## 3 FO Model of Dielectric Media  
In a homogeneous and isotropic medium, the electromagnetic field satisfies [PERSON]'s equations  
\[\mathbf{\
abla\cdot D}=\rho \tag{7}\]\[\mathbf{\
abla}\mathbf{\times}\mathbf{\mathrm{E}}=-\frac{\partial\mathbf{\mathrm{B}}}{\partial t} \tag{8}\]  
\[\mathbf{\
abla}\mathbf{\cdot}\mathbf{\mathrm{B}}=0 \tag{9}\]  
\[\mathbf{\
abla}\mathbf{\times}\mathbf{\mathrm{H}}=\mathbf{\mathrm{J}}+\frac{\partial\mathbf{ \mathrm{D}}}{\partial t} \tag{10}\]  
where \(\mathbf{\mathrm{D}}\) and \(\mathbf{\mathrm{B}}\) are, respectively, the electric- and magnetic-flux densities, \(\mathbf{\mathrm{E}}\) and \(\mathbf{\mathrm{H}}\) are, respectively, the electric- and magnetic-field intensities, \(\mathbf{\mathrm{J}}\) is the current density and \(\rho\) is the volume-charge density. Let us consider the electromagnetic medium described by the FO constitutive relations  
\[\varepsilon_{\beta}\mathbf{\mathrm{E}}=D_{t}^{1-\beta}\mathbf{\mathrm{D}} \tag{11}\]  
\[\mu_{\gamma}\mathbf{\mathrm{H}}=D_{t}^{1-\gamma}\mathbf{\mathrm{B}} \tag{12}\]  
\[\mathbf{\mathrm{J}}=\sigma_{\eta}D_{t}^{1-\eta}\mathbf{\mathrm{E}} \tag{13}\]  
where \(\varepsilon_{\beta}\), \(\mu_{\gamma}\), \(\sigma_{\eta}\) (\(\beta,\gamma,\eta\in(0,1]\)) are parameters corresponding to the permittivity, permeability and conductivity, with units \(\frac{\mathbf{\mathrm{F}}}{\mathbf{\mathrm{s}}^{1-\beta}\mathbf{\mathrm{m}}}\), \(\frac{\mathbf{\mathrm{H}}}{\mathbf{\mathrm{s}}^{1-\gamma}\mathbf{\mathrm{m}}}\) and \(\frac{\mathbf{\mathrm{S}}}{\mathbf{\mathrm{s}}^{1-\gamma}\mathbf{\mathrm{m}}}\) (\(F\)--Farad, \(H\)--Henr, \(S\)--Siemens), respectively. Assuming there is no energy dissipation due to Joule heating (\(\sigma_{\eta}=0\)), we can write the set of constitutive relations in the form  
\[\varepsilon_{\beta}D_{t}^{\beta}\mathbf{\mathrm{E}}=\frac{\partial\mathbf{\mathrm{D}} }{\partial t} \tag{14}\]  
\[\mu_{\gamma}D_{t}^{\gamma}\mathbf{\mathrm{H}}=\frac{\partial\mathbf{\mathrm{B}}}{ \partial t}. \tag{15}\]  
Using (14) and (15), one obtains from (7)-(10) [PERSON]'s equations with the TF derivatives as follows:  
\[\mathbf{\
abla}\mathbf{\cdot}\mathbf{\mathrm{E}}=0 \tag{16}\]  
\[\mathbf{\
abla}\mathbf{\times}\mathbf{\mathrm{E}}=-\frac{\partial\mathbf{\mathrm{B}}}{\partial t} \tag{8}\]0  
\[\mathbf{\
abla}\mathbf{\times}\mathbf{\mathrm{E}}=-\frac{\partial\mathbf{\mathrm{B}}}{\partial t} \tag{8}\]1  
\[\mathbf{\
abla}\mathbf{\times}\mathbf{\mathrm{E}}=-\frac{\partial\mathbf{\mathrm{B}}}{\partial t} \tag{8}\]2  
Let us calculate the curl of (17). Then, one obtains  
\[\mathbf{\
abla}\mathbf{\times}\mathbf{\mathrm{E}}=-\frac{\partial\mathbf{\mathrm{B}}}{\partial t} \tag{8}\]3  
Using the vector calculus identity \(\mathbf{\
abla}\mathbf{\times}\mathbf{\
abla}\mathbf{\times}\mathbf{\mathrm{E}}=\mathbf{\
abla}(\mathbf{ \
abla}\mathbf{\cdot}\mathbf{\mathrm{E}})-\
abla^{2}\mathbf{\mathrm{E}}\) and (16), one obtains from (20) the fractional diffusion-wave equation for the electric field  
\[\mathbf{\
abla}\mathbf{\times}\mathbf{\mathrm{E}}=-\frac{\partial\mathbf{\mathrm{B}}}{\partial t} \tag{8}\]4  
The fractional diffusion-wave equation interpolates between the diffusion and the wave processes, which are completely different with regard to their response to a localized disturbance [22]. That is, the diffusion equation describes a process in which a disturbance spreads infinitely fast, whereas the wave equation describes a process in which a disturbance propagates with a constant finite velocity. For the fractional diffusion-wave equation,a disturbance spreads with an infinite velocity, but its fundamental solution possesses a maximum which propagates with a finite velocity [26]. Of course, a detailed formulation of the problem in electromagnetics requires not only the Equation (21), but also boundary and initial conditions. While the boundary conditions result from the assumed behavior of the electromagnetic field on the boundary of the spatial domain, the initial conditions are assumed to be a zero electromagnetic field for \(t<0\).