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

**MSC:**- 65J10 Equations with linear operators (do not use 65Fxx)
- 65M12 Stability and convergence of numerical methods
- 65M15 Error bounds
- 46N20 Applications to differential and integral equations
- 46N40 Applications in numerical analysis [See also 65Jxx]
- 47N20 Applications to differential and integral equations
- 47N40 Applications in numerical analysis [See also 65Jxx]

**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*

**Upload:** 2005-06-27

**Update:** 2005

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