Introductory Green’s Functions

Abstract

In this chapter, we deal with various properties and characteristics of differential equations, especially first-order linear differential equations (FOLDEs) and second-order linear differential equations (SOLDEs). These differential equations are characterized by differential operators and boundary conditions (BCs). Of these, differential operators appearing in SOLDEs are particularly important. Under appropriate conditions, the said operators can be converted to Hermitian operators. The SOLDEs associated to classical orthogonal polynomials play a central role in many fields of mathematical physics including quantum mechanics and electromagnetism. We study the general principle of SOLDEs in relation to several specific SOLDEs we have studied in Part I and examine general features of an eigenvalue problem and an initial-value problem (IVP). In this context, Green’s functions provide a powerful tool for solving SOLDEs. For a practical purpose, we deal with actual construction of Green’s functions. In Sect. 8.8, we dealt with steady-state characteristics of electromagnetic waves in dielectrics in terms of propagation, reflection, and transmission. When we consider transient characteristics of electromagnetic and optical phenomena, we often need to deal with SOLDEs having constant coefficients. This is well known in connection with a motion of a damped harmonic oscillator. In the latter part of this chapter, we treat the initial value problem of a SOLDE of this type.