Green's Matrix for a Second Order Self-Adjoint Matrix Differential Operator

TL;DR

This paper presents a systematic method to construct the Green's matrix for second order self-adjoint matrix differential operators, especially when transform techniques are not applicable, by leveraging solutions of the homogeneous equation.

Contribution

It introduces a novel approach to derive the Green's matrix directly from solutions of the homogeneous system without relying on transform methods.

Findings

Provides a systematic construction method for Green's matrices

Applicable to cases where transform techniques fail

Enhances analytical tools for differential operators

Abstract

A systematic construction of the Green's matrix for a second order, self-adjoint matrix differential operator from the linearly independent solutions of the corresponding homogeneous differential equation set is carried out. We follow the general approach of extracting the Green's matrix from the Green's matrix of the corresponding first order system. This construction is required in the cases where the differential equation set cannot be turned to an algebraic equation set via transform techniques.