Calculus 



Integral calculus
 Definitions

 Integration by







 "Fractional derivative" redirects here.
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator
 $D\; =\; \backslash dfrac\{d\}\{dx\},$
and the integration operator J. (Usually J is used instead of I to avoid confusion with other Ilike glyphs and identities.)
In this context the term powers refers to iterative application or function composition, in the same sense that f^{2}(x) = f(f(x)). For example, one may ask the question of meaningfully interpreting
 $\backslash sqrt\{D\}\; =\; D^\{\backslash frac\{1\}\{2\}\}\; \backslash ,$
as a functional square root of the differentiation operator (an operator half iterated), i.e., an expression for some operator that when applied twice to a function will have the same effect as differentiation. More generally, one can look at the question of defining
 $D^a\; \backslash ,$
for realnumber values of a in such a way that when a takes an integer value n, the usual power of nfold differentiation is recovered for n > 0, and the −nth power of J when n < 0.
The motivation behind this extension to the differential operator is that the semigroup of powers D^{a} will form a continuous semigroup with parameter a, inside which the original discrete semigroup of D^{n} for integer n can be recovered as a subgroup. Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that fraction is then a misnomer for the exponent a, since it need not be rational; the use of the term fractional calculus is merely conventional.
Fractional differential equations are a generalization of differential equations through the application of fractional calculus.
Nature of the fractional derivative
Not to be confused with Fractal derivative.
An important point is that the fractional derivative at a point x is a local property only when a is an integer; in noninteger cases we cannot say that the fractional derivative at x of a function f 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. To use a metaphor, the fractional derivative requires some peripheral vision.
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. The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms.^{[1]}
Heuristics
A fairly natural question to ask is whether there exists an operator H, or halfderivative, such that
 $H^2\; f(x)\; =\; D\; f(x)\; =\; \backslash dfrac\{d\}\{dx\}\; f(x)\; =\; f\text{'}(x)$.
It turns out that there is such an operator, and indeed for any a > 0, there exists an operator P such that
 $(P\; ^\; a\; f)(x)\; =\; f\text{'}(x)\; \backslash ,$,
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
 $(\; J\; f\; )\; (\; x\; )\; =\; \backslash int\_0^x\; f(t)\; \backslash ;\; dt$.
Repeating this process gives
 $(\; J^2\; f\; )\; (\; x\; )\; =\; \backslash int\_0^x\; (\; J\; f\; )\; (\; t\; )\; dt\; =\; \backslash int\_0^x\; \backslash left(\; \backslash int\_0^t\; f(s)\; \backslash ;\; ds\; \backslash right)\; \backslash ;\; dt$,
and this can be extended arbitrarily.
The Cauchy formula for repeated integration, namely
 $(J^n\; f)\; (\; x\; )\; =\; \{\; 1\; \backslash over\; (n1)!\; \}\; \backslash int\_0^x\; (xt)^\{n1\}\; f(t)\; \backslash ;\; dt,$
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.
 $(J^\backslash alpha\; f)\; (\; x\; )\; =\; \{\; 1\; \backslash over\; \backslash Gamma\; (\; \backslash alpha\; )\; \}\; \backslash int\_0^x\; (xt)^\{\backslash alpha1\}\; f(t)\; \backslash ;\; dt$
This is in fact a welldefined operator.
