Editing Fractional calculus (section)

=== Other types ===
Classical fractional derivatives include:
* [[Grünwald–Letnikov derivative]]<ref name=deOliveira2014>{{cite journal |last1=de Oliveira |first1=Edmundo Capelas |last2=Tenreiro Machado |first2=José António |date=2014-06-10 |title=A Review of Definitions for Fractional Derivatives and Integral |journal=Mathematical Problems in Engineering |volume=2014 |pages=1–6 |language=en |doi=10.1155/2014/238459 |doi-access=free|hdl=10400.22/5497 |hdl-access=free }}</ref><ref name=Aslan2015>{{cite journal |last=Aslan |first=İsmail |date=2015-01-15 |title=An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation |journal=Mathematical Methods in the Applied Sciences |language=en |volume=38 |issue=1 |pages=27–36 |doi=10.1002/mma.3047 |bibcode=2015MMAS...38...27A |hdl=11147/5562 |s2cid=120881978 |hdl-access=free}}</ref>
* Sonin–Letnikov derivative<ref name=Aslan2015/>
* Liouville derivative<ref name=deOliveira2014/>
* [[Differintegral|Caputo derivative]]<ref name=deOliveira2014/>
* Hadamard derivative<ref name=deOliveira2014/><ref>{{cite journal |last1=Ma |first1=Li |last2=Li |first2=Changpin |date=2017-05-11 |title=On hadamard fractional calculus |journal=Fractals |volume=25 |issue=3 |pages=1750033–2980 |doi=10.1142/S0218348X17500335 |bibcode=2017Fract..2550033M |issn=0218-348X}}</ref>
* Marchaud derivative<ref name=deOliveira2014/>
* Riesz derivative<ref name=Aslan2015/>
* Miller–Ross derivative<ref name=deOliveira2014/>
* [[Weyl integral|Weyl derivative]]<ref>{{cite book |last=Miller |first=Kenneth S.  |chapter=The Weyl fractional calculus |date=1975 |title=Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974 |pages=80–89 |editor-last=Ross |editor-first=Bertram |series=Lecture Notes in Mathematics |volume=457 |publisher=Springer |language=en |doi=10.1007/bfb0067098 |isbn=978-3-540-69975-0}}</ref><ref>{{cite journal |last=Ferrari |first=Fausto |date=January 2018 |title=Weyl and Marchaud Derivatives: A Forgotten History |journal=Mathematics |language=en |volume=6 |issue=1 |pages=6 |doi=10.3390/math6010006 |doi-access=free|arxiv=1711.08070 }}</ref><ref name=deOliveira2014/>
* [[Erdelyi–Kober operator|Erdélyi–Kober derivative]]<ref name=deOliveira2014/>
* [[Fractal calculus|<math>F^{\alpha}</math>-derivative]]<ref name="Ali">{{cite book |last= Khalili Golmankhaneh|first= Alireza |date=2022 |title=Fractal Calculus and its Applications |url=https://worldscientific.com/worldscibooks/10.1142/12988#t=aboutBook|location=Singapore |publisher= World Scientific Pub Co Inc|page=328 |doi= 10.1142/12988 |isbn=978-981-126-110-7 |s2cid= 248575991 }}</ref>
New fractional derivatives include:
* Coimbra derivative<ref name=deOliveira2014/>
* [[Katugampola fractional operators|Katugampola derivative]]<ref>{{cite journal |last1=Anderson |first1=Douglas R. |last2=Ulness |first2=Darin J. |date=2015-06-01 |title=Properties of the Katugampola fractional derivative with potential application in quantum mechanics |journal=Journal of Mathematical Physics |volume=56 |issue=6 |pages=063502 |doi=10.1063/1.4922018 |bibcode=2015JMP....56f3502A |issn=0022-2488}}</ref>
* Hilfer derivative<ref name=deOliveira2014/>
* Davidson derivative<ref name=deOliveira2014/>
* Chen derivative<ref name=deOliveira2014/>
* [[Caputo Fabrizio derivative]]<ref name=Algahtani2016>{{cite journal |last=Algahtani |first=Obaid Jefain Julaighim |date=2016-08-01 |title=Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077916301059 |journal=Chaos, Solitons & Fractals |series=Nonlinear Dynamics and Complexity |language=en |volume=89 |pages=552–559 |doi=10.1016/j.chaos.2016.03.026 |bibcode=2016CSF....89..552A |issn=0960-0779}}</ref><ref>{{cite journal |last1=Caputo |first1=Michele |last2=Fabrizio |first2=Mauro |date=2016-01-01 |title=Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels |journal=Progress in Fractional Differentiation and Applications |volume=2 |issue=1 |pages=1–11 |doi=10.18576/pfda/020101 |issn=2356-9336}}</ref>
* Atangana–Baleanu derivative<ref name=Algahtani2016/><ref name="doiserbia.nb.rs"/>

