  We consider the Cauchy problem for a first-order evolution equation with
memory in a finite-dimensional Hilbert space when the integral term is related
to the time derivative of the solution. The main problems of the approximate
solution of such nonlocal problems are due to the necessity to work with the
approximate solution for all previous time moments. We propose a transformation
of the first-order integrodifferential equation to a system of local
evolutionary equations. We use the approach known in the theory of Voltaire
integral equations with an approximation of the difference kernel by the sum of
exponents. We formulate a local problem for a weakly coupled system of
equations with additional ordinary differential equations. We have given
estimates of the stability of the solution by initial data and the right-hand
side for the solution of the corresponding Cauchy problem. The primary
attention is paid to constructing and investigating the stability of two-level
difference schemes, which are convenient for computational implementation. The
numerical solution of a two-dimensional model problem for the evolution
equation of the first order, when the Laplace operator conditions the
dependence on spatial variables, is presented.

Numerical Analysis;Numerical Analysis