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Problems of the numerical solution of the Cauchy problem for a first-orderdifferential-operator equation are discussed. A fundamental feature of theproblem under study is that the equation includes a fractional power of theself-adjoint positive operator. In computational practice, rationalapproximations of the fractional power operator are widely used in variousversions. The purpose of this work is to construct special approximations intime when the transition to a new level in time provided a set of standardproblems for the operator and not for the fractional power operator. Stablesplitting schemes with weights parameters are proposed for the additiverepresentation of rational approximation for a fractional power operator.Possibilities of using similar time approximations for other problems arenoted. The numerical solution of a two-dimensional non-stationary problem witha fractional power of the Laplace operator is also presented.

Splitting Schemes for Non-Stationary Problems with a Rational Approximation for Fractional Powers of the Operator