We study the initial value problem (Formula presented.) where A is closed linear operator defined on a Banach space X, x belongs to the domain of A, and the kernel a is a particular discretization of an integrable kernel (Formula presented.) Assuming that A generates a resolvent family, we find an explicit representation of the solution to the initial value problem (*) as well as for its inhomogeneous version, and then we study the stability of such solutions. We also prove that for a special class of kernels a, it suffices to assume that A generates an immediately norm continuous C0-semigroup. We employ a new computational method based on the Poisson transformation.