Kesirli analiz

"Fraksiyonel türev" buraya yönlendirir.

Fraksiyonel türev matematiksel analizin bir branşı diferansiyel işlemcinin gerçel sayılar kuvvetleri veya karmaşık sayı kuvvetleri olasılığı çalışmalar alarak

$D={\dfrac {d}{dx}},$

ve integrasyon işlemcisi J. (genellikle J diğer I-yımsı kabartma ile karışıklığı önlemeye I yerine kullanılan ve özdeşlik.)

Bu konu içinde ardışık uygulamalar veya fonksyon düzenine kaynak kuvvet terimleri mantığı içinde f²(x) = f(f(x)). Örneğin, one anlamlı karşılaştırmanın bir soru sorabilir

${\sqrt {D}}=D^{\frac {1}{2}}\,$

as bir fonksiyonel kare kök of the differentiation operator (mathematics)işlemci (yarı ardışık bir işlemci ), yani, bir expression for bazı işlemci için bu when applied ikinci to bir fonksiyon diferansiyasyon olarak aynı etki olacak.Daha genel, tek tanımın konusunda bakılabilir

$D^{a}\,$

a nın gerçel-değerleri için böyle bir durumda bir yol a ise bir tamsayı n değeri alarak,n-kat diferansiyasyonun kuvveti kullanılarak n > 0 için,n < 0 ise Jnin ve −ninci kuvveti kurtarıyor.

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

Nature of the fractional derivative

Fractal derivative ile karıştırılmamalıdır.

An important point is that the fractional derivative at a point x is a local property only when a is an integer; in non-integer 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 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. 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.

Sezgiseller

Bir soru sormak oldukça doğaldır olup olmadığını burada varolan bir işlemci H dır, veya yarı-türev, böylece

$H^{2}f(x)=Df(x)={\dfrac {d}{dx}}f(x)=f'(x)$.

Böyle bir işleç olduğu çıkıyor, ve gerçekten herhangi a > 0, için burada varolan bir işleç P böylece

$(P^{a}f)(x)=f'(x)\,$,

veya dⁿy/dxⁿ'tanımı ile diğer tutulan yol n nin tüm gerçel değerlerine uzanabilir.

Diyelimki f(x) ,x > 0 için tanımlı bir fonksiyon olsun.0 dan xa tanımlı form. Bu kodlanır

$(Jf)(x)=\int _{0}^{x}f(t)\;dt$.

Bu süreci tekrar verir

$(J^{2}f)(x)=\int _{0}^{x}(Jf)(t)dt=\int _{0}^{x}\left(\int _{0}^{t}f(s)\;ds\right)\;dt$,

ve isteğe bağlı olarak uzatılabilir.

tekrarlı integrasyon için Cauchy formülü, yani

$(J^{n}f)(x)={1 \over (n-1)!}\int _{0}^{x}(x-t)^{n-1}f(t)\;dt,$

gerçek n için bir genellemeye basit bir yol içinde yer alır.

Faktöriyel fonksiyonunun gamma işlevini kullanarak ayrık doğasını kaldırmak için bize ayrılmaz operatörünün fraksiyonel uygulamaları için doğal bir aday verir.

$(J^{\alpha }f)(x)={1 \over \Gamma (\alpha )}\int _{0}^{x}(x-t)^{\alpha -1}f(t)\;dt$

Bu, aslında iyi tanımlanmış bir operatördür.

Bunu basitçe göstermek için J operatörü doyurucudur

$(J^{\alpha })(J^{\beta }f)(x)=(J^{\beta })(J^{\alpha }f)(x)=(J^{\alpha +\beta }f)(x)={1 \over \Gamma (\alpha +\beta )}\int _{0}^{x}(x-t)^{\alpha +\beta -1}f(t)\;dt$

