# About the Use of Generalized Forms of Derivatives in the Study of Electromagnetic Problems
## 3 Generalized Exponential Operators and Fractional Electrical Circuit  
In this section, we consider the extension to the fractional case of the differential equations characterizing an electric circuit. In the previous section we fixed the few elements of the formalism we will use in the following. We privileged, in our definition of fractional derivative, the Caputo form. The reason for this choice stems from the fact that we prefer to avoid ambiguity due to the non-vanishing of the derivative of a constant. The modeling of an electrical circuit in terms of non-integer differential equations does not appear, in general, well motivated and often the mathematical formalism is privileged with respect to the effective physical meaning. It should be however noted that the Ohm law is at the basis of the circital phenomena. This law summarizes, from a macroscopic point of view, the microscopic dynamic of charge carriers inside a conductor, along with the relevant interactions defining the charge drift velocity. Classical Newtonian mechanics is used to provide a link between the applied electric field and charge velocities, whereas the diffusion process may be governed by other mechanisms not strictly associated with classical diffusion and more similar to Levy type process which, according to the discussion of the previous section, requires fractional heat type equations. We assume, for the moment, that the fractional derivative can model an actual physical circuit and discuss some aspects of the relevant study, in terms of the solutions of the associated differential equations. **Example 3** (RC Circuit). : _An ordinary RC circuit is ruled by the first-order differential equation_  
\[ \begin{cases} R\frac{d}{dt}q(t)+\frac{1}{C}q(t)=V(t)\\ V(t)=V_{0}f(t)\end{cases} . \tag{38}\]  
_Equation (38), in non dimensional form, gives_  
\[\frac{d}{d\eta}Q(\eta)+Q(\eta)=f^{*}(\eta),\qquad\qquad\eta=\frac{t}{\tau}, \qquad\tau=RC,\qquad Q(\eta)=\frac{q(t)}{CV_{0}},\qquad f^{*}(\eta)=f(\tau\eta). \tag{39}\]  
_The non-homogeneous part of the solution can, according to the methods outlined in the previous section, be written as_  
\[Q(\eta)=\frac{1}{1+\partial_{\eta}}f^{*}(\eta)=\int_{0}^{\infty}e^{-s}e^{- \bar{\omega}\eta}f^{*}(\eta)ds=\int_{0}^{\infty}e^{-s}f^{*}(\eta-s)ds \tag{40}\]  
_and a graphical example is reported in Figure 7a, for \(f(\eta)=\cos(\sigma\eta)\), \(\sigma=\omega\tau\). The solution was checked numerically to be sure that it is consistent with the original Equation (38)._  
_Before proceeding further it should, however, be noted that, by standard means, the solution of first-order non-homogeneous ODE's of the type (39) gives_  
\[Q(\eta,\eta_{0})=Q^{*}e^{-(\eta-\eta_{0})}+\int_{\eta_{0}}^{\eta}e^{-(\eta- \bar{\xi})}f(\bar{\xi})d\bar{\xi} \tag{41}\]  
_or, after rearranging the integration variable,_  
\[Q(\eta,\eta_{0})=Q^{*}e^{-(\eta-\eta_{0})}+\int_{0}^{\eta-\eta_{0}}e^{-\sigma }f(\eta-\sigma)d\sigma. \tag{42}\]  
_We denoted by \(Q^{*}\) the initial condition (at \(\eta=0\)); the first term is therefore the solution of the homogeneous part, while the integral accounts for the non-homogeneous term. If we rearrange the integration variable in Equation (40), it is evident that the two solutions coincide for \(Q^{*}=0\) and \(\eta_{0}=-\infty\)._  
**Observation 3**. : _Whilst naive, in view of the extension to a fractional RC circuit, we note that Equation (38) is just the Kirchhoff voltage law [41] and can also be written as_  
\[V(t)=R\ C\frac{d}{dt}V_{\rm C}(t)+V_{\rm C}(t), \tag{43}\]  
_where \(V_{\rm C}\) is the voltage across the capacitor. Since RC has the physical dimensions of a time, Equation (43) is dimensionally consistent. This is a point of non-secondary importance whenever one is trying its extension to the fractional case, which can be written as_  
\[V(t)=\frac{R\ C}{\alpha^{1-\gamma}}\frac{d^{\gamma}}{dt^{\gamma}}V_{\rm C}(t )+V_{\rm C}(t)\, \tag{44}\]  
_where the constant \(\alpha^{1-\gamma}\) has been introduced to avoid problems with physical dimensions.