Numerical solutions of electromagnetic wave model of fractional derivative using class of finite difference scheme

TL;DR

This paper introduces a finite difference numerical scheme based on Hermite formulas for solving fractional PDEs modeling electromagnetic waves in dielectric media, demonstrating stability, convergence, and high accuracy through numerical experiments.

Contribution

It develops a new finite difference scheme for fractional PDEs in electromagnetic wave modeling, with proven stability, convergence, and improved accuracy over existing methods.

Findings

The scheme achieves an accuracy order of O(k^{(4-α)} + k^{(4-β)} + h^2).

Numerical experiments confirm the scheme's efficiency and accuracy.

Comparative studies show the proposed method outperforms some existing approaches.

Abstract

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.