ABSTRACT
This chapter begins with problems of applied mechanics that are modelled with ordinary differential equations. It describes partial differential equations for which the list of available Green's functions and matrices is very limited. The chapter presents the technique that was originally developed for the construction of Green's functions and matrices for elliptic equations in two dimensions. It considers a number of compact representations of Green's functions for Laplace's and Klein-Gordon equations expressed in various coordinate systems. The technique is based on the so-called method of eigenfunction expansion and has proven to be especially effective for a variety of problems in computational continuum mechanics, which reduce to Laplace's, Klein-Gordon, and biharmonic equations to Lame's system for the displacement formulation of the plane problem in the theory of elasticity. Two stages are distinguished within the scope of the suggested technique that represents Green's functions in terms of their Fourier expansions with respect to one of the independent variables.
