68 - Green's Functions

Abstract

This chapter describes the green's functions applicable to linear differential equations with linear boundary conditions. It yields an exact solution, in the form of an integral or an infinite series. Initially, the solution of the linear differential equation with a “point source” is determined. Then, using superposition, the “forcing function” (appearing in either the differential equation or the boundary condition) is treated as a collection of point sources. Green's functions can be calculated once then used repeatedly for different functions f (x) and h(x). The chapter illustrates two methods for constructing G(x; z) for the special case of a second-order linear ordinary differential equation. It also illustrates the construction process for g(x; z) for a partial differential equation. The chapter highlights that delta functions, in nonrectangular coordinate systems, are easily determined by a change of variables in the defining relation: ∫δ(z)dz = 1. In changing variables, the Jacobian of the transformation will then divide the delta function terms.