Fractional calculus

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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

,

and of the integration operator J

,[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 to a function, in some analogy to function composition acting on a variable, i.e. f ∘2(x) = f ∘ f (x) = f ( f (x) ).

For example, one may ask the question of meaningfully interpreting

as an analogue of the functional square root for the differentiation operator, i.e., 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 real-number values of the parameter a in such a way that, when a takes an integer value n ∈ ℤ, it coincides with the usual n-fold differentiation D if n > 0, and with the -n–th power of J when n < 0.

One of the motivations behind the introduction and study of such kind of extensions of the differentiation operator D is that the sets of operator powers Da | a ∈ ℝ } defined in this way are continuous semigroups with parameter a, of which the original discrete semigroup of Dn | n ∈ ℤ } for integer n is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, it is interesting to apply it to other branches of mathematics.

Fractional differential equations (also known as extraordinary differential equations) are a generalization of differential equations through the application of fractional calculus.

Historical notes

In applied mathematics and mathematical analysis, fractional derivative is a derivative of any arbitrary order, real or complex. The first appearance of the concept of a fractional derivative is found in a letter by the famous mathematician Leibniz in 1695 in a letter to Guillaume de l'Hôpital.[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]

Nature of the fractional derivative

The derivative of a function at a point x is a local property only when a is an integer; this is not the case for non-integer power derivatives. In other words, it is not correct to say that the fractional derivative at of a function depends only on values of very near , in the way that integer-power derivatives certainly do. Therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out.

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 half-derivative, such that

.

It turns out that there is such an operator, and indeed for any a > 0, there exists an operator P such that

or to put it another way, the definition of dny/dxn can be extended to all real values of n.

Let f(x) be a function defined for x > 0. Form the definite integral from 0 to x. Call this

.

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 well-defined operator.

It is straightforward to show that the J operator satisfies

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.[citation needed]

Fractional derivative of a basic power function

The half derivative (purple curve) of the function f(x) = x (blue curve) together with the first derivative (red curve).
The animation shows the derivative operator oscillating between the antiderivative (α=−1: y = 12x2) and the derivative (α = +1: y = 1) of the simple power function y = x continuously.

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 and , we obtain the half-derivative of the function as

Repeating this process yields

(because and )

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:[4]

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 notice that 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. Noting that

and

etc., we assert

.

For example

as expected. Indeed, given the convolution rule

and shorthanding p(x) = xα−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

Atangana–Baleanu fractional integral

Recently, using the generalized Mittag-Leffler function, Atangana and Baleanu suggested a new formulation of the fractional derivative with nonlocal and nonsingular kernel. They then constructed the following fractional integral operator:

where is a normalization function such that .[5][6]

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).

By contrast the Grünwald–Letnikov derivative starts with the derivative instead of the integral.

Hadamard fractional integral

The Hadamard fractional integral is introduced by J. Hadamard [7] and is given by the following formula,

Fractional derivatives

Not like 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 n-th order derivative over the integral of order (nα), the α order derivative is obtained. It is important to remark that n is the nearest integer greater than α ( that is, ).

Caputo fractional derivative

There is another option for computing fractional derivatives; the Caputo fractional derivative. It was introduced by M. Caputo in his 1967 paper.[8] 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.

The following list summarizes the fractional derivatives defined in the literature.[2]

Atangana–Baleanu derivative

There is a new option of computing a fractional derivative, this option is based on the general Mittag-Leffler function as kernel. This new version was introduced by Abdon Atangana and Dumitru Baleanu in 2016 in their work.[9] The authors introduced two versions, Atangana–Baleanu in Caputo sense (ABC) which is the convolution of a local derivative of a given function with the generalized Mittag-Leffler function. The second version is called Atangana–Baleanu fractional derivative in Riemann–Liouville sense (ABR) and is the derivative of a convolution of a given function non necessary differentiable with the generalized Mittag-Leffler function.[10] Atangana–Baleanu fractional derivative in Caputo sense is illustrated as follows.

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
  • Chen–Machado derivative
  • Coimbra derivative
  • Katugampola derivative
  • Caputo–Katugampola derivative
  • Hilfer derivative
  • Hilfer-Katugampola derivative
  • Davidson derivative
  • Chen derivative
  • Atangana–Baleanu derivative
  • Pichaghchi derivative

Generalizations

Erdélyi–Kober operator

The Erdélyi–Kober operator is an integral operator introduced by Arthur Erdélyi (1940).[11] and Hermann Kober (1940)[12] and is given by

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

Katugampola operators

A recent generalization introduced by Udita Katugampola (2011) 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][13]

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 Katugampola-type 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 pseudo-differential 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),[14] a fractional conservation of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is:

Groundwater flow problem

In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of derivative with fractional order.[15][16] In these works, The classical Darcy law is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow.

