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 Caputos 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^2right) $ 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.
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