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 $\alpha\in(0,1]$ 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, $\alpha=1.$ In turn, notice that the value of order $\alpha$ may be adjusted from the experimental X-ray diffraction data.