Green's function for Sturm-Liouville problem

TL;DR

This paper develops a new Green's function approach for a class of boundary value transmission problems associated with Sturm-Liouville equations on two intervals, expanding analytical tools for such differential equations.

Contribution

It introduces a modified inner product and constructs Green's function for the Sturm-Liouville boundary value transmission problem, providing a new method for solving these equations.

Findings

Constructed Green's function for the problem

Defined a symmetric linear operator in a modified space

Derived the resolvent function for inhomogeneous cases

Abstract

The purpose of this study is to investigate a new class of boundary value transmission problems (BVTP's) for Sturm-Liouville equation on two separate intervals. We introduce modified inner product in direct sum space $L_{2}[a,c)\oplus L_{2}(c,b]\oplus\mathbb{C}^{2}$ and define symmetric linear operator in it such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. Then by suggesting an own approaches we construct Green's function for problem under consideration and find the resolvent function for corresponding inhomogeneous problem.