}^{\beta }}{c_0^2}}{\displaystyle \frac{{\mathrm{\partial }}^{\beta +2}u}{\mathrm{\partial }{t}^{\beta +2}}}=0.}$$

See also Holm & Näsholm (2011)^[73] 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)^[74] and in the survey paper,^[75] as well as theAcoustic attenuation article. See Holm & Nasholm (2013)^[76] 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.^[77]

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.^[78] Interestingly, Pandey and Holm derivedLomnitz's law inseismology and Nutting's law innon-Newtonian rheology using the framework of fractional calculus.^[79] Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.^[78]

Thefractional Schrödinger equation, a fundamental equation offractional quantum mechanics, has the following form:^[80][81]$${\displaystyle i\hslash \frac{\mathrm{\partial }\psi (\mathbf{r},t)}{\mathrm{\partial }t}=D_{\alpha }{\left(-{\hslash }^2\mathrm{\Delta }\right)}^{\frac{\alpha }{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t).}$$

where the solution of the equation is thewavefunctionψ(r,t) – the quantum mechanicalprobability amplitude for the particle to have a givenposition vectorr at any given timet, andħ is thereduced Planck constant. Thepotential energy functionV(r,t) depends on the system.

Further,$\mathrm{\Delta }=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\mathbf{r}}^2}$ is theLaplace operator, andD_α is a scale constant with physicaldimension[D_α] = J^{1 −α}·m^α·s^−α = kg^{1 −α}·m^{2 −α}·s^{α − 2}, (atα = 2,$D_2=\frac{1}{2m}$ for a particle of massm), and the operator(−ħ²Δ)^α/2 is the 3-dimensional fractional quantum Riesz derivative defined by$${\displaystyle (-{\hslash }^2\mathrm{\Delta }{)}^{\frac{\alpha }{2}}\psi (\mathbf{r},t)=\frac{1}{(2\pi \hslash {)}^3}\int d^{3}p{e}^{\frac{i}{\hslash }\mathbf{p}\cdot \mathbf{r}}|\mathbf{p}{|}^{\alpha }\phi (\mathbf{p},t).}$$

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

As a natural generalization of thefractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena:^[82]$${\displaystyle i\hslash \frac{\mathrm{\partial }{\psi }^{\alpha (\mathbf{r})}(\mathbf{r},t)}{\mathrm{\partial }{t}^{\alpha (\mathbf{r})}}={\left(-{\hslash }^2\mathrm{\Delta }\right)}^{\frac{\beta (t)}{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t),}$$

where$\mathrm{\Delta }=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\mathbf{r}}^2}$ is theLaplace operator and the operator(−ħ²Δ)^β(t)/2 is the variable-order fractional quantum Riesz derivative.

• Acoustic attenuation
• Autoregressive fractionally integrated moving average
• Initialized fractional calculus
• Nonlocal operator

• Fractional-order system
• Fractional Fourier transform
• Prabhakar function

• Debnath, L. (2004). "A brief historical introduction to fractional calculus".International Journal of Mathematical Education in Science and Technology.35 (4):487–501.Bibcode:2004IJMES..35..487D.doi:10.1080/00207390410001686571.S2CID 122198977.

• Miller, Kenneth S.; Ross, Bertram, eds. (1993).An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons.ISBN 978-0-471-58884-9.
• Samko, S.; Kilbas, A.A.; Marichev, O. (1993).Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books.ISBN 978-2-88124-864-1.
• Carpinteri, A.; Mainardi, F., eds. (1998).Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag Telos.ISBN 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. Elsevier.ISBN 978-0-08-053198-4.
• Tarasov, V.E. (2010).Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Nonlinear Physical Science. Springer.doi:10.1007/978-3-642-14003-7.ISBN 978-3-642-14003-7.
• Li, Changpin; Cai, Min (2019).Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM.doi:10.1137/1.9781611975888.ISBN 978-1-61197-587-1.
• Herrmann, Richard (2025).Fractional Calculus: An Introduction for Physicists. World Scientific.doi:10.1142/14345.ISBN 978-9-819-81439-8.

• Fractional calculus
• Generalizations

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