by T. Ju. Bohonova, I. P. Gavrilyuk, V. L. Makarov, V. Vasylyk
Preprint series: 05-06, Reports on Numerical Mathematics
Preprint series: , Reports on Numerical Mathematics
Abstract: A suitable abstract setting of the initial value problem for the first order differential equation with an unbounded operator coefficient in a Banach space where the domain of the operator depends on the dependent variable t is introduced. A new exponentially convergent algorithm for such problems is proposed. This algorithm is based on a generalization of the Duhamel´s integral for vector-valued functions which allows to translate the initial problem into a boundary integral equation and then approximate it with exponential accuracy. Examples of boundary value problems for the heat equation with time-dependent boundary conditions are given which confirm and illustrate the theoretical results obtained.
Keywords: First order differential equations in Banach space, operator coefficient with a variable domain, Duhamel´s integral, operator exponential, exponentially convergent algorithms