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Fractional derivative analysis of Helmholtz and paraxial-wave equations

Turski A. J., Atamaniuk B., Turska E.

Journal of Technical Physics

2003

Vol. 44, no 2

193-206

Fundamental rules and definitions of Fractional Calculus are outlined. Factorizing 1-D and 2-D Helmholtz equations, four semi-differential eigenfunctions are determined. The functions exhibit incident and reHected plane waves as well as diffracted incident and reflected waves on 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 differential fractional solutions of Fresnel-integral type are determined. We arrived at two solutions, which are the mothers of known and new solutions.