Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
Part of a series of articles about  
Calculus  





Specialized 

and of the integration operator J[Note 1]
and developing a calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator D to a function f, that is, repeatedly composing D with itself, as in .
For example, one may ask for a meaningful interpretation of:
as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that when applied twice to any function will have the same effect as differentiation. More generally, one can look at the question of defining a linear functional
for every realnumber a in such a way that, when a takes an integer value n ∈ ℤ, it coincides with the usual nfold differentiation D if n > 0, and with the −nth power of J when n < 0.
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator D is that the sets of operator powers { D^{a} a ∈ ℝ } defined in this way are continuous semigroups with parameter a, of which the original discrete semigroup of { D^{n}  n ∈ ℤ } for integer n is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional differential equations, also known as extraordinary differential equations,[1] are a generalization of differential equations through the application of fractional calculus.
Historical notes
In applied mathematics and mathematical analysis, fractional derivative is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to Guillaume de l'Hôpital by Gottfried Wilhelm Leibniz in 1695.[2] As far as the existence of such a theory is concerned, the foundations of the subject were laid by Liouville in a paper from 1832.[3] The autodidact Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890.[4]
Nature of the fractional derivative
The ath derivative of a function f (x) at a point x is a local property only when a is an integer; this is not the case for noninteger power derivatives. In other words, it is not correct to say that the fractional derivative at x of a function f (x) depends only on values of f very near x, in the way that integerpower derivatives certainly do. Therefore, it is expected that the theory involves some sort of boundary conditions, involving information on the function further out.[5]
The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms.
Heuristics
A fairly natural question to ask is whether there exists a linear operator H, or halfderivative, such that
It turns out that there is such an operator, and indeed for any a > 0, there exists an operator P such that
or to put it another way, the definition of d^{n}y/dx^{n} can be extended to all real values of n.
Let f (x) be a function defined for x > 0. Form the definite integral from 0 to x. Call this
Repeating this process gives
and this can be extended arbitrarily.
The Cauchy formula for repeated integration, namely
leads in a straightforward way to a generalization for real n.
Using the gamma function to remove the discrete nature of the factorial function gives us a natural candidate for fractional applications of the integral operator.
This is in fact a welldefined operator.
It is straightforward to show that the J operator satisfies
Proof where in the last step we exchanged the order of integration and pulled out the f (s) factor from the t integration. Changing variables to r defined by t = s + (x − s)r,
The inner integral is the beta function which satisfies the following property:
Substituting back into the equation
Interchanging α and β shows that the order in which the J operator is applied is irrelevant and completes the proof.
This relationship is called the semigroup property of fractional differintegral operators. Unfortunately the comparable process for the derivative operator D is significantly more complex, but it can be shown that D is neither commutative nor additive in general.[6]
Fractional derivative of a basic power function
Let us assume that f (x) is a monomial of the form
The first derivative is as usual
Repeating this gives the more general result that
Which, after replacing the factorials with the gamma function, leads us to
For k = 1 and a = 1/2, we obtain the halfderivative of the function x as
To demonstrate that this is, in fact, the "half derivative" (where H^{2}f (x) = Df (x)), we repeat the process to get:
(because and Γ(1) = 1) which is indeed the expected result of
For negative integer power k, the gamma function is undefined and we have to use the following relation:[7]
This extension of the above differential operator need not be constrained only to real powers. For example, the (1 + i)th derivative of the (1 − i)th derivative yields the 2nd derivative. Also setting negative values for a yields integrals.
For a general function f (x) and 0 < α < 1, the complete fractional derivative is
For arbitrary α, since the gamma function is undefined for arguments whose real part is a negative integer and whose imaginary part is zero, it is necessary to apply the fractional derivative after the integer derivative has been performed. For example,
Laplace transform
We can also come at the question via the Laplace transform. Knowing that
and
and so on, we assert
 .
For example,
as expected. Indeed, given the convolution rule
and shorthanding p(x) = x^{α − 1} for clarity, we find that
which is what Cauchy gave us above.
Laplace transforms "work" on relatively few functions, but they are often useful for solving fractional differential equations.
Fractional integrals
Riemann–Liouville fractional integral
The classical form of fractional calculus is given by the Riemann–Liouville integral, which is essentially what has been described above. The theory for periodic functions (therefore including the 'boundary condition' of repeating after a period) is the Weyl integral. It is defined on Fourier series, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the unit circle whose integrals evaluate to 0). The RiemannLiouville integral exists in two forms, upper and lower. Considering the interval [a,b], the integrals are defined as
Where the former is valid for t > a and the latter is valid for t < b.[8]
By contrast the Grünwald–Letnikov derivative starts with the derivative instead of the integral.