It is straightforward to show that the J operator satisfies
 $(J^\backslash alpha)\; (J^\backslash beta\; f)(x)\; =\; (J^\backslash beta)\; (J^\backslash alpha\; f)(x)\; =\; (J^\{\backslash alpha+\backslash beta\}\; f)(x)\; =\; \{\; 1\; \backslash over\; \backslash Gamma\; (\; \backslash alpha\; +\; \backslash beta)\; \}\; \backslash int\_0^x\; (xt)^\{\backslash alpha+\backslash beta1\}\; f(t)\; \backslash ;\; dt$
Proof

 $$
\begin{align}
(J^\alpha) (J^\beta f)(x) & = \frac{1}{\Gamma(\alpha)} \int_0^x (xt)^{\alpha1} (J^\beta f)(t) \; dt \\
& = \frac{1}{\Gamma(\alpha) \Gamma(\beta)} \int_0^x \int_0^t (xt)^{\alpha1} (ts)^{\beta1} f(s) \; ds \; dt \\
& = \frac{1}{\Gamma(\alpha) \Gamma(\beta)} \int_0^x f(s) \left( \int_s^x (xt)^{\alpha1} (ts)^{\beta1} \; dt \right) ds
\end{align}
where in the last step we exchanged the order of integration and pulled out the f(s) factor from the t integration. Changing variables to r defined by t = s + (x − s)r,
 $$
(J^\alpha) (J^\beta f)(x) = \frac{1}{\Gamma(\alpha) \Gamma(\beta)} \int_0^x (xs)^{\alpha + \beta  1} f(s) \left( \int_0^1 (1r)^{\alpha1} r^{\beta1} \; dr \right) ds
The inner integral is the beta function which satisfies the following property:
 $$
\int_0^1 (1r)^{\alpha1} r^{\beta1} \; dr = B(\alpha, \beta) = \dfrac{\Gamma(\alpha)\,\Gamma(\beta)}{\Gamma(\alpha+\beta)}
Substituting back into the equation
 $$
(J^\alpha) (J^\beta f)(x) = \frac{1}{\Gamma(\alpha + \beta)} \int_0^x (xs)^{\alpha + \beta  1} f(s) \; ds = (J^{\alpha + \beta} f)(x)
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.
Fractional derivative of a basic power function
Let us assume that f(x) is a monomial of the form
 $f(x)=x^k\backslash ;.$
The first derivative is as usual
 $f\text{'}(x)=\backslash dfrac\{d\}\{dx\}f(x)=k\; x^\{k1\}\backslash ;.$
Repeating this gives the more general result that
 $\backslash dfrac\{d^a\}\{dx^a\}x^k=\backslash dfrac\{k!\}\{(ka)!\}x^\{ka\}\backslash ;,$
Which, after replacing the factorials with the gamma function, leads us to
 $\backslash dfrac\{d^a\}\{dx^a\}x^k=\backslash dfrac\{\backslash Gamma(k+1)\}\{\backslash Gamma(ka+1)\}x^\{ka\}\backslash ;.$
For $k=1$ and $\backslash textstyle\; a=\backslash frac\{1\}\{2\}$, we obtain the halfderivative of the function $x$ as
 $\backslash dfrac\{d^\{\backslash frac\{1\}\{2\}\}\}\{dx^\{\backslash frac\{1\}\{2\}\}\}x=\backslash dfrac\{\backslash Gamma(1+1)\}\{\backslash Gamma(1\backslash frac\{1\}\{2\}+1)\}x^\{1\backslash frac\{1\}\{2\}\}=\backslash dfrac\{1!\}\{\backslash Gamma(\backslash frac\{3\}\{2\})\}x^\{\backslash frac\{1\}\{2\}\}\; =$
\dfrac{2x^{\frac{1}{2}}}{\sqrt{\pi}}.
Repeating this process yields
 $\backslash dfrac\{d^\{\backslash frac\{1\}\{2\}\}\}\{dx^\{\backslash frac\{1\}\{2\}\}\}2\; \backslash pi^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}=2\; \backslash pi^\{\backslash frac\{1\}\{2\}\}\backslash dfrac\{\backslash Gamma(1+\backslash frac\{1\}\{2\})\}\{\backslash Gamma(\backslash frac\{1\}\{2\}\backslash frac\{1\}\{2\}+1)\}x^\{\backslash frac\{1\}\{2\}\backslash frac\{1\}\{2\}\}=2\; \backslash pi^\{\backslash frac\{1\}\{2\}\}\backslash dfrac\{\backslash Gamma(\backslash frac\{3\}\{2\})\}\{\backslash Gamma(1)\}x^\{0\}=\backslash dfrac\{2\; \backslash sqrt\{\backslash pi\}x^0\}\{2\; \backslash sqrt\{\backslash pi\}0!\}=1,$
which is indeed the expected result of
