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’.
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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
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: R_{1} =
r^{n} and R_{2} =
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
\[
A' (r) = \frac{1}{2n}\, f(r) \qquad\mbox{and} \qquad B' (r) = -\frac{1}{2n}\,
f(r) .
\]
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:
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 :
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 𝑎_{n}, 𝑎_{n-1}, … , 𝑎_{0} are continuous functions and 𝑎_{n}(x) ≠ 0. As usual, \( \displaystyle \texttt{D}^0 = \texttt{I} \) is the identical operator.
Correspondingly, we consider the nonhomogeneous differential equation
\begin{equation} \label{Eqgreen.1}
L \left[ x, \texttt{D} \right] y (x) = f(x) , \qquad x \in \Omega \subset \mathbb{R},
\end{equation}
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, x_{0}) satisfying the differential equation
\begin{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},
\end{equation}
where x_{0} is a fixed point from Ω. The function in the right-hand side the Dirac delta function. This means that away from the point
x_{0}
\begin{equation} \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}
\end{equation}
The Green's function for the differential operator L can be defined in another equivalent way. It is a function G(x, x_{0}) of two variables x and x_{0} 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 = x_{0}:
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 - x_{0} when all coefficients of the differential operator L are constants, so
\( L \left[ x, \texttt{D} \right] = L \left[ \texttt{D} \right] . \)
Duffy, D.G., Green's Functions with Applications (Advances in Applied Mathematics) 2nd Edition, Chapman and Hall/CRC; 2nd edition (August 14, 2018)
Greenberg, M., Applications of Green's Functions in Science and Engineering (Dover Books on Engineering), 2015.
Economou. E.N., Green’s Functions in Quantum Physics, Springer,
Kythe, P.K., Green's Functions and Linear Differential Equations: Theory, Applications, and Computation, Chapman & Hall/CRC, 2011. ISBN-13 : 978-1439840085
Roach, G.F., Green's Functions 2nd Edition, Cambridge University Press, 1982. ISBN 10: 0521282888 / ISBN 13: 9780521282888
Rother, T., Green’s Functions in Classical Physics,
Stakgold, I, Green's Functions and Boundary Value Problems,
John Wiley \& Sons, New York, 1979.
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