The One-Sided Green's Function

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Abstract
Let L be a linear differential operator of the nth order whose coefficients pi(x) are continuous in a semi-infinite interval I: [a, ∞). A function H(x, ζ) is said to be a one-sided Green's function for the operator L if it satisfies the four conditions: (1) H is continuous and its first n derivatives with respect to x are continuous in I. (2) Hα(ζ, ζ)=0 for α=0, 1, …, n−2. (3) Hn−1(ζ, ζ)=1/p0(ζ). (4) LH=0. (The subscript on the H refers to partial differentiation with respect to the first argument, and p0(x) is the coefficient of dn/dxn in the expression for L.) It is shown that H is unique and if u(x)=∫ax H(x, ζ)f(ζ)dζ, then Lu=f(x) and u(α)(a)=0, α=0, 1, …, n−1. Furthermore, if H is given, a fundamental system of solutions of Lu=0 can be written down explicitly in terms of H and its derivatives evaluated at the end point a. The converse problem is trivial. Other properties of H are also considered, for example, its relation to the impulsive response of a network.

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