Numerical method for abstract Cauchy problem with nonlinear nonlocal condition

TL;DR

This paper presents a numerical method for solving a class of first-order differential equations with nonlinear nonlocal conditions in Banach spaces, utilizing operator reduction, collocation, and Sinc-based evaluation techniques.

Contribution

It introduces a novel numerical approach combining reduction to Hammerstein equations, collocation discretization, and Sinc-based operator exponential evaluation.

Findings

Method is justified under strong positivity of operator A.

Converges under existence and uniqueness conditions.

Efficient numerical implementation demonstrated.

Abstract

Problem for the first order differential equation with an unbounded operator coefficient in Banach space and nonlinear nonlocal condition is considered. A numerical method is proposed and justified for the solution of this problem under assumptions that the mentioned operator coefficient $A$ is strongly positive and some existence and uniqueness conditions are fulfilled. The method is based on the reduction of the given problem to an abstract Hammerstein equation. The later one is discretized by collocation and then solved via the fixed-point iteration method. Each iteration of the method involves Sinc-based numerical evaluation of the operator exponential represented by a Dunford-Cauchy integral along hyperbola enveloping the spectrum of $A$.