Fast convergent method for the $m$-point problem in Banach space

TL;DR

This paper introduces an exponentially convergent algorithm for solving the $m$-point nonlocal differential problem in Banach spaces, leveraging operator function representations and resolvent quadratures, with demonstrated numerical efficiency.

Contribution

It presents a novel, rapidly converging method for the $m$-point problem in Banach spaces using Dunford-Cauchy integrals and resolvent sums, under specific operator conditions.

Findings

Algorithm achieves exponential convergence.

Numerical examples confirm efficiency.

Applicable under strong positivity and existence conditions.

Abstract

The $m$-point nonlocal problem for the first order differential equation with an operator coefficient in a Banach space $X$ is considered. An exponentially convergent algorithm is proposed and justified provided that the operator coefficient $A$ is strongly positive and some existence and uniqueness conditions are fulfilled. This algorithm is based on representations of operator functions by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of $A$ and on the proper quadratures involving short sums of resolvents. The efficiency of the proposed algorithms is demonstrated by numerical examples.