Local Computational Algorithms for a Systemof First-Order Equations with Memory Effects

We consider the Cauchy problem for a system of first-order integro-differential equations with difference kernels in a finite-dimensional Hilbert space. This class of equations arises in the mathematical modeling of a wide range of nonstationary processes taking into account memory effects, including, in particular, the system of Maxwell equations. For the numerical solution, a method of reducing the original nonlocal problem to an equivalent system of local first-order differential equations on the basis of the approximation of kernels by a finite sum of exponential functions is proposed. Two-level operator-difference schemes are proposed, for which the stability with respect to the initial data and the right-hand side is analyzed. The theoretical analysis performed demonstrates the well-posedness of the approach proposed.