Proof

${\begin{aligned}(J^{\alpha })(J^{\beta }f)(x)&={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}(x-t)^{\alpha -1}(J^{\beta }f)(t)\;dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\int _{0}^{t}(x-t)^{\alpha -1}(t-s)^{\beta -1}f(s)\;ds\;dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}f(s)\left(\int _{s}^{x}(x-t)^{\alpha -1}(t-s)^{\beta -1}\;dt\right)ds\end{aligned}}$ son adımda biz entegrasyon sırasını değiş tokuş ve t entegrasyonundan burada faktör f (s) çıkarılır. r değişken değiştirme tarafından tanımlanan t = s + (x − s)r, $(J^{\alpha })(J^{\beta }f)(x)={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}(x-s)^{\alpha +\beta -1}f(s)\left(\int _{0}^{1}(1-r)^{\alpha -1}r^{\beta -1}\;dr\right)ds$ İçsel integral beta fonksiyonu bunun aşağıdaki tatmin edici özelliğidir : $\int _{0}^{1}(1-r)^{\alpha -1}r^{\beta -1}\;dr=B(\alpha ,\beta )={\dfrac {\Gamma (\alpha )\,\Gamma (\beta )}{\Gamma (\alpha +\beta )}}$ Denklemin içine geriye koyma $(J^{\alpha })(J^{\beta }f)(x)={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}(x-s)^{\alpha +\beta -1}f(s)\;ds=(J^{\alpha +\beta }f)(x)$ 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

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) and the derivative (α=1) of the simple power function y=x continuously.

Let us assume that f(x) is a monomial of the form

$f(x)=x^{k}\;.$

The first derivative is as usual

$f'(x)={\dfrac {d}{dx}}f(x)=kx^{k-1}\;.$

Repeating this gives the more general result that

${\dfrac {d^{a}}{dx^{a}}}x^{k}={\dfrac {k!}{(k-a)!}}x^{k-a}\;,$

Which, after replacing the factorials with the gamma function, leads us to

${\dfrac {d^{a}}{dx^{a}}}x^{k}={\dfrac {\Gamma (k+1)}{\Gamma (k-a+1)}}x^{k-a}\;$ for $k\geq 0$

For $k=1$ and $\textstyle a={\frac {1}{2}}$, we obtain the half-derivative of the function $x$ as

${\dfrac {d^{\frac {1}{2}}}{dx^{\frac {1}{2}}}}x={\dfrac {\Gamma (1+1)}{\Gamma (1-{\frac {1}{2}}+1)}}x^{1-{\frac {1}{2}}}={\dfrac {1!}{\Gamma ({\frac {3}{2}})}}x^{\frac {1}{2}}={\dfrac {2x^{\frac {1}{2}}}{\sqrt {\pi }}}.$

Repeating this process yields

${\dfrac {d^{\frac {1}{2}}}{dx^{\frac {1}{2}}}}2\pi ^{-{\frac {1}{2}}}x^{\frac {1}{2}}=2\pi ^{-{\frac {1}{2}}}{\dfrac {\Gamma (1+{\frac {1}{2}})}{\Gamma ({\frac {1}{2}}-{\frac {1}{2}}+1)}}x^{{\frac {1}{2}}-{\frac {1}{2}}}=2\pi ^{-{\frac {1}{2}}}{\dfrac {\Gamma ({\frac {3}{2}})}{\Gamma (1)}}x^{0}={\dfrac {2{\sqrt {\pi }}x^{0}}{2{\sqrt {\pi }}0!}}=1,$

Nitekim beklenen sonuçlar verecek şekilde

$\left({\dfrac {d^{\frac {1}{2}}}{dx^{\frac {1}{2}}}}{\dfrac {d^{\frac {1}{2}}}{dx^{\frac {1}{2}}}}\right)x={\dfrac {d}{dx}}x=1.$

Negatif tamsayı kuvveti k için, gama fonksiyonu tanımsız ve aşağıdaki ilişkiyi kullanmak zorunda

${\dfrac {d^{a}}{dx^{a}}}x^{-k}=(-1)^{a}{\dfrac {\Gamma (k+a)}{\Gamma (k)}}x^{-(k+a)}$ for $k<0$

Yukarıdaki diferansiyel operatörün bu uzantısı sadece gerçek güçlere kısıtlı olması gerekmez. For example, the (1 + i)th derivative of the (1 − i)th derivative yields the 2nd derivative.Ayrıca a" için negatif değerler bağlamında integral veren fark.

For a general function f(x) and 0 < α < 1, the complete fractional derivative is

$D^{\alpha }f(x)={\frac {1}{\Gamma (1-\alpha )}}{\frac {d}{dx}}\int _{0}^{x}{\frac {f(t)}{(x-t)^{\alpha }}}dt$

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,

$D^{\frac {3}{2}}f(x)=D^{\frac {1}{2}}D^{1}f(x)=D^{\frac {1}{2}}{\frac {d}{dx}}f(x)$

Laplace dönüşümü

Ayrıca sorudan Laplace dönüşümü yoluyla alınabilir. Unutmadan

${\mathcal {L}}\left\{Jf\right\}(s)={\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}(s)={\frac {1}{s}}({\mathcal {L}}\left\{f\right\})(s)$

and

${\mathcal {L}}\left\{J^{2}f\right\}={\frac {1}{s}}({\mathcal {L}}\left\{Jf\right\})(s)={\frac {1}{s^{2}}}({\mathcal {L}}\left\{f\right\})(s)$

vs, varsayalım;

$J^{\alpha }f={\mathcal {L}}^{-1}\left\{s^{-\alpha }({\mathcal {L}}\{f\})(s)\right\}$.

