Green's function
Green's function
In mathematics, a Green's function of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions is its impulse response.
This means that if L is the linear differential operator, then
the Green's function G is the solution of the equation LG = δ, where δ is Dirac's delta function;
the solution of the initial-value problem Ly = f is the convolution (G * f), where G is the Green's function.
Through the superposition principle, given a linear ordinary differential equation (ODE), L(solution) = source, one can first solve L(green) = δ**s, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.
Green's functions are named after the British mathematician George Green, who first developed the concept in the 1830s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead.
Under many-body theory, the term is also used in physics, 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 roles of propagators.
Definition and uses
(1)
where δ is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form
(2)
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 the type of boundary conditions satisfied, by a Green's function number. Also, Green's functions in general are distributions, not necessarily functions of a real variable.
Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, the 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,
This definition does not significantly change any of the properties of the Green's function due to the evenness of the Dirac delta function.
If the operator is translation invariant, that is, when L has constant coefficients with respect to x, then the Green's function can be taken to be a convolution operator, that is,
In this case, the Green's function is the same as the impulse response of linear time-invariant system theory.
Motivation
Loosely speaking, if such a function G can be found for the operator L, then, if we multiply the equation (1) for the Green's function by f (s), and then integrate with respect to s, we obtain,
This means that
(3)
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 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)); 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.
Green's functions for solving inhomogeneous 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 be the Sturm–Liouville operator, a linear differential operator of the form
and let D be the boundary conditions operator
Let f(x) be a continuous function in [ 0, ℓ ]. Further suppose that the problem
is regular, i.e., only the trivial solution exists for the homogeneous problem.
Theorem
There is one and only one solution u(x) that satisfies
and it is given by
where G(x,s) is a Green's function satisfying the following conditions:
is continuous in and .
For , .
For , .
Derivative "jump": .
Symmetry: .
Advanced and retarded Green's functions
Sometimes the Green's function can be split into a sum of two functions. One with the variable positive (+) and the other with the variable negative (-). These are the advanced and retarded Green's functions, and when the equation under study depends on time, one of the parts is causal and the other anti-causal. In these problems usually the causal part is the important one. These are frequently the solutions to the inhomogeneous electromagnetic wave equation.
Finding Green's functions
Units
While it doesn't uniquely fix the form the Green's function will take, performing a dimensional analysis to find the units a Green's function must have is an important sanity check on any Green's function found through other means. A quick examination of the defining equation,
Eigenvalue expansions
If a differential operator L admits a set of eigenvectors Ψn(x) (i.e., a set of functions Ψn and scalars λ**n such that LΨn = λ**n 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,
Then the following holds,
Applying the operator L to each side of this equation results in the completeness relation, which was assumed.
The general study of the 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 (Cole 2011).
Combining Green's functions
Table of Green's functions
Differential operatorL | Green's functionG | Example of application |
---|---|---|
with | 1D damped harmonic oscillator | |
2D Laplace operator | with | 2D Poisson equation |
3D Laplace operator | with | Poisson equation |
Helmholtz operator | stationary 3D Schrödinger equation for free particle | |
indimensions | Yukawa potential, Feynman propagator | |
1D wave equation | ||
2D wave equation | ||
D'Alembert operator | 3D wave equation | |
1D diffusion | ||
2D diffusion | ||
3D diffusion | ||
1D Klein–Gordon equation | ||
2D Klein–Gordon equation | ||
3D Klein–Gordon equation | ||
telegrapher's equation | ||
2D relativistic heat conduction | ||
3D relativistic heat conduction |
Green's functions for the Laplacian
Green's functions for linear differential operators involving the Laplacian may be readily put to use using the second of Green's identities.
To derive Green's theorem, begin with the divergence theorem (otherwise known as Gauss's theorem),
Plugging this into the divergence theorem produces Green's theorem,
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
reduces to simply φ(x) due to the defining property of the Dirac delta function and we have
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.
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, the Green's function is chosen such that its normal derivative vanishes on the bounding surface, as it would seem to be the most logical choice. (See Jackson J.D. classical electrodynamics, page 39). 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. (Again, see Jackson J.D. classical electrodynamics, page 39 for this and the following argument).
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
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
Example
Example. Find the Green function for the following problem, whose Green's function number is X11:
First step: The Green's function for the linear operator at hand is defined as the solution to
To summarize the results thus far:
So the Green's function for this problem is:
Further examples
Let n = 1 and let the subset be all of ℝ. Let L be . Then, the Heaviside step function H(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
Let , and all three are elements of the real numbers. Then, for any function from reals to reals, , with an th derivative that is integrable over the interval :
- The Green's function in the above equation,, is not unique. How is the equation modified ifis added to, wheresatisfiesfor all(for example,with)? Also, compare the above equation to the form of aTaylor seriescentered at.
See also
Bessel potential
Discrete Green's functions – defined on graphs and grids
Impulse response – the analog of a Green's function in signal processing
Fundamental solution
Green's function in many-body theory
Correlation function
Propagator
Green's identities
Parametrix
Volterra integral equation
Resolvent formalism
Keldysh formalism
Spectral theory