 $\backslash left(\backslash dfrac\{d^\{\backslash frac\{1\}\{2\}\}\}\{dx^\{\backslash frac\{1\}\{2\}\}\}\backslash dfrac\{d^\{\backslash frac\{1\}\{2\}\}\}\{dx^\{\backslash frac\{1\}\{2\}\}\}\backslash right)x=\backslash dfrac\{d\}\{dx\}x=1.$
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
 $D^\{\backslash alpha\}f(x)=\backslash frac\{1\}\{\backslash Gamma(1\backslash alpha)\}\backslash frac\{d\}\{dx\}\backslash int\_\{0\}^\{x\}\backslash frac\{f(t)\}\{(xt)^\{\backslash alpha\}\}dt$
For arbitrary α, since the gamma function is undefined for arguments whose real part is a negative integer, it is necessary to apply the fractional derivative after the integer derivative has been performed. For example,
 $D^\{\backslash frac\{3\}\{2\}\}f(x)=D^\{\backslash frac\{1\}\{2\}\}D^\{1\}f(x)=D^\{\backslash frac\{1\}\{2\}\}\backslash frac\{d\}\{dx\}f(x)$
Laplace transform
We can also come at the question via the Laplace transform. Noting that
 $\backslash mathcal\; L\; \backslash left\backslash \{Jf\backslash right\backslash \}(s)\; =\; \backslash mathcal\; L\; \backslash left\backslash \{\backslash int\_0^t\; f(\backslash tau)\backslash ,d\backslash tau\backslash right\backslash \}(s)=\backslash frac1s(\backslash mathcal\; L\backslash left\backslash \{f\backslash right\backslash \})(s)$
and
 $\backslash mathcal\; L\; \backslash left\backslash \{J^2f\backslash right\backslash \}=\backslash frac1s(\backslash mathcal\; L\; \backslash left\backslash \{Jf\backslash right\backslash \}\; )(s)=\backslash frac1\{s^2\}(\backslash mathcal\; L\backslash left\backslash \{f\backslash right\backslash \})(s)$
etc., we assert
 $J^\backslash alpha\; f=\backslash mathcal\; L^\{1\}\backslash left\backslash \{s^\{\backslash alpha\}(\backslash mathcal\; L\backslash \{f\backslash \})(s)\backslash right\backslash \}$.
For example
 $$
\begin{array}{lcr}
J^\alpha\left(t^k\right) &= &\mathcal L^{1}\left\{\dfrac{\Gamma(k+1)}{s^{\alpha+k+1}}\right\}\\
&= &\dfrac{\Gamma(k+1)}{\Gamma(\alpha+k+1)}t^{\alpha+k}
\end{array}
as expected. Indeed, given the convolution rule
 $\backslash mathcal\; L\backslash \{f*g\backslash \}=(\backslash mathcal\; L\backslash \{f\backslash \})(\backslash mathcal\; L\backslash \{g\backslash \})$
and shorthanding p(x) = x^{α − 1} for clarity, we find that
 $$
\begin{array}{rcl}
(J^\alpha f)(t) &= &\frac{1}{\Gamma(\alpha)}\mathcal L^{1}\left\{\left(\mathcal L\{p\}\right)(\mathcal L\{f\})\right\}\\
&=&\frac{1}{\Gamma(\alpha)}(p*f)\\
&=&\frac{1}{\Gamma(\alpha)}\int_0^t p(t\tau)f(\tau)\,d\tau\\
&=&\frac{1}{\Gamma(\alpha)}\int_0^t(t\tau)^{\alpha1}f(\tau)\,d\tau\\
\end{array}
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).
 $\_aD\_t^\{\backslash alpha\}\; f(t)=\{\_aI\_t^\backslash alpha\}f(t)=\backslash frac\{1\}\{\backslash Gamma(\backslash alpha)\}\backslash int\_a^t\; (t\backslash tau)^\{\backslash alpha1\}f(\backslash tau)d\backslash tau$
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 ^{[2]} and is given by the following formula,
 $\_a\backslash mathbf\{D\}\_t^\{\backslash alpha\}\; f(t)\; =\; \backslash frac\{1\}\{\backslash Gamma(\backslash alpha)\}\backslash int\_a^t\; \backslash Bigg(\backslash log\backslash frac\{t\}\{\backslash tau\}\backslash Bigg)^\{\backslash alpha\; 1\}\; f(\backslash tau)\backslash frac\{d\backslash tau\}\{\backslash tau\}$
for t > a.
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 bigger than α.
