Draft:Coimbra derivative

The Coimbra derivative is the most versatile variable order derivative, and consequently it is the most often used for physical modeling: For $$q(t) < 1 $$ $$ \begin{align} ^{\mathbb{C}}_{ a}\mathbb{D}^{q(t)} f(t)=\frac{1}{\Gamma[1-q(t)]} \int_{0^+}^t (t-\tau)^{-q(t)}\frac{d\,f(\tau)}{d\tau}d\tau\,+\,\frac{(f(0^+)-f(0^-))\,t^{-q(t)}}{\Gamma(1-q(t))}, \end{align}$$ where the lower limit $$a$$ can be taken as either $$0^-$$ or $$-\infty$$ as long as $$f(t)$$ is identically zero from or $$-\infty$$ to $$0^-$$. Note that this operator returns the correct fractional derivatives for all values of $$t$$ and can be applied to either the dependent function itself $$ f(t)$$ with a variable order of the form $$q(f(t))$$ or to the independent variable with a variable order of the form $$q(t)$$.$$^{[1]}$$

The Coimbra derivative defined above returns the correct value of the zero-th order derivative $$(^{\mathbb{C}}\mathbb{D}^{0} f(t) = f(t))$$ when $$q(t)=0$$ and returns the first derivative $$(^{\mathbb{C}}\mathbb{D}^{1} f(t) = f'(t))$$ when $$q(t)\rightarrow 1$$. Also, the operator returns the $$p$$-th derivative of $$f(t)$$ when $$q(t) = p$$, a necessary property for dynamical modeling that is often overlooked when the focus is placed on mathematical properties that resemble fixed order differential operators (for example, whether the operators abide by the exponent rule, etc.). Also of great importance to dynamic modeling is the fact that this operator is dynamically consistent with causal behavior in the initial conditions. In other words, when $$f(t)$$ is a true constant from $$-\infty$$ to the initial time $$(t = 0+)$$, the Coimbra operator returns zero for all values of $$q (f (t ))$$. However, if $$f(t)$$ is discontinuous between $$t=0^{-}$$ and $$ t=0^{+}$$, the operator returns the appropriate Heaviside contribution to the integral value of $$^{\mathbb{C}}\mathbb{D}^{q(t)}(t)$$. In accordance with this causal definition, we take the value of the physical variable $$f(t)$$ to be identically null from $$-\infty$$ to $$0^-$$  as a representation of dynamic equilibrium. A nonzero initial condition is treated as a Heaviside function at $$t = 0$$, and, therefore, it is included in the second term of the definition of the operator above. This way, the Coimbra operator allows for discontinuities at the initial condition, as long as there is dynamic equilibrium for $$ -\infty < t\leq 0^-$$, i.e., when $$f(t \leq 0^-) = 0$$. Since one is free to place the initial counter of time at any given instant, this operator is both causal and allows for discontinuous initial conditions.:

The Coimbra derivative can be generalized to any order, leading to the Coimbra Generalized Order Differintegration Operator (GODO) For $$q(t) < m $$ $$ \begin{align} ^{\mathbb{\quad C}}_{\,\,-\infty}\mathbb{D}^{q(t)} f(t)=\frac{1}{\Gamma[ m-q(t)]} \int_{0^+}^t (t-\tau)^{m-1-q(t)}\frac{d^m f(\tau)}{d\tau^m}d\tau\,+\,\sum^{m-1}_{n = 0} \frac{(\frac{d^n f(t)}{dt^n }|_{0^+}-\frac{d^n f(t)}{dt^n}|_{0^-})\,t^{n -q(t)}}{\Gamma[n+1-q(t)]}, \end{align}$$ where $$m$$ is an integer larger than the larger value of $$q(t)$$ for all values of $$t$$. Note that the second (summation) term on the right side of the definition above can be expressed as

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

Applications and Numerical Implementation
A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,      as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors