# Preface

Green's functions are named after the British mathematician and physicist George Green born in Nottingham in 1793 who ‘invented’ the Green function in 1828. This invention was developed in an essay entitled Mathematical Analysis to the Theories of Electricity and Magnetism originally published in Nottingham in 1828 and reprinted by the George Green Memorial Committee to mark the bicentenary of the birth of George Green in 1993 when he was finally given the recognition he deserved.

The Green function is a powerful mathematical tool that was successfully applied to classical electromagnetism and acoustics in the late Nineteenth Century. More recently, the Green function has been the working tool of calculations in particle physics, condensed matter and solid state physics, quantum mechanics and many other topics of applied mathematics and mathematical physics. In particular, the development of the Feynman diagram was based on the Green function. In fact, the Feynman diagram can be considered to be a pictorial representation of a Green function (a Green function associated with wave operators) --- what Richard Feynman referred to as a ‘propagator’.

# Green's functions

Green's function method provides connections between differential operator and integral-operator for the description of physics problems. The essential feature of any Green’s function method is that it provides a way to describe the response of the differential equation solution to an arbitrary source term (in the presence of the boundary conditions).

Example: Consider the Euler equation on the finite interval

$r^2 R'' (r) + r\, R' (r) - n^2 R(r) = f(r) , \qquad 0 < r < \ell ,$
where n is an integer and f is a given function. We rewrite the above Euler equation in a self-adjoint form:
$r\,\frac{\text d}{{\text d}r} \left( r\, \frac{{\text d}R}{{\text d}r} \right) - n^2 R(r) = f(r) , \qquad 0 < r < \ell .$
Since the corresponding homogeneous equation $$r^2 R'' (r) + r\, R' (r) - n^2 R(r) = 0$$ has two linearly independent solutions for n > 0: R1 = rn and R2 = r-n, we seek its solution in the form
$R(r) = A(r)\, r^n + B(r)\, r^{-n} ,$
with some unknown yet smooth functions A and B. Finishing the variation of parameters, we come to two equations
$A' (r) = \frac{1}{2n}\, f(r) \qquad\mbox{and} \qquad B' (r) = -\frac{1}{2n}\, f(r) .$
Integration yeilds
$A(r) = - \frac{1}{2n}\, \int_r^{\ell} f(r) \,r^{-n+1}\,{\text d}r , \quad B(r) = - \frac{1}{2n}\, \int_0^r f(r) \,r^{n+1}\,{\text d}r$
Therefore, we get
$R(r) = - \frac{1}{2n}\,r^n \int_r^{\ell} f(r) \,r^{-n+1}\,{\text d}r - \frac{1}{2n}\,r^{-n} \int_0^r f(r) \,r^{n+1}\,{\text d}r .$
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The Fokas Method

Let us consider anormalized linear differential operator of second order
$L \left[ \texttt{D} \right] = \texttt{D}^2 + p\,\texttt{D} + q\, \texttt{I} , \qquad \texttt{D} = {\text d}/{\text d}x , \quad \texttt{D}^0 = \texttt{I},$
where p, q are constants and I is the identical operator.

We demonstrate application of the Fokas method to determination of the Green function on the example of second order differential equation with the boundary conditions of the third kind:

$$\label{EqFokas.1} L \left[ \texttt{D} \right] y = f(x) , \quad 0 < x < \ell , \qquad y(0) - b_0 y' (0) = c_0 , \quad y(\ell ) + b_{\ell} y' (\ell ) = c_{\ell} .$$
For simplicitly, we assume that the characteristic equation λ² + pλ + q = 0 has two distince real roots:
$\lambda_1 = -\frac{p}{2} + \frac{1}{2}\, \sqrt{p^2 - 4q} , \qquad \lambda_2 = -\frac{p}{2} - \frac{1}{2}\, \sqrt{p^2 - 4q} , \qquad p^2 - 4q > 0.$
Multiplying the differential equation by the exponential function $$e^{\lambda x} \ne 0 ,$$ where λ is one of the roots of the characteristic equation, we can reduce it to the divergent form :
$L \left[ \texttt{D} \right] y \,e^{\lambda x} = \texttt{D} \left( \,e^{\lambda x} \texttt{D}\,y \right) + \left( p - \lambda \right) \texttt{D} \left( y\,e^{\lambda x} \right) \,e^{\lambda x} - 2p\,\lambda y\,e^{\lambda x}$
Integration with respect to x first time in the interval (0, x) and then in the interval (x, ℓ), we obtain