Hadamard fractional integral
The Hadamard fractional integral is introduced by Jacques Hadamard[9] and is given by the following formula,
Atangana–Baleanu fractional integral
Recently, using the generalized MittagLeffler function, Atangana and Baleanu suggested a new formulation of the fractional derivative with a nonlocal and nonsingular kernel. The integral is defined as:
where AB(α) is a normalization function such that AB(0) = AB(1) = 1.[10]
Fractional derivatives
Unlike classical Newtonian derivatives, a fractional derivative is defined via a fractional integral.
Riemann–Liouville fractional derivative
The corresponding derivative is calculated using Lagrange's rule for differential operators. Computing nth order derivative over the integral of order (n − α), the α order derivative is obtained. It is important to remark that n is the smallest integer greater than α ( that is, n = ⌈α⌉). Similar to the definitions for the RiemannLiouville integral, the derivative has upper and lower variants.[11]
Caputo fractional derivative
Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his 1967 paper.[12] 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.
There is the Caputo fractional derivative defined as:
which has the advantage that is zero when f (t) is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative. Moreover, there is the Caputo fractional derivative of distributed order defined as
where φ(ν) is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.
Atangana–Baleanu derivative
Like the integral, there is also a fractional derivative using the general MittagLeffler function as a kernel.[10] The authors introduced two versions, the Atangana–Baleanu in Caputo sense (ABC) derivative, which is the convolution of a local derivative of a given function with the generalized MittagLeffler function, and the Atangana–Baleanu in Riemann–Liouville sense (ABR) derivative, which is the derivative of a convolution of a given function that is not differentiable with the generalized MittagLeffler function.[13] The AtanganaBaleanu fractional derivative in Caputo sense is defined as:
And the Atangana–Baleanu fractional derivative in Riemann–Liouville is defined as:
Other types
Classical fractional derivatives include:
 Grünwald–Letnikov derivative
 Sonin–Letnikov derivative
 Liouville derivative
 Caputo derivative
 Hadamard derivative
 Marchaud derivative
 Riesz derivative
 Riesz–Miller derivative
 Miller–Ross derivative
 Weyl derivative
 Erdélyi–Kober derivative
New fractional derivatives include:
 Machado derivative (This derivative does not exist anywhere in the literature)
 Chen–Machado derivative
 Coimbra derivative
 Katugampola derivative
 Caputo–Katugampola derivative
 Hilfer derivative
 Hilfer–Katugampola derivative
 Davidson derivative
 Chen derivative
 Caputo Fabrizio derivative
 Atangana–Baleanu derivative
 Pichaghchi derivative
Generalizations
Erdélyi–Kober operator
The Erdélyi–Kober operator is an integral operator introduced by Arthur Erdélyi (1940).[16] and Hermann Kober (1940)[17] and is given by
which generalizes the Riemann–Liouville fractional integral and the Weyl integral.
Katugampola operators
A recent generalization introduced by Udita Katugampola is the following, which generalizes the Riemann–Liouville fractional integral and the Hadamard fractional integral. The integral is now known as the Katugampola fractional integral and is given by,[2][18]
Even though the integral operator in question is a close resemblance of the famous Erdélyi–Kober operator, it is not possible to obtain the Hadamard fractional integral as a direct consequence of the Erdélyi–Kober operator. Also, there is a Katugampolatype fractional derivative, which generalizes the Riemann–Liouville and the Hadamard fractional derivatives.[2] As with the case of fractional integrals, the same is not true for the Erdélyi–Kober operator.[2]
Functional calculus
In the context of functional analysis, functions f (D) more general than powers are studied in the functional calculus of spectral theory. The theory of pseudodifferential operators also allows one to consider powers of D. The operators arising are examples of singular integral operators; and the generalisation of the classical theory to higher dimensions is called the theory of Riesz potentials. So there are a number of contemporary theories available, within which fractional calculus can be discussed. See also Erdélyi–Kober operator, important in special function theory (Kober 1940), (Erdélyi 1950–51).