 $\_aD\_t^\backslash alpha\; f(t)=\backslash frac\{d^n\}\{dt^n\}\{\_aD\_t^\{(n\backslash alpha)\}\}f(t)=\backslash frac\{d^n\}\{dt^n\}\{\_aI\_t^\{n\backslash alpha\}\}f(t)$
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.^{[3]} 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.
 $\{\_a^CD\_t^\backslash alpha\}\; f(t)=\backslash frac\{1\}\{\backslash Gamma(n\backslash alpha)\}\backslash int\_a^t\; \backslash frac\{f^\{(n)\}(\backslash tau)d\backslash tau\}\{(t\backslash tau)^\{\backslash alpha+1n\}\}$
Generalizations
Erdélyi–Kober operator
The Erdélyi–Kober operator is an integral operator introduced by Arthur Erdélyi (1940) and Hermann Kober (1940) and is given by
 $\backslash frac\{x^\{\backslash nu\backslash alpha+1\}\}\{\backslash Gamma(\backslash alpha)\}\backslash int\_0^x\; (tx)^\{\backslash alpha1\}t^\{\backslash alpha\backslash nu\}f(t)\; dt$
which generalizes the Riemann fractional integral and the Weyl integral. A recent generalization is the following, which generalizes the RiemannLiouville fractional integral and the Hadamard fractional integral. It is given by,^{[4]}
 $(\{\}^\backslash rho\; \backslash mathcal\{I\}^\backslash alpha\_\{a+\}f)(x)\; =\; \backslash frac\{\backslash rho^\{1\; \backslash alpha\; \}\}\{\backslash Gamma(\{\backslash alpha\})\}\; \backslash int^x\_a\; \backslash frac\{\backslash tau^\{\backslash rho1\}\; f(\backslash tau)\; \}\{(x^\backslash rho\; \; \backslash tau^\backslash rho)^\{1\backslash alpha\}\}\backslash ,\; d\backslash tau,$
for x > a.
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),^{[5]} 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:
 $\backslash rho\; (\backslash nabla^\{\backslash alpha\}\; \backslash cdot\; \backslash vec\{u\})\; =\; \backslash Gamma(\backslash alpha\; +1)\backslash Delta\; x^\{1\backslash alpha\}\backslash rho(\backslash beta\_s+\backslash phi\; \backslash beta\_w)\; \backslash frac\{\backslash part\; p\}\{\backslash part\; t\}$
Fractional advection dispersion equation
This equation has been shown useful for modeling contaminant flow in heterogenous porous media.^{[6]}^{[7]}^{[8]}
Timespace fractional diffusion equation models
Anomalous diffusion processes in complex media can be well characterized by using fractionalorder diffusion equation models.^{[9]}^{[10]} 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
 $$
\frac{\partial^\alpha u}{\partial t^\alpha}=K (\triangle)^\beta u.
A simple extension of fractional derivative is the variableorder fractional derivative, the α, β are changed into α(x, t), β(x, t). Its applications in anomalous diffusion modeling can be found in reference.^{[11]}
Structural damping models
Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.^{[12]}
Acoustical wave equations for complex media
The propagation of acoustical waves in complex media, e.g. biological tissue, commonly implies attenuation obeying a frequency powerlaw. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:
 $$
{\nabla^2 u \dfrac 1{c_0^2}\frac{\partial^2 u}{\partial t^2} + \tau_\sigma^\alpha \dfrac{\partial^\alpha}{\partial t^\alpha}\nabla^2 u  \dfrac {\tau_\epsilon^\beta}{c_0^2} \dfrac{\partial^{\beta+2} u}{\partial t^{\beta+2}} = 0.}
See also ^{[13]} 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 ^{[14]} and in the survey paper,^{[15]} as well as the acoustic attenuation article.
Fractional Schrödinger equation in quantum theory
The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics discovered by Nick Laskin,^{[16]} has the following form:^{[17]}
 $i\backslash hbar\; \backslash frac\{\backslash partial\; \backslash psi\; (\backslash mathbf\{r\},t)\}\{\backslash partial\; t\}=D\_\backslash alpha\; (\backslash hbar^2\backslash Delta\; )^\{\backslash alpha\; /2\}\backslash psi\; (\backslash mathbf\{r\},t)+V(\backslash mathbf\{r\},t)\backslash psi\; (\backslash mathbf\{r\},t).$
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}] = erg^{1 − α}·cm^{α}·sec^{−α}, (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
 $$
(\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar
)^3}\int d^3pe^{i \mathbf{p}\cdot\mathbf{r}/\hbar }\mathbf{p}^\alpha \varphi (
\mathbf{p},t)\,.
