Fractional calculus
Part of a series of articles about  
Calculus  





Specialized


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 realnumber values of the parameter 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 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 { 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, 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.
Contents
 1 Historical notes
 2 Nature of the fractional derivative
 3 Heuristics
 4 Fractional derivative of a basic power function
 5 Laplace transform
 6 Fractional integrals
 7 Fractional derivatives
 8 Generalizations
 9 Functional calculus

10 Applications
 10.1 Fractional conservation of mass
 10.2 Groundwater flow problem
 10.3 Fractional advection dispersion equation
 10.4 Timespace fractional diffusion equation models
 10.5 Structural damping models
 10.6 PID controllers
 10.7 Acoustical wave equations for complex media
 10.8 Fractional Schrödinger equation in quantum theory
 10.9 Variableorder fractional Schrödinger equation
 11 See also
 12 Notes
 13 References
 14 External links
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 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]}
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 noninteger 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 integerpower derivatives certainly do. Therefore, it is expected that the theory involves some sort of boundary conditions, involving information on the function further out.^{[4]}
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.^{[citation needed]}
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 and , we obtain the halfderivative of the function as
To demonstrate that this is, in fact, the "Half Derivative" (where ), we repeat the process to get:
 (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:^{[5]}
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. Knowing that
and
etc., 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).
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 ^{[6]} 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 nonlocal and nonsingular kernel. They then constructed the following fractional integral operator:
where is a normalization function such that .^{[7]}^{[8]}
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 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 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.^{[9]} 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.
Atangana–Baleanu derivative
There is a new option of computing a fractional derivative, this option is based on the general MittagLeffler function as kernel. This new version was introduced by Abdon Atangana and Dumitru Baleanu in 2016 in their work.^{[10]} 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 MittagLeffler 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 MittagLeffler function.^{[11]} Atangana–Baleanu fractional derivative in Caputo sense is illustrated as follows.
And the Atangana–Baleanu fractional derivative in Riemann–Liouville is defined as:
The AtanganaBaleanu fractional derivatives brought new weapons into applied mathematics to model complex realworld problems more accurately because the derivatives have the following properties:.^{[12]}^{[13]}^{[14]}
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
 HilferKatugampola 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).^{[15]} and Hermann Kober (1940)^{[16]} 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]}^{[17]}
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),^{[18]} 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.^{[19]}^{[20]} 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^{[clarification needed]} has been shown useful for modeling contaminant flow in heterogenous porous media.^{[21]}^{[22]}^{[23]}
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–Nicolson method. 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^{[24]}
Timespace fractional diffusion equation models
Anomalous diffusion processes in complex media can be well characterized by using fractionalorder diffusion equation models.^{[25]}^{[26]} 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, the α, β are changed into α(x, t), β(x, t). Its applications in anomalous diffusion modeling can be found in reference.^{[24]}^{[27]}^{[28]}
Structural damping models
Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.^{[29]}
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 nonnegative, denote the coefficients for the proportional, integral, and derivative terms, respectively (sometimes denoted P, I, and D).^{[30]}
Acoustical wave equations for complex media
The propagation of acoustical waves in complex media, e.g. 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 ^{[31]} 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 ^{[32]} and in the survey paper,^{[33]} as well as the acoustic attenuation article. See ^{[34]} for a recent paper which compares fractional wave equations which model powerlaw attenuation.
Fractional Schrödinger equation in quantum theory
The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form:^{[35]}
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 ^{[36]}
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 is used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like glyphs, e.g. identities.
 ^ ^{a} ^{b} ^{c} ^{d} Katugampola, U.N., "A New Approach To Generalized Fractional Derivatives", Bull. Math. Anal. App. Vol 6, Issue 4, 15 October 2014, pages 1–15
 ^ 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)
 ^ "Fractional Calculus". www.mathpages.com. Retrieved 20180103.
 ^ Bologna, Mauro, Short Introduction to Fractional Calculus (PDF), Universidad de Tarapaca, Arica, Chile
