Computation of Green's function of the bounded solutions problem

TL;DR

This paper presents a method to compute Green's function for bounded solutions of linear differential equations using Newton interpolation, including sensitivity analysis of the solution.

Contribution

It introduces a novel representation of Green's function via Newton interpolating polynomial for approximate computation.

Findings

Effective approximation of Green's function demonstrated.

Sensitivity estimates for the bounded solutions problem provided.

Applicable to systems with eigenvalues off the imaginary axis.

Abstract

It is well known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a square matrix, has a unique bounded solution $x$ for any bounded continuous free term $f$, provided the coefficient $A$ has no eigenvalues on the imaginary axis. This solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(t-s)x(s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In the paper, a representation of Green's function in the form of the Newton interpolating polynomial is used for approximate calculation of $\mathcal G$. An estimate of the sensitivity of the problem is given.