n-th order differential operators

Let us consider the n-th order linear differential operator $L \left[ x, \texttt{D} \right] = a_n (x)\,\texttt{D}^n + a_{n-1} (x)\,\texttt{D}^{n-1} + \cdots + a_1 (x)\, \texttt{D} + a_0 (x) \texttt{I} , \qquad \texttt{D} = \frac{\text d}{{\text d}x} ,$ where all coefficients 𝑎n, 𝑎n-1, … , 𝑎0 are continuous functions and 𝑎n(x) ≠ 0. As usual, $$\displaystyle \texttt{D}^0 = \texttt{I}$$ is the identical operator. Correspondingly, we consider the nonhomogeneous differential equation $$\label{Eqgreen.1} L \left[ x, \texttt{D} \right] y (x) = f(x) , \qquad x \in \Omega \subset \mathbb{R},$$ subject to some kind of boundary conditions, which we will initially suppose are homogeneous. The Green's function corresponding to Eq.\eqref{Eqgreen.1} is a function G(x, x0) satisfying the differential equation $$\label{Eqgreen.2} L \left[ x, \texttt{D} \right] G \left( x, x_0 \right) = \delta \left( x- x_0 \right) , \qquad x \in \Omega \subset \mathbb{R},$$ where x0 is a fixed point from Ω. The function in the right-hand side the Dirac delta function. This means that away from the point x0
$$\label{Eqgreen.3} \begin{split} L \left[ x, \texttt{D} \right] G \left( x, x_0 \right) = 0 \qquad\mbox{for } x < x_0 , \\ L \left[ x, \texttt{D} \right] G \left( x, x_0 \right) = 0 \qquad\mbox{for } x > x_0 . \end{split}$$
The Green's function for the differential operator L can be defined in another equivalent way. It is a function G(x, x0) of two variables x and x0 that satisfies the differential equation
$L \left[ x, \texttt{D} \right] G \left( x, x_0 \right) = 0 \qquad x \ne x_0 ,$
and its (n-1)-th derivative suffers a discontinuous jump at x = x0:
$$\label{Eqgreen.4} \lim_{x \to x_0 +0} \,\frac{\partial^{n-1} G}{\partial x^{n-1}} - \lim_{x \to x_0 -0} \,\frac{\partial^{n-1} G}{\partial x^{n-1}} = \frac{1}{a_n (x_0 )} ,$$
but for all other its derivatives its are continuous:
$$\label{Eqgreen.5} \lim_{x \to x_0 +0} \,\frac{\partial^{k} G}{\partial x^{k}} = \lim_{x \to x_0 -0} \,\frac{\partial^{k} G}{\partial x^{k}} , \qquad k=0,1,2,\ldots , n-2.$$
The problem for determining the Green’s function is now very concrete, and simply uses elementary ODE techniques. First, equations \eqref{Eqgreen.3} are solved separately. Then the general solution must be made to satisfy the right-hand boundary conditions only, whereas the solution must satisfy the left-hand boundary conditions. This will leave free constants in the piecewise solution for G, and these are uniquely determined by demanding that the connection conditions \eqref{Eqgreen.4}, \eqref{Eqgreen.5} are all met. Note that the Green's function depends on the dfference x - x0 when all coefficients of the differential operator L are constants, so $$L \left[ x, \texttt{D} \right] = L \left[ \texttt{D} \right] .$$

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