Green's Functions for Stieltjes Boundary Problems

TL;DR

This paper extends the transformation of Green's operators into Green's functions to the broader class of Stieltjes boundary problems, which include more complex boundary conditions and may involve ill-posed cases.

Contribution

It generalizes the classical transformation process to Stieltjes boundary problems, addressing cases with multiple evaluation points, derivatives, and integral terms.

Findings

Transformation extends to all Stieltjes boundary problems

Ill-posed problems involve distributional terms

Structure of Green's function remains unaffected

Abstract

Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation points. (2) They allow derivatives of arbitrary order. (3) Global terms in the form of definite integrals are allowed. Assuming the Stieltjes boundary problem is regular (a unique solution exists for every forcing function), there are symbolic methods for computing the associated Green's operator. In the classical case of well-posed two-point boundary value problems, it is known how to transform the Green's operator into the so-called Green's function, the representation usually preferred by physicists and engineers. In this paper we extend this transformation to the whole class of Stieltjes boundary problems. It turns out that the extension (1) leads to more case distinction, (2) implies ill-posed problems and hence distributional terms, (3) has apparently no effect on the structure of the Green's function.