Fractional advection dispersion equation

This equation[clarification needed] has been shown useful for modeling contaminant flow in heterogenous porous media.[17][18][19]

Atangana and Kilicman extended fractional advection dispersion equation to variable order fractional advection dispersion equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of variational order derivative. The modified equation was numerically solved via the Crank-Nicholson scheme. The stability and convergence of the scheme in this case were presented. The numerical simulations showed that, the modified equation is more reliable in predicting the movement of pollution in the deformable aquifers, than the constant fractional and integer derivatives[20]

Time-space fractional diffusion equation models

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.[21][22] The time derivative term is corresponding to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation can be written as

A simple extension of fractional derivative is the variable-order fractional derivative, the α, β are changed into α(x, t), β(x, t). Its applications in anomalous diffusion modeling can be found in reference.[20][23]

Structural damping models

Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.[24]

PID controllers

Generalizing PID controllers to use fractional orders can increase their degree of freedom. The new equation relating the control variable in terms of a measured error value can be written as

where and are positive fractional orders and , , and , all non-negative, denote the coefficients for the proportional, integral, and derivative terms, respectively (sometimes denoted P, I, and D).[25]

Acoustical wave equations for complex media

The propagation of acoustical waves in complex media, e.g. biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:

See also [26] 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 [27] and in the survey paper,[28] as well as the acoustic attenuation article. See [29] for a recent paper which compares fractional wave equations which model power-law attenuation.

Fractional Schrödinger equation in quantum theory

The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form:[30]

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/r2 is the Laplace operator, and Dα is a scale constant with physical dimension [Dα] = J1 − α·mα·sα = Kg1 − α·m2−α·sα−2, (at α = 2, D2 = 1/2m for a particle of mass m), and the operator (−ħ2Δ)α/2 is the 3-dimensional fractional quantum Riesz derivative defined by

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

Variable-order fractional Schrödinger equation

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena [31]

where Δ = 2/r2 is the Laplace operator and the operator (−ħ2Δ)β (t)/2 is the variable-order fractional quantum Riesz derivative.

See also

Other fractional theories

Notes

  1. ^ The symbol J is commonly is used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like glyphs, e.g. identities.
  2. ^ a b c d e Katugampola, U.N., "A New Approach To Generalized Fractional Derivatives", Bull. Math. Anal. App. Vol 6, Issue 4, 15 October 2014, pages 1–15
  3. ^ 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)
  4. ^ Bologna, Mauro, Short Introduction to Fractional Calculus (PDF), Universidad de Tarapaca, Arica, Chile 
  5. ^ Badr Saad T. Alkahtani, " Chua's circuit model with Atangana–Baleanu derivative with fractional order Chaos, Solitons & Fractals", Volume 89, August 2016, Pages 547–551
  6. ^ Obaid Jefain Julaighim Algahtani. Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn modelChaos, Solitons & Fractals, Volume 89, August 2016, Pages 552–559.
  7. ^ Hadamard, J., Essai sur l'étude des fonctions données par leur développement de Taylor, Journal of pure and applied mathematics, vol. 4, no. 8, pp. 101–186, 1892.
  8. ^ Caputo, Michele (1967). "Linear model of dissipation whose Q is almost frequency independent-II". Geophysical Journal International. 13 (5): 529–539. doi:10.1111/j.1365-246x.1967.tb02303.x – via Oxford University Press. (Subscription required (help)). .
  9. ^ Atangana A., Dumitru B. "New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, Year 2016, Vol. 20, No. 2, pp. 763–769". 
  10. ^ Atangana A., Koca I. (2016). "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos, Solitons & Fractals, Volume 89, August 2016, Pages 447–454". Chaos, Solitons & Fractals. 89: 447–454. doi:10.1016/j.chaos.2016.02.012. 
  11. ^ Erdélyi, Arthur (1950–51). "On some functional transformations". Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino. 10: 217–234. MR 0047818. 
  12. ^ Kober, Hermann (1940). "On fractional integrals and derivatives". The Quarterly Journal of Mathematics (Oxford Series). 11 (1): 193–211. doi:10.1093/qmath/os-11.1.193. 
  13. ^ Katugampola, Udita N. (2011). "New approach to a generalized fractional integral". Applied Mathematics and Computation. 218: 860–865. doi:10.1016/j.amc.2011.03.062. 
  14. ^ Wheatcraft, S., Meerschaert, M., (2008). "Fractional Conservation of Mass." Advances in Water Resources 31, 1377–1381.
  15. ^ 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. 
  16. ^ Atangana, Abdon; Vermeulen, P. D. (2014). "Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation". Abstract and Applied Analysis. 2014: 1–11. doi:10.1155/2014/381753. 
  17. ^ Benson, D., Wheatcraft, S., Meerschaert, M., (2000). "Application of a fractional advection-dispersion equation." Water Resources Res 36, 1403–1412.
  18. ^ Benson, D., Wheatcraft, S., Meerschaert, M., (2000). "The fractional-order governing equation of Lévy motion." Water Resources Res 36, 1413–1423.
  19. ^ Benson, D., Schumer, R., Wheatcraft, S., Meerschaert, M., (2001). "Fractional dispersion, Lévy motion, and the MADE tracer tests." Transport Porous Media 42, 211–240.
  20. ^ a b 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. 
  21. ^ Metzler, R., Klafter, J., (2000). "The random walk's guide to anomalous diffusion: a fractional dynamics approach." Phys. Rep., 339, 1–77.
  22. ^ F. Mainardi, Y. Luchko, G. Pagnini, "The fundamental solution of the space-time fractional diffusion equation", Fractional Calculus and Applied Analysis, Vol. 4, No 2 (2001). 153–192 arXiv:cond-mat/0702419
  23. ^ R. Gorenflo, F. Mainardi, "Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk", Springer Lecture Notes in Physics, No 621, Berlin 2003, pp. 148–166 arXiv:0709.3990
  24. ^ Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. by F. Mainardi, Imperial College Press, 2010.
  25. ^ Tenreiro Machado JA, et al. (2009). "Some Applications of Fractional Calculus in Engineering". Mathematical Problems in Engineering. 2010: 1–34. doi:10.1155/2010/639801. 
  26. ^ Holm, S.; Näsholm, S. P. (2011). "A causal and fractional all-frequency wave equation for lossy media". Journal of the Acoustical Society of America. 130 (4): 2195–2201. 
  27. ^ Näsholm, S. P.; Holm, S. (2011). "Linking multiple relaxation, power-law attenuation, and fractional wave equations". Journal of the Acoustical Society of America. 130 (5): 3038–3045. 
  28. ^ S. P. Näsholm and S. Holm, "On a Fractional Zener Elastic Wave Equation," Fract. Calc. Appl. Anal. Vol. 16, No 1 (2013), pp. 26–50, DOI: 10.2478/s13540-013--0003-1 Link to e-print
  29. ^ Holm S., Näsholm, S. P., "Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography," Ultrasound Med. Biol., 40(4), pp. 695–703, DOI: 10.1016/j.ultrasmedbio.2013.09.033 [1]
  30. ^ N. Laskin, (2002), Fractional Schrödinger equation, Physical Review E66, 056108 7 pages. (also available online: http://arxiv.org/abs/quant-ph/0206098)
  31. ^ Bhrawy, A.H.; Zaky, M.A. (2017). "An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations". Applied Numerical Mathematics. 111: 197–218. doi:10.1016/j.apnum.2016.09.009. 

