Helmholtz和傍轴波动方程的分数阶导数分析
Fractional Derivative Analysis of Helmholtz and Paraxial-Wave Equations
摘要：
概述了分数阶微分积分的基本规则和定义。通过分解一维和二维Helmholtz方程，确定了四个分数阶本征函数.该函数表现出入射和反射平面波以及半平面边缘的衍射入射和反射波。它们允许构建索末菲半平面衍射解决方案。本文对近轴传播的抛物波方程（PWE，Leontovich-Fock）进行了因式分解，导出了菲涅耳积分型的离散分数阶解。我们得出了两个解决方案，它们是已知解决方案和新解决方案的母方。
英文摘要：
Fundamental rules and definitions of Fractional Differintegrals are outlined. Factorizing 1-D and 2-D Helmholtz equations four fractional eigenfunctions are determined. The functions exhibit incident and reflected plane waves as well as diffracted incident and reflected waves of the half-plane edge. They allow to construct the Sommerfeld half-plane diffraction solutions. Parabolic-Wave Equation (PWE, Leontovich-Fock) for paraxial propagation is factorized and differetial fractional solutions of Fresnel-integral type are derived. We arrived at two solutions, which are the mothers of known and new solutions.
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