örneğin

${\begin{array}{lcr}J^{\alpha }\left(t^{k}\right)&=&{\mathcal {L}}^{-1}\left\{{\dfrac {\Gamma (k+1)}{s^{\alpha +k+1}}}\right\}\\&=&{\dfrac {\Gamma (k+1)}{\Gamma (\alpha +k+1)}}t^{\alpha +k}\end{array}}$

beklendiği gibi. yani, verilen evrişim kuralı

${\mathcal {L}}\{f*g\}=({\mathcal {L}}\{f\})({\mathcal {L}}\{g\})$

ve kısael p(x) = x^{α − 1} için netlik, şunu buluruz

${\begin{array}{rcl}(J^{\alpha }f)(t)&=&{\frac {1}{\Gamma (\alpha )}}{\mathcal {L}}^{-1}\left\{\left({\mathcal {L}}\{p\}\right)({\mathcal {L}}\{f\})\right\}\\&=&{\frac {1}{\Gamma (\alpha )}}(p*f)\\&=&{\frac {1}{\Gamma (\alpha )}}\int _{0}^{t}p(t-\tau )f(\tau )\,d\tau \\&=&{\frac {1}{\Gamma (\alpha )}}\int _{0}^{t}(t-\tau )^{\alpha -1}f(\tau )\,d\tau \\\end{array}}$

burada Cauchy yukarıda bize bunları verdi.

Laplace nispeten az sayıda fonksiyonlar üzerinde "iş" dönüştürür, ancak sık sık fraksiyonel diferansiyel denklemlerin çözümü için yararlıdır.

Fraksiyonel integraller

Riemann–Liouville fraksiyonel integrali

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).

$_{a}D_{t}^{-\alpha }f(t)={_{a}I_{t}^{\alpha }}f(t)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}(t-\tau )^{\alpha -1}f(\tau )d\tau$

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

Hadamard fraksiyonel integrali

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

$_{a}\mathbf {D} _{t}^{-\alpha }f(t)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}{\Bigg (}\log {\frac {t}{\tau }}{\Bigg )}^{\alpha -1}f(\tau ){\frac {d\tau }{\tau }}$

for t > a.

Fraksiyonel türevler

Not like classical Newtonian derivatives, a fractional derivative is defined via a fractional integral.

Riemann–Liouville fraksiyonel türevi

The corresponding derivative is calculated using Lagrange's rule for differential operators. Computing n-th 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.

$_{a}D_{t}^{\alpha }f(t)={\frac {d^{n}}{dt^{n}}}{_{a}D_{t}^{-(n-\alpha )}}f(t)={\frac {d^{n}}{dt^{n}}}{_{a}I_{t}^{n-\alpha }}f(t)$

Caputo fraksiyonel türevi

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

${_{a}^{C}D_{t}^{\alpha }}f(t)={\frac {1}{\Gamma (n-\alpha )}}\int _{a}^{t}{\frac {f^{(n)}(\tau )d\tau }{(t-\tau )^{\alpha +1-n}}}$

Genelleme

Erdélyi–Kober işlemcisi

Erdélyi–Kober işlemcisi bir integral işlemci Arthur Erdélyi (1940) ve Hermann Kober (1940) tarafından tanıtıldı ve ile verilen

${\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0}^{x}(t-x)^{\alpha -1}t^{-\alpha -\nu }f(t)dt$

which generalizes the Riemann fractional integral and the Weyl integral. A recent generalization is the following, which generalizes the Riemann-Liouville fractional integral and the Hadamard fractional integral. It is given by,

$({}^{\rho }{\mathcal {I}}_{a+}^{\alpha }f)(x)={\frac {\rho ^{1-\alpha }}{\Gamma ({\alpha })}}\int _{a}^{x}{\frac {\tau ^{\rho -1}f(\tau )}{(x^{\rho }-\tau ^{\rho })^{1-\alpha }}}\,d\tau ,$

for x > a.

