Green's functions for reducible functional differential equations

TL;DR

This paper develops a general method to derive Green's functions for reducible functional differential equations involving reflection and Hilbert transforms, simplifying their solutions by reducing them to ordinary differential equations.

Contribution

It introduces a novel, systematic approach for obtaining Green's functions for a class of functional differential equations involving reflection and transforms.

Findings

Method successfully derives Green's functions for various boundary value problems.

Reduces complex functional differential equations to simpler ODEs.

Provides explicit examples demonstrating the method's effectiveness.

Abstract

In this work we study differential problems in which the reflection operator and the Hilbert transform are involved. We reduce these problems to ODEs in order to solve them. Also, we describe a general method for obtaining the Green's function of reducible functional differential equations and illustrate it with the case of homogeneous boundary value problems with reflection and several specific examples.