Approximate solution to abstract differential equations with variable domain

TL;DR

This paper introduces an exponentially convergent algorithm for solving abstract first-order differential equations with unbounded operator coefficients and variable domains, using a generalized Duhamel integral approach.

Contribution

It presents a novel algorithm that achieves exponential convergence for complex differential equations with variable domains, extending existing methods.

Findings

Algorithm demonstrates exponential accuracy in approximations.

Theoretical results validated through heat transfer boundary value problem examples.

Provides a new approach for differential equations with unbounded operators.

Abstract

A new exponentially convergent algorithm is proposed for an abstract the first order differential equation with unbounded operator coefficient possessing a variable domain. The algorithm is based on a generalization of the Duhamel integral for vector-valued functions. This technique translates the initial problem to a system of integral equations. Then the system is approximated with exponential accuracy. The theoretical results are illustrated by examples associated with the heat transfer boundary value problems.