Chapter 3 - Green's Functions

Abstract

This chapter discusses a nonhomogeneous linear second-order ordinary differential equation, with given boundary conditions, by presenting the solution in terms of an integral. The function G(x,t) is called Green's function after the English mathematician George Green, who pioneered work in this area in the 1830s. The chapter demonstrates three methods of constructing Green's functions—that is, by using the Dirac-delta function, variation of parameters, and eigenfunction expansions. A second method for constructing Green's function is based on the theorem that uses the technique called variation of parameters to find a particular solution to certain second-order differential equations. The chapter also discusses the Fredholm alternative and Green's function for the Laplacian in higher dimensions. Then, the fundamental solution or the free-space Green's function (the situation with no boundary conditions) for the negative Laplacian in two and three dimensions is derived in the chapter.