  In this article, a numerical scheme is introduced for solving the fractional
partial differential equation (FPDE) arising from electromagnetic waves in
dielectric media (EMWDM) by using an efficient class of finite difference
methods. The numerical scheme is based on the Hermite formula. The Caputo's
fractional derivatives in time are discretized by a finite difference scheme of
order $\mathcal{O}(k^{(4-\alpha)})$ \& $\mathcal{O}(k^{(4-\beta)})$, $1<\beta
<\alpha \leq 2$. The stability and the convergence analysis of the proposed
methods are given by a procedure similar to the standard von Neumann stability
analysis under mild conditions. Also for FPDE, accuracy of order
$\mathcal{O}\left( k^{(4-\alpha)}+k^{(4-\beta)}+h^2\right) $ is investigated.
Finally, several numerical experiments with different fractional-order
derivatives are provided and compared with the exact solutions to illustrate
the accuracy and efficiency of the scheme. A comparative numerical study is
also done to demonstrate the efficiency of the proposed scheme.
