We consider the problem of the cosmic ray spectrum formation assuming that cosmic rays are produced by galactic sources. The fractional diffusion equation proposed in our recent papers is used to describe the cosmic rays propagation in interstellar medium. We show that in the framework of this approach it is possible to explain the locally observed basic features of the cosmic rays in the energy region 10¹⁰ ÷ 10²⁰ eV: difference between spectral exponents of protons and other nuclei, mass composition variation, “knee” problem, flattening of the primary spectrum for E ≥ 10¹⁸ ÷ 10¹⁹ eV.