In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
Article
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if
L
{\displaystyle L}
is a linear differential operator, then
the Green's function
G
{\displaystyle G}
is the solution of the equation
L
G
=
δ
,
{\displaystyle LG=\delta ,}
where
δ
{\displaystyle \delta }
is Dirac's delta function;
the solution of the inhomogeneous problem
L
y
=
f
{\displaystyle Ly=f}
is the convolution,
y
=
(
G
∗
f
)
.
{\displaystyle y=(G\ast f).}
By the superposition principle, given a linear ordinary differential equation (ODE),
L
y
=
f
{\displaystyle Ly=f}
, one can first solve
L
G
=
δ
s
{\displaystyle LG=\delta _{s}}
, for each s. If the source is a sum of delta functions, then the solution is a sum of Green's functions as well due to linearity of L. This means that the integral, viewed as a continuous sum, can reconstruct a wide class of sources,
f
{\displaystyle f}
, through the convolution integral. Whenever the integral of
f
{\displaystyle f}
with
G
{\displaystyle G}
converges, then the solution to the inhomogeneous equation,
L
y
=
f
{\displaystyle Ly=f}
, is given by
y
=
G
∗
f
{\displaystyle y=G\ast f}
.
Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead, which take into account the modern language of the theory of distributions or generalized functions.
Building off of the superposition principle in many-body theory, the term is also used in physics and engineering, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the role of propagators, also referred to as two-point (correlation) functions.
Contents
Definition and uses
A Green's function, G(x,s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
, at a point s, is any solution of
where δ is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form
If the kernel of L is non-trivial, then the Green's function is not unique. However, in practice, some combination of symmetry, boundary conditions and/or other externally imposed criteria will give a unique Green's function. Green's functions may be categorized by a Green's function number according to the type of boundary conditions being satisfied. Green's functions are not necessarily functions of a real variable but are generally understood in the sense of distributions.
Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, Green's function of the Hamiltonian is a key concept with important links to the concept of density of states.
The Green's function as used in physics is usually defined with the opposite sign, instead. That is,
L
G
(
x
,
s
)
=
δ
Motivation
Loosely speaking, if such a function G can be found for the operator L, then, if we multiply equation 1 for the Green's function by f(s), and then integrate with respect to s, we obtain,
Green's functions for solving non-homogeneous boundary value problems
The primary use of Green's functions in mathematics is to solve non-homogeneous boundary value problems. In modern theoretical physics, Green's functions are also usually used as propagators in Feynman diagrams; the term Green's function is often further used for any correlation function.
Framework
Let
L
{\displaystyle L}
be a Sturm–Liouville operator, a linear differential operator of the form
Green's function is not necessarily unique since the addition of any solution of the homogeneous equation to one Green's function results in another Green's function. Therefore, if the homogeneous equation has nontrivial solutions, multiple Green's functions exist. Certain boundary value or initial value problems involve finding a Green's function that is nonvanishing only for
s
≤
x
{\displaystyle s\leq x}
; in this case, the solution is sometimes called a retarded Green's function. Similarly, a Green's function that is nonvanishing only for
s
≥
x
{\displaystyle s\geq x}
is called an advanced Green's function. In such cases, any linear combination of the two Green's functions is also a valid Green's function. Both the advanced and retarded Green's functions are called one-sided, while a Green's function that is nonvanishing for all
x
{\displaystyle x}
in the domain of definition is called two-sided.
The terminology advanced and retarded is especially useful when the variable x corresponds to time. In such cases, the solution provided by the use of the retarded Green's function depends only on the past sources and is causal whereas the solution provided by the use of the advanced Green's function depends only on the future sources and is acausal. In these problems, it is often the case that the causal solution is the physically important one. However, the advanced Green's function is useful in finding solutions to certain inverse problems where sources are to be found from boundary data. The use of advanced and retarded Green's function is especially common for the analysis of solutions of the inhomogeneous electromagnetic wave equation.
Finding Green's functions
Eigenvalue expansions
If a differential operator L admits a set of eigenvectors Ψn(x) (i.e., a set of functions Ψn and scalars λn such that LΨn = λn Ψn ) that is complete, then it is possible to construct a Green's function from these eigenvectors and eigenvalues.
