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We consider the Cauchy problem for a first-order evolution equation withmemory in a finite-dimensional Hilbert space when the integral term is relatedto the time derivative of the solution. The main problems of the approximatesolution of such nonlocal problems are due to the necessity to work with theapproximate solution for all previous time moments. We propose a transformationof the first-order integrodifferential equation to a system of localevolutionary equations. We use the approach known in the theory of Voltaireintegral equations with an approximation of the difference kernel by the sum ofexponents. We formulate a local problem for a weakly coupled system ofequations with additional ordinary differential equations. We have givenestimates of the stability of the solution by initial data and the right-handside for the solution of the corresponding Cauchy problem. The primaryattention is paid to constructing and investigating the stability of two-leveldifference schemes, which are convenient for computational implementation. Thenumerical solution of a two-dimensional model problem for the evolutionequation of the first order, when the Laplace operator conditions thedependence on spatial variables, is presented.

Approximate solution of the Cauchy problem for a first-order integrodifferential equation with solution derivative memory