Splitting schemes for one class of operator differential equations

At present, splitting schemes of various types are available for evolution equations of the first and second order in the case when the basic elliptic operator of the problem has an additive representation. Numerous applications lead to boundary value problems for nonstationary Sobolev-type equations with an elliptic operator at the time derivative. When splitting schemes are used to find an approximate solution of such problems, it is necessary to use an additive representation for both the basic elliptic operator and the operator at the time derivative. This paper deals with the Cauchy problem for a first-order evolution equation in the special case when the operator at the derivative can be represented in terms of the basic operator. The equation is written as a differential-algebraic system of two equations. Unconditionally stable multicomponent splitting schemes are constructed.