The Wronskian and the variation of parameters method in the theory of linear Stieltjes differential equations of second order

TL;DR

This paper extends the concepts of Wronskian and variation of parameters to second order linear differential equations with Stieltjes derivatives, providing new tools for solving such equations with applications like the Helmholtz equation.

Contribution

It introduces the notions of Wronskian and simplified Wronskian for Stieltjes derivatives and develops a variation of parameters method for equations with g-continuous coefficients.

Findings

Defined Wronskian and simplified Wronskian for Stieltjes derivatives

Developed variation of parameters method for second order equations

Applied theory to Helmholtz equation with piecewise-constant coefficients

Abstract

In this work, we define the notions of Wronskian and simplified Wronskian for Stieltjes derivatives and study some of their properties in a similar manner to the context of time scales or the usual derivative. Later, we use these tools to investigate second order linear differential equations with Stieltjes derivatives to find linearly independent solutions, as well as to derive the variation of parameters method for problems with $g$-continuous coefficients. This theory is later illustrated with some examples such as the study of the one-dimensional linear Helmholtz equation with piecewise-constant coefficients.