一类finite difference格式求解分数阶导数电磁波模型
Numerical solutions of electromagnetic wave model of fractional derivative using class of finite difference scheme
摘要：
本文介绍了一种用有效的有限差分法求解介质电磁波（EMWDM）引起的分数阶偏微分方程（FPDE）的数值方法。数值格式基于Hermite公式。用$\mathcal{O}(k^{(4-α)})$ \& $\mathcal{O}(k^{(4-β)})$，$1<β<α\leq 2$阶有限差分格式离散Caputo分数阶导数.所提出的方法的稳定性和收敛性分析给出了一个类似于标准的冯诺依曼稳定性分析在温和的条件下的程序。对于FPDE，研究了阶数$\mathcal{O}\left( k^{(4-α)}+k^{(4-β)}+h^2\right) $的精度。最后，通过几个不同分数阶导数的数值实验，与精确解进行了比较，验证了格式的精度和效率。最后，通过数值实验验证了该方法的有效性.
英文摘要：
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 $\mathcal{O}(k^{(4-α)})$ \& $\mathcal{O}(k^{(4-β)})$, $1<β<α\leq 2$. 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-α)}+k^{(4-β)}+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.
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