Green's Function

A Green's function is an integral kernel that can be used to solve differential equations from a large number of families including simpler examples such as ordinary differential equations with initial conditions or boundary conditions, as well as more difficult examples such as inhomogeneous partial differential equations (PDE) with boundary conditions. Important for a number of reasons, Green's functions allow for visual interpretations of the actions associated to a source of force or to a charge concentrated at a point (Qin 2014), thus making them particularly useful in areas of applied mathematics. In particular, Green's function methods are widely used in, e.g., physics, and engineering.

More precisely, given a linear differential operator acting on the collection of distributions over a subset of some Euclidean space , a Green's function at the point corresponding to is any solution of

where denotes the delta function. The motivation for defining such a function is widespread, but multiplying the above identity by a function and integrating with respect to yields

The right-hand side reduces merely to due to properties of the delta function, and because is a linear operator acting only on and not on , the left-hand side can be rewritten as

This reduction is particularly useful when solving for in differential equations of the form

where the above arithmetic confirms that

The integral gives a particular solution , subject to convergence and differentiation under the integral sign. The general solution includes a homogeneous solution with :

When compatible homogeneous boundary conditions give a unique solution and satisfies them, .

The figure above illustrates both the intuitive physical interpretation of a Green's function as well as a relatively simple associated differential equation with which to compare the above definition (Hartmann 2013). In particular, it shows a taut rope of length suspended between two walls, held into place by an identical horizontal force applied on each of its ends, and a unit lateral point load placed at an interior point on the rope. Let denote the deflection at position . A point load gives the piecewise linear function displayed below. More generally, for a distributed load density , the differential equation is

for with . For constant load density , its solution is

The Green's function for a unit point load at is

As demonstrated in the above figure, the displaced rope has the piecewise linear format given by above, thus confirming the claim that the Green's function associated to this system represents the action of the horizontal rope corresponding to the application of a unit point load.

A Green's function taking a pair of arguments is sometimes referred to as a two-point Green's function. This is in contrast to multi-point Green's functions which are of particular importance in the area of many-body theory.

As an elementary example of a two-point function as defined above, consider the problem of determining the potential generated by a charge distribution whose charge density is , whereby applications of Poisson's equation and Coulomb's law to the potential at produced by each element of charge yields a solution

which holds, under certain conditions, over the region where . Because the right-hand side can be viewed as an integral operator converting into , one can rewrite this solution in terms of a Green's function having the form

whereby the solution can be rewritten:

(Arfken et al. 2012).

The above figure shows the Green's function associated to the solution of the - equation discussed above where here, and , respectively , is plotted on the -, respectively -, axis.

A somewhat comprehensive list of Green's functions corresponding to various differential equations is maintained online by Kevin Cole (Cole 2000).

Due to the multitude of literature written on Green's functions, several different notations and definitions may emerge, some of which are topically different than the above but which in general do not affect the important properties of the results. As the above example illustrates, for instance, some authors prefer to denote the variables and in terms of vectors and to emphasize the fact that they're elements of for some which may be larger than 1 (Arfken 1985). It is also relatively common to see the definition with a negative sign so that is defined to be the function for which

and the corresponding particular solution is then . The sign convention must be used consistently. Several other notations are also known to exist for a Green's function, some of which include the use of a lower-case in place of (Stakgold 1979) as well as the inclusion of a vertical line instead of a comma, e.g., (Duffy 2001).

In other instances, literature presents definitions which are intimately connected to the contexts in which they're presented. For example, some authors define Green's functions to be functions which satisfy a certain set of conditions, e.g., existence on a special kind of domain, association with a very particular differential operator , or satisfaction of a precise set of boundary conditions. One of the most common such examples can be found in notes by, e.g., Speck, where a Green's function is defined to satisfy for points and for all points lying in the boundary of (Speck 2011). This particular definition presents an integral kernel corresponding to the solution of a generalized Poisson's equation and would therefore face obvious limitations when being adapted to a more general setting. For the preceding Poisson problem, the fundamental and regular decomposition can be written, after adapting the sign convention of Hartmann (2013), as

where, with the preceding sign convention, and for the Laplacian . In such situations, is known as the fundamental solution of the underlying differential equation and is known as its regular solution; as such, and are sometimes called the fundamental and regular parts of , respectively.

Several fundamental properties of a general Green's function follow immediately (or almost so) from its definition and carry over to all particular instances. For example, if the kernel of the operator is nontrivial, then there may be several Green's functions associated to a single operator; as a result, one must exhibit caution when referring to "the" Green's function. For inverse kernels with respect to Lebesgue measure and compatible adjoint boundary conditions, Green's functions satisfy an adjoint operator relation in their two arguments so that

where here, is defined to be the solution of the equation

Here, is the adjoint operator of , and the bar denotes complex conjugation. One immediate corollary of this fact is that for real self-adjoint boundary value problems with a unique inverse and matching boundary conditions, the corresponding is symmetric:

This identity is often called the reciprocity principle and says, in physical terms, that the response at caused by a unit source at is the same as the response at due to a unit force at (Stakgold 1979).

The essential property of any Green's function is that it provides a way to describe the response of an arbitrary differential equation solution to some kind of source term in the presence of some number of boundary conditions (Arfken et al. 2012). Some authors consider a Green's function to serve roughly an analogous role in the theory of partial differential equations as do Fourier series in the solution of ordinary differential equations (Mikula and Kos 2006).

For more abstract scenarios, a number of concepts exist which serve as context-specific analogues to the notion of a Green's function. For instance, in functional analysis, it is often useful to consider a so-called generalized Green's function which has many analogous properties when integrated abstractly against functionals rather than functions. Indeed, such generalizations have yielded an entirely analogous branch of theoretical PDE analysis and are themselves the focus of a large amount of research.