Green's Function For Linear Differential Operators In One Variable

TL;DR

This paper derives a general series integral formula for the causal Green's function of linear differential operators in one variable, including methods for explicit solutions and boundary condition considerations.

Contribution

It provides a unified integral series approach for Green's functions of linear differential operators, including explicit formulas and boundary condition handling.

Findings

Derived a general integral series formula for Green's functions.

Showed multiplicative property of Green's functions.

Presented a method for arbitrary boundary conditions.

Abstract

General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic formula for fundamental solutions of corresponding homogenous linear differential equation as series of integrals. Furthermore, multiplicative property of causal Green's functions is shown and by which explicit formulas for causal Green's functions of some classes of decomposable linear differential operators are given. A method to find Green's function of general linear differential operator of given degree in one variable with arbitrary boundary condition according to coefficient functions of differential operator is demonstrated.