A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

TL;DR

This paper introduces a unified Hilbert space framework for solving differential equations with delay and memory terms, encompassing integro-differential, neutral, and delay differential equations, ensuring causality of solutions.

Contribution

It develops a novel approach by defining a time derivative as a normal operator in Hilbert spaces, unifying various types of differential equations with memory.

Findings

Causal solution operators are established for the class of equations.

The framework covers integro-differential, neutral, and delay differential equations.

A new solution theory simplifies analysis of equations with memory in Hilbert spaces.

Abstract

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.