 ^ 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.
 ^ 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
 ^ 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.
 ^ Caputo, Michele (1967). "Linear model of dissipation whose Q is almost frequency independentII". Geophysical Journal International. 13 (5): 529–539. Bibcode:1967GeoJ...13..529C. doi:10.1111/j.1365246x.1967.tb02303.x – via Oxford University Press. (Subscription required (help))..
 ^ Atangana A., Dumitru B. "New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model, Thermal Science, Year 2016, Vol. 20, No. 2, pp. 763–769".
 ^ 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.
 ^ Ervin Lenzi K., Tateishi Angel A., Haroldo Ribeiro V. (2017). "The Role of Fractional TimeDerivative Operators on Anomalous Diffusion". Frontiers in Physics. 5: 1–9. doi:10.3389/fphy.2017.00052.
 ^ Abdon, Atangana (2018). "Non validity of index law in fractional calculus: A fractional differential operator with Markovian and nonMarkovian properties". Physica A: Statistical Mechanics and its Applications. 505: 688–706. doi:10.1016/j.physa.2018.03.056.
 ^ Atangana A., J.F.Gomez Aguila. (2018). "Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena". The European Physical Journal Plus. 133: 166. doi:10.1140/epjp/i2018120213.
 ^ Erdélyi, Arthur (1950–51). "On some functional transformations". Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino. 10: 217–234. MR 0047818.
 ^ Kober, Hermann (1940). "On fractional integrals and derivatives". The Quarterly Journal of Mathematics (Oxford Series). 11 (1): 193–211. Bibcode:1940QJMat..11..193K. doi:10.1093/qmath/os11.1.193.
 ^ Katugampola, Udita N. (2011). "New approach to a generalized fractional integral". Applied Mathematics and Computation. 218: 860–865. arXiv:1010.0742 . doi:10.1016/j.amc.2011.03.062.
 ^ Wheatcraft, S., Meerschaert, M., (2008). "Fractional Conservation of Mass." Advances in Water Resources 31, 1377–1381.
 ^ 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.
 ^ Atangana, Abdon; Vermeulen, P. D. (2014). "Analytical Solutions of a SpaceTime Fractional Derivative of Groundwater Flow Equation". Abstract and Applied Analysis. 2014: 1–11. doi:10.1155/2014/381753.
 ^ Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "Application of a fractional advectiondispersion equation". Water Resources Res. 36: 1403–1412. Bibcode:2000WRR....36.1403B. doi:10.1029/2000wr900031.
 ^ Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "The fractionalorder governing equation of Lévy motion". Water Resources Res. 36: 1413–1423. Bibcode:2000WRR....36.1413B. doi:10.1029/2000wr900032.
 ^ Benson, D., Schumer, R., Wheatcraft, S., Meerschaert, M., (2001). "Fractional dispersion, Lévy motion, and the MADE tracer tests." Transport Porous Media 42, 211–240.
 ^ ^{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.
 ^ Metzler, R.; Klafter, J. (2000). "The random walk's guide to anomalous diffusion: a fractional dynamics approach". Phys. Rep. 339: 1–77. Bibcode:2000PhR...339....1M. doi:10.1016/s03701573(00)000703.
 ^ Mainardi, F.; Luchko, Y.; Pagnini, G. (2001). "The fundamental solution of the spacetime fractional diffusion equation". Fractional Calculus and Applied Analysis. 4 (2): 153–192. arXiv:condmat/0702419 . Bibcode:2007cond.mat..2419M.
 ^ 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
 ^ M. J. Colbrook, X. Ma P. F. Hopkins J. Squire, "Scaling laws of passivescalar diffusion in the interstellar medium", MNRAS, No 2, 2017, pp. 24212429 [1]
 ^ Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. by F. Mainardi, Imperial College Press, 2010.
 ^ 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.
 ^ 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.
 ^ 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.
 ^ 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/s1354001300031 Link to eprint
 ^ 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 [2]
 ^ N. Laskin, (2002), Fractional Schrödinger equation, Physical Review E66, 056108 7 pages. (also available online: https://arxiv.org/abs/quantph/0206098)
 ^ 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.
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. Bibcode:2011CNSNS..16.1140M. 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/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.
Review articles
 Ortigueira, M.D.; Machado, J.A. Tenreiro (2015). "What is a fractional derivative?". Journal of Computational Physics. 293: 4–13. Bibcode:2015JCoPh.293....4O. 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 0125255500.
 Miller, Kenneth S.; Ross, Bertram, eds. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. ISBN 0471588849.
 Samko, S.; Kilbas, A.A.; Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications. Taylor & Francis Books. ISBN 2881248640.
 Carpinteri, A.; Mainardi, F., eds. (1998). Fractals and Fractional Calculus in Continuum Mechanics. SpringerVerlag Telos. ISBN 321182913X.
 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 0125588402.
 West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003). Physics of Fractal Operators. Springer Verlag. ISBN 0387955542.
 Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 0444518320.
 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.
 Daftardargejji, 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 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 (R) Toolbox, a Matlab and Simulink Toolbox dedicated to fractional calculus, which is freely downloadable
 [3]
 Závada, Petr (1998). "Operator of Fractional Derivative in the Complex Plane". Communications in Mathematical Physics. 192: 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: 163–197. doi:10.1155/S1110757X02110102.