====Coimbra derivative====
The '''Coimbra derivative''' is used for physical modeling:<ref> C. F. M. Coimbra (2003) "Mechanics with Variable Order Differential Equations," Annalen der Physik (12), No. 11-12, pp. 692-703.</ref> A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,<ref>L. E. S. Ramirez, and C. F. M. Coimbra (2007) "A Variable Order Constitutive Relation for Viscoelasticity"– Annalen der Physik (16) 7-8, pp. 543-552.</ref><ref>H. T. C. Pedro, M. H. Kobayashi, J. M. C. Pereira, and C. F. M. Coimbra (2008) "Variable Order Modeling of Diffusive-Convective Effects on the Oscillatory Flow Past a Sphere" – Journal of Vibration and Control, (14) 9-10, pp. 1569-1672.</ref><ref>G. Diaz, and C. F. M. Coimbra (2009) "Nonlinear Dynamics and Control of a Variable Order Oscillator with Application to the van der Pol Equation" – Nonlinear Dynamics, 56, pp. 145—157.</ref><ref>L. E. S. Ramirez, and C. F. M. Coimbra (2010) "On the Selection and Meaning of Variable Order Operators for Dynamic Modeling"– International Journal of Differential Equations Vol. 2010, Article ID 846107.</ref><ref> L. E. S. Ramirez and C. F. M. Coimbra (2011) "On the Variable Order Dynamics of the Nonlinear Wake Caused by a Sedimenting Particle," Physica D (240) 13, pp. 1111-1118.</ref><ref>E. A. Lim, M. H. Kobayashi and C. F. M. Coimbra (2014) "Fractional Dynamics of Tethered Particles in Oscillatory Stokes Flows," Journal of Fluid Mechanics (746) pp. 606-625.</ref><ref>J. Orosco and C. F. M. Coimbra (2016) "On the Control and Stability of Variable Order Mechanical Systems" Nonlinear Dynamics, (86:1), pp. 695–710.</ref> as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors<ref>E. C. de Oliveira, J. A. Tenreiro Machado (2014), "A Review of Definitions for Fractional Derivatives and Integral", Mathematical Problems in Engineering, vol. 2014, Article ID 238459.</ref><ref>S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh (2012) "Numerical techniques for the variable order time fractional diffusion equation" Applied Mathematics and Computation
Volume 218, Issue 22, pp. 10861-10870.</ref><ref>H. Zhang and S. Shen, "The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation," Numer. Math. Theor. Meth. Appl. Vol. 6, No. 4, pp. 571-585.</ref><ref>H. Zhang, F. Liu, M. S. Phanikumar, and M. M. Meerschaert (2013) "A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model," Computers & Mathematics with Applications, 66, issue 5, pp. 693–701.</ref>

 For <math>q(t) < 1 </math>
 <math display="block"> \begin{align}
^{\mathbb{C}}_{ a}\mathbb{D}^{q(t)} f(t)=\frac{1}{\Gamma[1-q(t)]}  \int_{0^+}^t (t-\tau)^{-q(t)}\frac{d\,f(\tau)}{d\tau}d\tau\,+\,\frac{(f(0^+)-f(0^-))\,t^{-q(t)}}{\Gamma(1-q(t))},
\end{align}</math>
where the lower limit <math>a</math> can be taken as either <math>0^-</math> or <math>-\infty</math> as long as <math>f(t)</math> is identically zero from or <math>-\infty</math> to <math>0^-</math>. Note that this operator returns the correct fractional derivatives for all values of <math>t</math> and can be applied to either the dependent function itself <math> f(t)</math> with a variable order of the form <math>q(f(t))</math> or to the independent variable with a variable order of the form <math>q(t)</math>.<math>^{[1]}</math>  

The Coimbra derivative can be generalized to any order,<ref> C. F. M. Coimbra "Methods of using generalized order
differentiation and integration of input variables to forecast
trends," U.S. Patent Application 13,641,083 (2013). </ref> leading to the Coimbra Generalized Order Differintegration Operator (GODO)<ref>J. Orosco and C. F. M. Coimbra (2018) "Variable-order Modeling of Nonlocal Emergence in Many-body Systems: Application to Radiative Dispersion," Physical Review E (98), 032208.</ref>
 For <math>q(t) < m </math>
 <math display="block"> \begin{align}
^{\mathbb{\quad C}}_{\,\,-\infty}\mathbb{D}^{q(t)} f(t)=\frac{1}{\Gamma[
m-q(t)]}  \int_{0^+}^t (t-\tau)^{m-1-q(t)}\frac{d^m f(\tau)}{d\tau^m}d\tau\,+\,\sum^{m-1}_{n = 0} \frac{(\frac{d^n f(t)}{dt^n }|_{0^+}-\frac{d^n f(t)}{dt^n}|_{0^-})\,t^{n -q(t)}}{\Gamma[n+1-q(t)]},
\end{align}</math>
where <math>m</math> is an integer larger than the larger value of <math>q(t)</math> for all values of <math>t</math>. Note that the second (summation) term on the right side of the definition above can be expressed as 

 <math display="block"> \begin{align}
\frac{1}{\Gamma[m-q(t)]}\sum^{m-1}_{n = 0} \{[\frac{d^n\!f(t)}{dt^n}|_{0^+}-\frac{d^n\!f(t)}{dt^n }|_{0^-}]\,t^{n -q(t)} \prod^{m-1}_{j=n+1} [j- q(t)]\}
\end{align}</math>
so to keep the denominator on the positive branch of the Gamma (<math>\Gamma</math>) function and for ease of numerical calculation.