Green's Function of a generalized boundary value transmission problem

TL;DR

This paper develops a Green's function approach for a class of generalized boundary value transmission problems associated with Sturm-Liouville equations on two intervals, providing a new operator framework and solution methods.

Contribution

It introduces a modified inner product and constructs Green's function for the generalized BVTP, offering a novel operator-theoretic approach to these problems.

Findings

Constructed Green's function for the generalized BVTP

Defined a symmetric operator in a modified inner product space

Derived the resolvent function for the inhomogeneous problem

Abstract

The aim of this study is to investigate a class of generalized 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.