Chapter 10
Introductory Green’s Functions
In this chapter, we deal with various properties and characteristics of differential
equations, especially first-order linear differential equations (FOLDEs) and secondorder 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.
10.1 Second-Order Linear Differential Equations
(SOLDEs)
A general form of n-th order linear differential equations has the following form:
a n x
ð Þ
d
n u
dx n þ a nÀ1 x
ð Þ
d
nÀ1 u
dx nÀ1 þ Á Á Á þ a 1 x
ð Þ
du
dx
þ a 0 x
ð Þu ¼ d x
ð Þ:
ð10:1Þ
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_10
379
Introductory Green’s Functions
In this chapter, we deal with various properties and characteristics of differential
equations, especially first-order linear differential equations (FOLDEs) and secondorder 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.
10.1 Second-Order Linear Differential Equations
(SOLDEs)
A general form of n-th order linear differential equations has the following form:
a n x
ð Þ
d
n u
dx n þ a nÀ1 x
ð Þ
d
nÀ1 u
dx nÀ1 þ Á Á Á þ a 1 x
ð Þ
du
dx
þ a 0 x
ð Þu ¼ d x
ð Þ:
ð10:1Þ
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_10
379