References

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. 457: 1–36. 
  • 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. doi:10.1016/j.cnsns.2010.05.027. 
  • 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/s13540-013-0030-y. 
  • 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/s11192-013-1032-6. 

Review articles

  • Ortigueira, M.D.; Machado, J.A. Tenreiro (2015). "What is a fractional derivative?". Journal of Computational Physics. 293: 4–13. doi:10.1016/j.jcp.2014.07.019. 
  • U.N. Katugampola, Correction to ``What is a fractional derivative? by Ortigueira and Machado [Journal of Computational Physics, Volume 293, 15 July 2015, Pages 4–13. Special issue on Fractional PDEs], Submitted for publication in J. Computational Physics (2015)
  • V.E. Tarasov "No violation of the Leibniz rule. No fractional derivative." Communications in Nonlinear Science and Numerical Simulation Vol.18. No.11. (2013) pp. 2945–2948. arXiv:1402.7161

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 0-12-525550-0. 
  • Miller, Kenneth S.; Ross, Bertram, eds. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. ISBN 0-471-58884-9. 
  • Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN 2-88124-864-0. 
  • Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag Telos. ISBN 3-211-82913-X. 
  • Podlubny, Igor (1998). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. 198. Academic Press. ISBN 0-12-558840-2. 
  • West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003). Physics of Fractal Operators. Springer Verlag. ISBN 0-387-95554-2. 
  • Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 0-444-51832-0. 
  • 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. Springer. 
  • Zhou, Y. (2010). Basic Theory of Fractional Differential Equations. Singapore: World Scientific. 
  • Uchaikin, V.V. (2012). Fractional Derivatives for Physicists and Engineers. Higher Education Press. 
  • Daftardar-gejji, Varsha (2013). Fractional Calculus: Theory and Applications. Narosa Publishing House. 
  • Herrmann, R. (2014). Fractional Calculus - An Introduction for Physicists. Singapore: World Scientific. 
  • Srivastava, Hari M (2014). Special Functions in Fractional Calculus and Related Fractional Differintegral Equations. Singapore: World Scientific. 
  • Li, C P; Zeng, F H (2015). Numerical Methods for Fractional Calcuus. USA: CRC Press. 

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)[permanent dead link]
  • Specialized journal: Progress in Fractional Differentiation and Applications
  • Specialized journal: Communications in Fractional Calculus (ISSN 2218-3892)
  • 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 (R) Toolbox, a Matlab and Simulink Toolbox dedicated to fractional calculus, which is freely downloadable
  • [2]
  • Závada, Petr (1998). "Operator of Fractional Derivative in the Complex Plane". Communications in Mathematical Physics. 192: 261–285. arXiv:funct-an/9608002Freely accessible. doi:10.1007/s002200050299. 
  • Závada, Petr (2002). "Relativistic wave equations with fractional derivatives and pseudodifferential operators". Journal of Applied Mathematics. 2: 163–197. doi:10.1155/S1110757X02110102. 
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