The index α in the fractional Schrödinger equation is the Lévy index, 1 < α ≤ 2.
See also
Notes
References
 Fractional Integrals and Derivatives: Theory and Applications, by Samko, S.; Kilbas, A.A.; and Marichev, O. Hardcover: 1006 pages. Publisher: Taylor & Francis Books. ISBN 2881248640
 Theory and Applications of Fractional Differential Equations, by Kilbas, A. A.; Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Elsevier, February 2006. http://www.elsevier.com/wps/find/bookdescription.cws_home/707212/description#description)
 An Introduction to the Fractional Calculus and Fractional Differential Equations, by Kenneth S. Miller, Bertram Ross (Editor). Hardcover: 384 pages. Publisher: John Wiley & Sons; 1 edition (May 19, 1993). ISBN 0471588849
 The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V), by Keith B. Oldham, Jerome Spanier. Hardcover. Publisher: Academic Press; (November 1974). ISBN 0125255500
 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, vol. 198), by Igor Podlubny. Hardcover. Publisher: Academic Press; (October 1998) ISBN 0125588402
 Fractional Calculus. An Introduction for Physicists, by Richard Herrmann. Hardcover. Publisher: World Scientific, Singapore; (February 2011) http://www.worldscientific.com/worldscibooks/10.1142/8072)
 Fractals and quantum mechanics, by N. Laskin. Chaos Vol.10, pp. 780–790 (2000). (http://link.aip.org/link/?CHAOEH/10/780/1)
 Fractals and Fractional Calculus in Continuum Mechanics, by A. Carpinteri (Editor), F. Mainardi (Editor). Paperback: 348 pages. Publisher: SpringerVerlag Telos; (January 1998). ISBN 321182913X
 Physics of Fractal Operators, by Bruce J. West, Mauro Bologna, Paolo Grigolini. Hardcover: 368 pages. Publisher: Springer Verlag; (January 14, 2003). ISBN 0387955542
 Fractional Calculus and the TaylorRiemann Series, RoseHulman Undergrad. J. Math. Vol.6(1) (2005).
 Operator of fractional derivative in the complex plane, by Petr Zavada, Commun.Math.Phys.192, pp. 261–285,1998. arXiv preprint)
 Relativistic wave equations with fractional derivatives and pseudodifferential operators, by Petr Zavada, Journal of Applied Mathematics, vol. 2, no. 4, pp. 163–197, 2002. arXiv preprint)
 Fractional differentiation by neocortical pyramidal neurons, by Brian N Lundstrom, Matthew H Higgs, William J Spain & Adrienne L Fairhall, Nature Neuroscience, vol. 11 (11), pp. 1335 – 1342, 2008. abstract)
 Equilibrium points, stability and numerical solutions of fractionalorder predatorprey and rabies models, by Ahmed E., A.M.A. ElSayed, H.A.A. ElSaka. 2007. Jour. Math. Anal. Appl. 325,452.


 Recent history of fractional calculus by J.T. Machado, V. Kiryakova, F. Mainardi,
External links
 MathWorld — A Wolfram Web Resource.
 MathWorld  Fractional calculus
 MathWorld  Fractional derivative
 Fractional Calculus at MathPages
 Specialized journal: Fractional Calculus and Applied Analysis
 Specialized journal: Fractional Differential Equations (FDE)
 Specialized journal: Communications in Fractional Calculus (ISSN 22183892)
 www.nasatech.com
 unr.edu (Broken Link)
 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
 University of Notre Dame
 Fractional Calculus Modelling
 Introductory Notes on Fractional Calculus
 Pseudodifferential operators and diffusive representation in modeling, control and signal
 Power Law & Fractional Dynamics
 The CRONE (R) Toolbox, a Matlab and Simulink Toolbox dedicated to fractional calculus, which is freely downloadable
 Introduction to fractional derivatives
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.