↑ arvau

Wikipedia / Green's function

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.

is a linear differential operator, then

By the superposition principle, given a linear ordinary differential equation (ODE), one can first solve for each . 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 , through the convolution with converges, then the solution to the inhomogeneous equation, , is given by

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.

Definition and uses

A Green's function, , of a linear differential operator acting on distributions over a subset of the Euclidean space at a point , is any solution of where 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 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.

This definition does not significantly change any of the properties of Green's function due to the evenness of the Dirac delta function.

If the operator is translation invariant has constant coefficients with respect to , then the Green's function can be taken to be a convolution kernel In this case, Green's function is the same as the impulse response of linear time-invariant system theory.

Motivation

Loosely speaking, if such a function can be found for the operator , then, if we multiply for the Green's function by , and then integrate with respect to is linear and acts only on the variable (and not on the variable of integration This means that

Thus, one may obtain the function through knowledge of the Green's function in and the source term on the right-hand side in . This process relies upon the linearity of the operator .

In other words, the solution of , , can be determined by the integration given in . Although is known, this integration cannot be performed unless is also known. The problem now lies in finding the Green's function that satisfies . For this reason, the Green's function is also sometimes called the fundamental solution associated to the operator .

admits a Green's function. A Green's function can also be thought of as a right inverse of . Aside from the difficulties of finding a Green's function for a particular operator, the integral in may be quite difficult to evaluate. However the method gives a theoretically exact result.

This can be thought of as an expansion of according to a Dirac delta function basis (projecting over 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 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

be a Sturm–Liouville be the vector-valued boundary conditions

be a continuous function in for all is

Theorem

that satisfies is a Green's function satisfying the following conditions:

  1. .

  2. For

  3. For

  4. Derivative "jump":

  5. Symmetry:

Advanced and retarded Green's functions

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; in this case, the solution is sometimes called a retarded Green's function. is called an advanced Green's function. 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 admits a set of eigenvectors (i.e., a set of functions and scalars such that ) 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 satisfies the following completeness relation

Then the following holds, represents complex conjugation.

Applying the operator 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.

Representations in terms of the Wronskian

. We write

together form a basis of linearly independent and its derivative "jump", can be written as

is known as the Wronskian. 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).

Combining Green's functions

and to be the representation of the right operator inverse of analogous to how for the invertible linear operator defined by is represented by its matrix elements

A further identity follows for differential operators that are scalar polynomials of the derivative, The fundamental theorem of algebra commutes with itself are the zeros of Taking the Fourier transform The fraction can then be split into a sum using a partial fraction decomposition is the operator in the polynomial).

Table of Green's functions

The following table gives an overview of Green's functions of frequently appearing differential operators, where is the Heaviside step function is a Bessel function is a modified Bessel function of the first kind is a modified Bessel function of the second kind. Where time () appears in the first column, the retarded (causal) Green's function is listed.

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

and substitute into Gauss' law.

Plugging this into the divergence theorem produces Green's theorem

Suppose that the linear differential operator is the Laplacian, ∇2, and that there is a Green's function

in Green's second identity, see Green's identities

Using this expression, it is possible to solve Laplace's equation or Poisson's equation , subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for everywhere inside a volume where either (1) the value of is specified on the bounding surface of the volume (Dirichlet boundary conditions), or (2) the normal derivative of is specified on the bounding surface (Neumann boundary conditions).

Suppose the problem is to solve for reduces to simply due to the defining property of the Dirac delta function

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, is interpreted as the electric potential, as electric charge density as the normal component of the electric field.

If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that vanishes when either or is on the bounding surface. Thus only one of the two terms in the surface integral 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 , where 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

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

Find the Green function for the following problem, whose Green's function number

First step: The Green's function for the linear operator at hand is defined as the solution to

.

is skipped for similar reasons.

Second step: and

One can ensure proper discontinuity in the first derivative by integrating the defining differential equation (i.e.,

Further examples

See also

Footnotes

References

Cited works

  • Chapter5 contains a very readable account of using Green's functions to solve boundary value problems in electrostatics.

  • Textbook on Green's function with worked-out steps.