Applications
Fractional conservation of mass
As described by Wheatcraft and Meerschaert (2008),[19] a fractional conservation of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is nonlinear. In the referenced paper, the fractional conservation of mass equation for fluid flow is:
Groundwater flow problem
In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of derivative with fractional order.[20][21] In these works, The classical Darcy law is generalized by regarding the water flow as a function of a noninteger 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
This equation has been shown useful for modeling contaminant flow in heterogenous porous media.[22][23][24]
Atangana and Kilicman extended the fractional advection dispersion equation to a variable order equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of a variational order derivative. The modified equation was numerically solved via the Crank–Nicolson method. The stability and convergence in numerical simulations showed that the modified equation is more reliable in predicting the movement of pollution in deformable aquifers than equations with constant fractional and integer derivatives[25]
Timespace fractional diffusion equation models
Anomalous diffusion processes in complex media can be well characterized by using fractionalorder diffusion equation models.[26][27] The time derivative term is corresponding to longtime heavy tail decay and the spatial derivative for diffusion nonlocality. The timespace fractional diffusion governing equation can be written as
A simple extension of fractional derivative is the variableorder fractional derivative, α and β are changed into α(x, t) and β(x, t). Its applications in anomalous diffusion modeling can be found in reference.[25][28][29]
Structural damping models
Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.[30]
PID controllers
Generalizing PID controllers 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
where α and β are positive fractional orders and K_{p}, K_{i}, and K_{d}, all nonnegative, denote the coefficients for the proportional, integral, and derivative terms, respectively (sometimes denoted P, I, and D).[31]
Acoustical wave equations for complex media
The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency powerlaw. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:
See also Holm & Näsholm (2011)[32] 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)[33] and in the survey paper,[34] as well as the acoustic attenuation article. See Holm & Nasholm (2013)[35] for a paper which compares fractional wave equations which model powerlaw attenuation. This book on powerlaw attenuation also covers the topic in more detail.[36]
Fractional Schrödinger equation in quantum theory
The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form:[37][38]
where the solution of the equation is the wavefunction ψ(r, t) – the quantum mechanical probability amplitude for the particle to have a given position vector r at any given time t, and ħ is the reduced Planck constant. The potential energy function V(r, t) depends on the system.
Further, Δ = ∂^{2}/∂r^{2} is the Laplace operator, and D_{α} is a scale constant with physical dimension [D_{α}] = J^{1 − α}·m^{α}·s^{−α} = kg^{1 − α}·m^{2 − α}·s^{α − 2}, (at α = 2, D_{2} = 1/2m for a particle of mass m), and the operator (−ħ^{2}Δ)^{α/2} is the 3dimensional fractional quantum Riesz derivative defined by
The index α in the fractional Schrödinger equation is the Lévy index, 1 < α ≤ 2.
Variableorder fractional Schrödinger equation
As a natural generalization of the fractional Schrödinger equation, the variableorder fractional Schrödinger equation has been exploited to study fractional quantum phenomena:[39]
where Δ = ∂^{2}/∂r^{2} is the Laplace operator and the operator (−ħ^{2}Δ)^{β(t)/2} is the variableorder fractional quantum Riesz derivative.
See also
Other fractional theories
Notes
 The symbol J is commonly used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like glyphs, such as identities.
References
 Daniel Zwillinger (12 May 2014). Handbook of Differential Equations. Elsevier Science. ISBN 9781483220963.
 Katugampola, Udita N. (15 October 2014). "A New Approach To Generalized Fractional Derivatives" (PDF). Bulletin of Mathematical Analysis and Applications. 6 (4): 1–15. arXiv:1106.0965. Bibcode:2011arXiv1106.0965K.
 For the history of the subject, see the thesis (in French): Stéphane Dugowson, Les différentielles métaphysiques (histoire et philosophie de la généralisation de l'ordre de dérivation), Thèse, Université Paris Nord (1994)
 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: 75–89. doi:10.1016/03150860(77)900398.
 "Fractional Calculus". www.mathpages.com. Retrieved 20180103.
 Kilbas, Srivastava & Trujillo 2006, p. 75 (Property 2.4)
 Bologna, Mauro, Short Introduction to Fractional Calculus (PDF), Universidad de Tarapaca, Arica, Chile
 Hermann, Richard (2014). Fractional Calculus: An Introduction for Physicists (2nd ed.). New Jersey: World Scientific Publishing. p. 46. Bibcode:2014fcip.book.....H. doi:10.1142/8934. ISBN 9789814551076.
 Hadamard, J. (1892). "Essai sur l'étude des fonctions données par leur développement de Taylor" (PDF). Journal de Mathématiques Pures et Appliquées. 4 (8): 101–186.
