RESUMEN

A suitable abstract setting of an initial value problem for the first order operador differential equation is introduced for the case when both the unbounded operator coefficient and its domain can depend on the differentiation variable t. A new exponentially convergent algorithm is proposed. The algorithm is based on a generalization of the Duhamel integral for vector-valued functions. The Duhamel-like technique makes it possible to translate the initial problem to an integral equation and then approximate it with exponential accuracy. The theoretical results are illustrated by examples associated with the heat transfer boundary value problems.