  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.
