Two-points problem for an evolutional first order equation in Banach space

TL;DR

This paper introduces an exponentially convergent algorithm for solving two-point nonlocal problems in Banach spaces involving first order evolution equations with operator coefficients, demonstrating efficiency through numerical examples.

Contribution

It proposes a new algorithm with exponential convergence for a class of evolution equations in Banach spaces, under specific operator conditions.

Findings

Algorithm achieves exponential convergence in time.

Numerical examples confirm the efficiency of the method.

System of linear equations solved by fixed-point iteration.

Abstract

Two-points nonlocal problem for the first order differential evolution equation with an operator coefficient in a Banach space $X$ is considered. An exponentially convergent algorithm is proposed and justified in assumption that the operator coefficient is strongly positive and some existence and uniqueness conditions are fulfilled. This algorithm leads to a system of linear equations that can be solved by fixed-point iteration. The algorithm provides exponentially convergence in time that in combination with fast algorithms on spatial variables can be efficient treating such problems. The efficiency of the proposed algorithms is demonstrated by numerical examples.