If one knows the solution to a differential equation subject to a point source and the differential operator is linear, then one can superpose them to build the solution for a general source .
This means that if is a linear differential operator, then
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.
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 Green's function due to the evenness of the Dirac delta function.
Loosely speaking, if such a function G can be found for the operator L, then, if we multiply the equation1 for the Green's function by f(s), and then integrate with respect to s, we obtain,
Because the operator is linear and acts only on the variable x (and not on the variable of integration s), one may take the operator outside of the integration, yielding
This means that
(3)
is a solution to the equation
Thus, one may obtain the function u(x) through knowledge of the Green's function in equation1 and the source term on the right-hand side in equation2. This process relies upon the linearity of the operator L.
In other words, the solution of equation2, u(x), can be determined by the integration given in equation3. 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 equation1. For this reason, the Green's function is also sometimes called the fundamental solution associated to the operator L.
Not every operator 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 equation3 may be quite difficult to evaluate. However the method gives a theoretically exact result.
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. In some cases, it is possible to find one Green's function that is nonvanishing only for , which is called a retarded Green's function, and another Green's function that is nonvanishing only for , which 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. 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. The use of advanced and retarded Green's function is especially common for the analysis of solutions of the inhomogeneous electromagnetic wave equation.
Units
While it does not 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,
shows that the units of depend not only on the units of but also on the number and units of the space of which the position vectors and are elements. This leads to the relationship:
where is defined as, "the physical units of ", and is the volume element of the space (or spacetime).
For example, if and time is the only variable then:
If , the d'Alembert operator, and space has 3 dimensions then:
Eigenvalue expansions
If a differential operatorL admits a set of eigenvectorsΨn(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,
Then the following holds,
where represents complex conjugation.
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.
If the differential operator can be factored as then the Green's function of can be constructed from the Green's functions for and :
The above identity follows immediately from taking 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, combined with the fact that commutes with itself, guarantees that the polynomial can be factored, putting in the form:
where are the zeros of . Taking the Fourier transform of with respect to both and gives:
The fraction can then be split into a sum using a partial fraction decomposition before Fourier transforming back to and space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if then one form for its Green's function is:
While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial (for example, when is the operator in the polynomial).
Compute and apply the product rule for the ∇ operator,
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, 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.[3]
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
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.
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
First step: The Green's function for the linear operator at hand is defined as the solution to
(Eq. *)
If , then the delta function gives zero, and the general solution is
For , the boundary condition at implies
if and .
For , the boundary condition at implies
The equation of is skipped for similar reasons.
To summarize the results thus far:
Second step: The next task is to determine and .
Ensuring continuity in the Green's function at implies
One can ensure proper discontinuity in the first derivative by integrating the defining differential equation (i.e., Eq. *) from to and taking the limit as goes to zero. Note that we only integrate the second derivative as the remaining term will be continuous by construction.
The two (dis)continuity equations can be solved for and to obtain
;\quad c_{3}=-{\frac {\sin ks}{k}}}
So Green's function for this problem is:
Let n = 1 and let the subset be all of R. Let L be . Then, the Heaviside step functionΘ(x − x0) is a Green's function of L at x0.
Let , and all three are elements of the real numbers. Then, for any function 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 if is added to , where satisfies for all (for example, with )? Also, compare the above equation to the form of a Taylor series centered at .
Cole, K.D.; Beck, J.V.; Haji-Sheikh, A.; Litkouhi, B. (2011). "Methods for obtaining Green's functions". Heat Conduction Using Green's Functions. Taylor and Francis. pp.101–148. ISBN978-1-4398-1354-6.
some examples taken from Schulz, Hermann (2001). Physik mit Bleistift: das analytische Handwerkszeug des Naturwissenschaftlers (4. Aufled.). Frankfurt am Main: Deutsch. ISBN978-3-8171-1661-4.
Jackson, John David (1998-08-14). Classical Electrodynamics. John Wiley & Sons. p.39.
Bayin, S.S. (2006). Mathematical Methods in Science and Engineering. Wiley. Chapters 18 and 19.
Eyges, Leonard (1972). The Classical Electromagnetic Field. New York, NY: Dover Publications. ISBN0-486-63947-9. Chapter5 contains a very readable account of using Green's functions to solve boundary value problems in electrostatics.
Barton, Gabriel (1989). Elements of Green's functions and propagation: potentials, diffusion, and waves. Oxford science publications. Oxford: New York: Clarendon Press; Oxford University Press. ISBN978-0-19-851988-1. Textbook on Green's function with worked-out steps.
Polyanin, A.D. (2002). Handbook of Linear Partial Differential Equations for Engineers and Scientists. Boca Raton, FL: Chapman & Hall/CRC Press. ISBN1-58488-299-9.
Mathews, Jon; Walker, Robert L. (1970). Mathematical methods of physics (2nded.). New York: W. A. Benjamin. ISBN0-8053-7002-1.
Folland, G.B.Fourier Analysis and its Applications. Mathematics Series. Wadsworth and Brooks/Cole.
Green, G (1828). An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham, England: T. Wheelhouse. pages 10-12.