Felix N. Chukhovskii | Searcharxiv

arXiv2024

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 $α$, $α\in (0, 1].$

math.GM↗

arXiv2022

Towards to solution of the fractional Takagi-Taupin equations. The Green function method

Developing the comprehensive theory of the X-ray diffraction by distorted crystals remains to be topical of the mathematical physics. Up to now, the X-ray diffraction theory grounded on the Takagi-Taupin equations with the first-order partial derivatives over the two coordinates within the X-ray scattering plane. In the work, the theoretical approach based on the first-order fractional Takagi-Taupin equations with the 'quasi-time variable' of the order $α\in(0,1]$ along the crystal depth has been suggested and the corresponding X-ray Cauchy issue is formulated. Accordingly, using the Green function method in the scope of the Cauchy issue, the fractional Takagi-Taupin equations in the integral form have been derived. In the case of the inhomogeneous incident X-ray beam, the solution of the Cauchy issue of the X-ray diffraction by perfect crystal has been obtained and compared with the corresponding one based on the solution of the conventional Takagi-Taupin equations, $α=1.$ In turn, notice that the value of order $α$ may be adjusted from the experimental X-ray diffraction data.

math.AP↗