Editing Fractional calculus (section)

====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.