Fonksiyonel hesap

fonksiyonel analizin konuları içinde, 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.

=Uygulamalar

Fraksiyonel kütle korunumu

As described by Wheatcraft and Meerschaert (2008), 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:

$-\rho (\nabla ^{\alpha }\cdot {\vec {u}})=\Gamma (\alpha +1)\Delta x^{1-\alpha }\rho (\beta _{s}+\phi \beta _{w}){\frac {\partial p}{\partial t}}$

Fraksiyonel adveksiyon dağılım denklemi

This equation has been shown useful for modeling contaminant flow in heterogenous porous media.

Zaman-uzay fraksiyonel difüzyon denklemi modelleri

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models. 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

${\frac {\partial ^{\alpha }u}{\partial t^{\alpha }}}=K(-\triangle )^{\beta }u.$

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.

Yapısal sönümleme modelleri

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

Karmaşık ortam için akustik dalga denklemleri

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:

${\nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial t^{\alpha }}}\nabla ^{2}u-{\dfrac {\tau _{\epsilon }^{\beta }}{c_{0}^{2}}}{\dfrac {\partial ^{\beta +2}u}{\partial t^{\beta +2}}}=0.}$

See also 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 and in the survey paper, as well as the acoustic attenuation article.

Kuantum teorisinde fraksiyonel Schrödinger denklemi

The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics discovered by Nick Laskin, has the following form:

$i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}=D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t).$

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/∂r² 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, 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

$(-\hbar ^{2}\Delta )^{\alpha /2}\psi (\mathbf {r} ,t)={\frac {1}{(2\pi \hbar )^{3}}}\int d^{3}pe^{i\mathbf {p} \cdot \mathbf {r} /\hbar }|\mathbf {p} |^{\alpha }\varphi (\mathbf {p} ,t)\,.$

Fraksiyonel Schrödinger denkleminde α indisi Lévy indisi, 1 < α ≤ 2.

Ayrıca bakınız

• Akustik zayıflama
• Differintegral
• Diferansiyel denklem
• Fraksiyonel dinamikleri
• Fraksiyonel Fourier dönüşümü
• Neopolarogram
• Fraksiyonel Schrödinger denklemi
• Özbağlanımlı kesirli bütünleşik hareketli ortalama

Kaynakça

• Fractional Integrals and Derivatives: Theory and Applications, by Samko, S.; Kilbas, A.A.; and Marichev, O. Hardcover: 1006 pages. Publisher: Taylor & Francis Books. ISBN 2-88124-864-0
• Theory and Applications of Fractional Differential Equations, by Kilbas, A. A.; Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Elsevier, February 2006. ISBN 0-444-51832-0 (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 0-471-58884-9
• 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 0-12-525550-0
• 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 0-12-558840-2
• Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. by F. Mainardi, Imperial College Press, 2010. 368 pages.
• Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. by V.E. Tarasov, Springer, 2010. 450 pages.
• Fractional Derivatives for Physicists and Engineers by V.V. Uchaikin, Springer, Higher Education Press, 2012, 385 pages.
• Fractional Calculus - An Introduction for Physicists by R. Herrmann, World Scientific, Singapore 2014. 500 pages.
• 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: Springer-Verlag Telos; (January 1998). ISBN 3-211-82913-X
• Physics of Fractal Operators, by Bruce J. West, Mauro Bologna, Paolo Grigolini. Hardcover: 368 pages. Publisher: Springer Verlag; (January 14, 2003). ISBN 0-387-95554-2
• Fractional Calculus and the Taylor-Riemann Series, Rose-Hulman 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. DOI:10.1007/s002200050299 (available online or as the 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. DOI:10.1155/S1110757X02110102 (available online or as the 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. DOI:10.1038/nn.2212 (abstract)
• Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, by Ahmed E., A.M.A. El-Sayed, H.A.A. El-Saka. 2007. Jour. Math. Anal. Appl. 325,452.
• 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. 
• Erdélyi, Arthur (1950–51). "On some functional transformations". Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino. 10: 217–234. MR 0047818. 
• Recent history of fractional calculus by J.T. Machado, V. Kiryakova, F. Mainardi,

Dış bağlantılar

• 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
• Specialized journal: Fractional Differential Equations (FDE)
• Specialized journal: Communications in Fractional Calculus (ISSN 2218-3892)
• 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
• History, Definitions, and Applications for the Engineer (PDF), by Adam Loverro, 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