Green’s Functions and Integral Equations

Abstract

In this and subsequent chapters we shall be concerned with the vibration of, and in particular the inverse problems for, three systems with continuously distributed mass: the taut vibrating string, and the rod in longitudinal or torsional vibration. In the first three sections of this chapter we shall state the governing differential equations, discuss some transformations linking them, derive some elementary properties of their solutions, and formulate them as integral equations, using the concept of a Green’s function. Section 8.4 lists some classical properties of integral equations with symmetric kernels while Section 8.5 introduces the concept of an oscillatory kernel. The physical meaning of this term is given in Theorem 8.5.7. Section 8.6 is concerned with classical results on completeness while Sections 8.7, 8.8 lay the groundwork for a description of the oscillatory properties of eigenfunctions. Section 8.9 introduces Perron’s theorem and the concept of an associted kernel. The reader may note that the presentation mirrors that given earlier for oscillatory matrices. In Section 8.10 we discuss the interlacing of eigenvalues, i.e, how the eigenvalues corresponding to one set of boundary conditions lie in between those for another. Section 8.11 is concerned with asymptotic properties, while Section 8.12 is virtually separate from the earlier part of the chapter and is concerned with impulse responses, i.e., the behaviour of the system in the time, rather than frequency, domain. This description is taken up again in Section 9.8.

Mathematicians who are only mathematicians have exact minds, provided all things are explained to them by means of definitions and axioms; otherwise they are inaccurate and unsufferable, for they are only right when the principles are quite clear.

Pascal’s Pensées