On a numerical method for solving the Cauchy problem for an operator differential equation

We study a projection-difference method for solving the Cauchy problem for an operator differential equation in a Hilbert space with the principal selfadjoint operator $A(t)$ and the subordinate linear operator $K(t)$. For approximation equations constructed with the Faedo–Galerkin method we discretize with respect to time using the Crank–Nicolson scheme. We estimate the errors of approximate solutions and the errors for fractional powers of the principal operator $A(t)$. We apply the method to solving an initial boundary value problem.