Fractional-Diffraction-Optics Cauchy Problem: Resolvent-Function Solution of the Matrix Integral Equation

The fractional diffraction optics theory has been elaborated using the Green function technique. The optics-fractional equation describing the diffraction X-ray scattering by imperfect crystals has been derived as the fractional matrix integral Fredholm--Volterra equation of the second kind. In the paper, to solve the Cauchy problems, the Liouville--Neumann-type series formalism has been used to build up the matrix Resolvent-function solution. In the case when the imperfect crystal-lattice elastic displacement field is the linear function $f({\bf R}) = a x+b$, $a, b = const,$ the explicit solution of the diffraction-optics Cauchy problem has been obtained and analyzed for arbitrary fractional-order-parameter $\alpha$, $\alpha\in (0, 1].$