  Problems of the numerical solution of the Cauchy problem for a first-order
differential-operator equation are discussed. A fundamental feature of the
problem under study is that the equation includes a fractional power of the
self-adjoint positive operator. In computational practice, rational
approximations of the fractional power operator are widely used in various
versions. The purpose of this work is to construct special approximations in
time when the transition to a new level in time provided a set of standard
problems for the operator and not for the fractional power operator. Stable
splitting schemes with weights parameters are proposed for the additive
representation of rational approximation for a fractional power operator.
Possibilities of using similar time approximations for other problems are
noted. The numerical solution of a two-dimensional non-stationary problem with
a fractional power of the Laplace operator is also presented.
