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In the current paper, we study a Petrov-Galerkin method for a Cauchy problem for an operator-differential equation with a leading self-adjoint operator  A  and a subordinate linear operator  K ( t ) in a Hilbert space. Error estimates for the approximate solutions are obtained. We consider the full equation discretization based on a two-level difference scheme. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The method is applied to the model parabolic problems.

Asymptotic estimates of a projection-difference method for an operator-differential equation