"Complete" means that the set of functions {Ψn} satisfies the following completeness relation,
First step: The Green's function for the linear operator at hand is defined as the solution to
Further examples
Let n = 1 and let the subset be all of R. Let L be
d
d
x
{\textstyle {\frac {d}{dx}}}
. Then, the Heaviside step function Θ(x − x0) is a Green's function of L at x0.
Let n = 2 and let the subset be the quarter-plane {(x, y) : x, y ≥ 0} and L be the Laplacian. Also, assume a Dirichlet boundary condition is imposed at x = 0 and a Neumann boundary condition is imposed at y = 0. Then the X10Y20 Green's function is
G
(
x
,
y
,
x
0
,
y
0
)
=
1
2
π
[
ln
(
x
−
x
0
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
Thus, one may obtain the function u(x) through knowledge of the Green's function in equation 1 and the source term on the right-hand side in equation 2. This process relies upon the linearity of the operator L.
In other words, the solution of equation 2, u(x), can be determined by the integration given in equation 3. Although f(x) is known, this integration cannot be performed unless G is also known. The problem now lies in finding the Green's function G that satisfies equation 1. For this reason, the Green's function is also sometimes called the fundamental solution associated to the operator L.
Not every operator
L
{\displaystyle L}
admits a Green's function. A Green's function can also be thought of as a right inverse of L. Aside from the difficulties of finding a Green's function for a particular operator, the integral in equation 3 may be quite difficult to evaluate. However the method gives a theoretically exact result.
This can be thought of as an expansion of f according to a Dirac delta function basis (projecting f over
δ
(
x
−
s
)
{\displaystyle \delta (x-s)}
; and a superposition of the solution on each projection. Such an integral equation is known as a Fredholm integral equation, the study of which constitutes Fredholm theory.
Applying the operator L to each side of this equation results in the completeness relation, which was assumed.
The general study of Green's function written in the above form, and its relationship to the function spaces formed by the eigenvectors, is known as Fredholm theory.
There are several other methods for finding Green's functions, including the method of images, separation of variables, and Laplace transforms.
. Though this is a somewhat limited case, the Wronskian frequently appears in other sets of boundary value problems that require a one-sided (advanced/retarded) Green's function as well, including those with conditions on boundary derivatives (Neumann conditions) or a pair of conditions on a function and its normal derivative on a single boundary (Cauchy conditions).
Suppose that the linear differential operator L is the Laplacian, ∇2, and that there is a Green's function G for the Laplacian. The defining property of the Green's function still holds,
Using this expression, it is possible to solve Laplace's equation ∇2φ(x) = 0 or Poisson's equation ∇2φ(x) = −ρ(x), subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for φ(x) everywhere inside a volume where either (1) the value of φ(x) is specified on the bounding surface of the volume (Dirichlet boundary conditions), or (2) the normal derivative of φ(x) is specified on the bounding surface (Neumann boundary conditions).
Suppose the problem is to solve for φ(x) inside the region. Then the integral
This form expresses the well-known property of harmonic functions, that if the value or normal derivative is known on a bounding surface, then the value of the function inside the volume is known everywhere.
In electrostatics, φ(x) is interpreted as the electric potential, ρ(x) as electric charge density, and the normal derivative
If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that G(x,x′) vanishes when either x or x′ is on the bounding surface. Thus only one of the two terms in the surface integral remains. If the problem is to solve a Neumann boundary value problem, it might seem logical to choose Green's function so that its normal derivative vanishes on the bounding surface. However, application of Gauss's theorem to the differential equation defining the Green's function yields
meaning the normal derivative of G(x,x′) cannot vanish on the surface, because it must integrate to 1 on the surface.
The simplest form the normal derivative can take is that of a constant, namely 1/S, where S is the surface area of the surface. The surface term in the solution becomes
is the average value of the potential on the surface. This number is not known in general, but is often unimportant, as the goal is often to obtain the electric field given by the gradient of the potential, rather than the potential itself.
With no boundary conditions, the Green's function for the Laplacian (Green's function for the three-variable Laplace equation) is
Supposing that the bounding surface goes out to infinity and plugging in this expression for the Green's function finally yields the standard expression for electric potential in terms of electric charge density as
If
x
≠
s
{\displaystyle x\neq s}
, then the delta function gives zero, and the general solution is