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 Hermann, Richard, ed. (2014). Fractional Calculus. Fractional Calculus: An Introduction for Physicists (2nd ed.). New Jersey: World Scientific Publishing Co. p. 54. Bibcode:2014fcip.book.....H. doi:10.1142/8934. ISBN 9789814551076.
 Caputo, Michele (1967). "Linear model of dissipation whose Q is almost frequency independent. II". Geophysical Journal International. 13 (5): 529–539. Bibcode:1967GeoJ...13..529C. doi:10.1111/j.1365246x.1967.tb02303.x..
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 Wheatcraft, Stephen W.; Meerschaert, Mark M. (October 2008). "Fractional conservation of mass" (PDF). Advances in Water Resources. 31 (10): 1377–1381. Bibcode:2008AdWR...31.1377W. doi:10.1016/j.advwatres.2008.07.004. ISSN 03091708.
 Atangana, Abdon; Bildik, Necdet (2013). "The Use of Fractional Order Derivative to Predict the Groundwater Flow". Mathematical Problems in Engineering. 2013: 1–9. doi:10.1155/2013/543026.
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 Wheatcraft, Stephen W.; Meerschaert, Mark M.; Schumer, Rina; Benson, David A. (20010101). "Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests". Transport in Porous Media. 42 (1–2): 211–240. CiteSeerX 10.1.1.58.2062. doi:10.1023/A:1006733002131. ISSN 15731634.
 Atangana, Abdon; Kilicman, Adem (2014). "On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative". Mathematical Problems in Engineering. 2014: 9. doi:10.1155/2014/542809.
 Metzler, R.; Klafter, J. (2000). "The random walk's guide to anomalous diffusion: a fractional dynamics approach". Phys. Rep. 339 (1): 1–77. Bibcode:2000PhR...339....1M. doi:10.1016/s03701573(00)000703.
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 Holm, S.; Näsholm, S. P. (2011). "A causal and fractional allfrequency wave equation for lossy media". Journal of the Acoustical Society of America. 130 (4): 2195–2201. Bibcode:2011ASAJ..130.2195H. doi:10.1121/1.3631626. PMID 21973374.
 Näsholm, S. P.; Holm, S. (2011). "Linking multiple relaxation, powerlaw attenuation, and fractional wave equations". Journal of the Acoustical Society of America. 130 (5): 3038–3045. Bibcode:2011ASAJ..130.3038N. doi:10.1121/1.3641457. PMID 22087931.
 Näsholm, S. P.; Holm, S. (2012). "On a Fractional Zener Elastic Wave Equation". Fract. Calc. Appl. Anal. arXiv:1212.4024. Bibcode:2012arXiv1212.4024N. doi:10.2478/s1354001300031.
 Holm, S.; Näsholm, S. P. (2013). "Comparison of fractional wave equations for power law attenuation in ultrasound and elastography". Ultrasound in Medicine & Biology. 40 (4): 695–703. arXiv:1306.6507. Bibcode:2013arXiv1306.6507H. CiteSeerX 10.1.1.765.120. doi:10.1016/j.ultrasmedbio.2013.09.033. PMID 24433745.
 Holm, S. (2019). Waves with PowerLaw Attenuation. Springer and Acoustical Society of America Press.
 Laskin, N. (2002). "Fractional Schrodinger equation". Phys. Rev. E. 66 (5): 056108. arXiv:quantph/0206098. Bibcode:2002PhRvE..66e6108L. CiteSeerX 10.1.1.252.6732. doi:10.1103/PhysRevE.66.056108. PMID 12513557.
 Laskin, Nick (2018). Fractional Quantum Mechanics. CiteSeerX 10.1.1.247.5449. doi:10.1142/10541. ISBN 9789813223790.
 Bhrawy, A.H.; Zaky, M.A. (2017). "An improved collocation method for multidimensional space–time variableorder fractional Schrödinger equations". Applied Numerical Mathematics. 111: 197–218. doi:10.1016/j.apnum.2016.09.009.
Sources
 Kilbas, Anatolii Aleksandrovich; Srivastava, Hari Mohan; Trujillo, Juan J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 9780444518323.
Further reading
Articles regarding the history of fractional calculus
 Ross, B. (1975). A brief history and exposition of the fundamental theory of fractional calculus. Fractional Calculus and Its Applications. Lecture Notes in Mathematics. Lecture Notes in Mathematics. 457. pp. 1–36. doi:10.1007/BFb0067096. ISBN 9783540071617.
