Exponentially convergent method for integral nonlocal problem for the first order differential equation with unbounded coefficient in Banach space

TL;DR

This paper introduces an exponentially convergent numerical method for solving first-order differential equations with unbounded operator coefficients in Banach spaces, using Dunford-Cauchy integrals and quadrature techniques.

Contribution

It proposes a new algorithm that ensures exponential convergence for integral nonlocal problems with unbounded operators, under certain positivity and existence conditions.

Findings

Algorithm demonstrates exponential convergence in numerical experiments

Method effectively handles unbounded operator coefficients

Numerical examples confirm high efficiency and accuracy

Abstract

Problem for the first order differential equation with an unbounded operator coefficient in Banach space and integral nonlocal condition is considered. An exponentially convergent algorithm is proposed and justified for the numerical solution of this problem in assumption that an operator coefficient $A$ is strongly positive and some existence and uniqueness conditions are fulfilled. This algorithm is based on the 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 several numerical examples.