 Debnath, L. (2004). "A brief historical introduction to fractional calculus". International Journal of Mathematical Education in Science and Technology. 35 (4): 487–501. doi:10.1080/00207390410001686571.
 Tenreiro Machado, J.; Kiryakova, V.; Mainardi, F. (2011). "Recent history of fractional calculus". Communications in Nonlinear Science and Numerical Simulation. 16 (3): 1140–1153. Bibcode:2011CNSNS..16.1140M. doi:10.1016/j.cnsns.2010.05.027. hdl:10400.22/4149.
 Tenreiro Machado, J.A.; Galhano, A.M.; Trujillo, J.J. (2013). "Science metrics on fractional calculus development since 1966". Fractional Calculus and Applied Analysis. 16 (2): 479–500. doi:10.2478/s135400130030y.
 Tenreiro Machado, J.A.; Galhano, A.M.S.F.; Trujillo, J.J. (2014). "On development of fractional calculus during the last fifty years". Scientometrics. 98 (1): 577–582. doi:10.1007/s1119201310326. hdl:10400.22/3769.
Books
 Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering. V. Academic Press. ISBN 9780125255509.
 Miller, Kenneth S.; Ross, Bertram, eds. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. ISBN 9780471588849.
 Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN 9782881248641.
 Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. SpringerVerlag Telos. ISBN 9783211829134.
 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 9780080531984.
 West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003). Physics of Fractal Operators. Physics Today. 56. Springer Verlag. p. 65. Bibcode:2003PhT....56l..65W. doi:10.1063/1.1650234. ISBN 9780387955544.
 Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press.
 Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Nonlinear Physical Science. Springer. ISBN 9783642140037.
 Zhou, Y. (2010). Basic Theory of Fractional Differential Equations. Singapore: World Scientific. doi:10.1142/9069. ISBN 9789814579896.
 Uchaikin, V.V. (2012). Fractional Derivatives for Physicists and Engineers. Fractional Derivatives for Physicists and Engineers: Background and Theory. Nonlinear Physical Science. Higher Education Press. Bibcode:2013fdpe.book.....U. doi:10.1007/9783642339110. ISBN 9783642339103.
 Daftardargejji, Varsha (2013). Fractional Calculus: Theory and Applications. Narosa Publishing House. ISBN 9788184873337.
 Srivastava, Hari M (2014). Special Functions in Fractional Calculus and Related Fractional Differintegral Equations. Singapore: World Scientific. doi:10.1142/8936. ISBN 9789814551106.
 Li, C P; Zeng, F H (2015). Numerical Methods for Fractional Calcuus. USA: CRC Press.
 Herrmann, R. (2018). Fractional Calculus  An Introduction for Physicists (3rd ed.). Singapore: World Scientific. doi:10.1142/11107. ISBN 9789813274570.
External links
 Eric W. Weisstein. "Fractional Differential Equation." From MathWorld — A Wolfram Web Resource.
 MathWorld – Fractional calculus
 MathWorld – Fractional derivative
 Fractional Calculus at MathPages
 Specialized journal: Fractional Calculus and Applied Analysis (1998–2014) and Fractional Calculus and Applied Analysis (from 2015)
 Specialized journal: Fractional Differential Equations (FDE)
 Specialized journal: Communications in Fractional Calculus (ISSN 22183892)
 Specialized journal: Journal of Fractional Calculus and Applications (JFCA)
 www.nasatech.com
 Igor Podlubny's collection of related books, articles, links, software, etc.
 GigaHedron – Richard Herrmann's collection of books, articles, preprints, etc.
 s.dugowson.free.fr
 History, Definitions, and Applications for the Engineer (PDF), by Adam Loverro, University of Notre Dame
 Fractional Calculus Modelling
 Introductory Notes on Fractional Calculus
 Power Law & Fractional Dynamics
 The CRONE Toolbox, a Matlab and Simulink Toolbox dedicated to fractional calculus, which is freely downloadable
 Závada, Petr (1998). "Operator of Fractional Derivative in the Complex Plane". Communications in Mathematical Physics. 192 (2): 261–285. arXiv:functan/9608002. Bibcode:1998CMaPh.192..261Z. doi:10.1007/s002200050299.
 Závada, Petr (2002). "Relativistic wave equations with fractional derivatives and pseudodifferential operators". Journal of Applied Mathematics. 2 (4): 163–197. arXiv:hepth/0003126. doi:10.1155/S1110757X02110102.