Impulse response of an inhomogeneous linear differential operator
If one knows the solution to a differential equation subject to a point source ${\textstyle {\hat {L}}(x)G(x,y)=\delta (x-y)}$ and the differential operator ${\textstyle {\hat {L}}(x)}$ is linear, then one can superpose them to find the solution ${\textstyle u(x)=\int f(y)G(x,y)\,dy}$ for a general source ${\textstyle {\hat {L}}(x)u(x)=f(x)}$.
A Green's function, G(x,s), of a linear differential operator$\operatorname {L} =\operatorname {L} (x)$ acting on distributions over a subset of the Euclidean space$\mathbb {R} ^{n}$, at a point s, is any solution of
$\operatorname {L} \,G(x,s)=\delta (s-x)\,,$
(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
$\operatorname {L} \,u(x)=f(x)~.$
(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.
Loosely speaking, if such a function G can be found for the operator $\operatorname {L}$, then, if we multiply the equation (1) for the Green's function by f(s), and then integrate with respect to s, we obtain,
Because the operator $\operatorname {L} =\operatorname {L} (x)$ is linear and acts only on the variable x (and not on the variable of integration s), one may take the operator $\operatorname {L}$ outside of the integration, yielding
is a solution to the equation $\operatorname {L} u(x)=f(x)~.$
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 $\operatorname {L}$.
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 $\operatorname {L}$.
Not every operator $\operatorname {L}$ admits a Green's function. A Green's function can also be thought of as a right inverse of $\operatorname {L}$. 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 $\delta (x-s)$; 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.
Green's functions for solving inhomogeneous boundary value problems
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,
$LG(x,s)=\delta (x-s),$
shows that the units of $G$ depend not only on the units of $L$ but also on the number and units of the space of which the position vectors $x$ and $s$ are elements. This leads to the relationship:
$[[G]]=[[L]]^{-1}[[dx]]^{-1},$
where $[[G]]$ is defined as, "the physical units of $G$", and $dx$ is the volume element of the space (or spacetime).
For example, if $L=\partial _{t}^{2}$ and time is the only variable then:
$[[L]]=[[{\text{time}}]]^{-2},$
$[[dx]]=[[{\text{time}}]],\ {\text{and}}$
$[[G]]=[[{\text{time}}]].$
If $L=\square ={\frac {1}{c^{2}}}\partial _{t}^{2}-\nabla ^{2}$, the d'Alembert operator, and space has 3 dimensions then:
If a differential operatorL admits a set of eigenvectorsΨ_{n}(x) (i.e., a set of functions Ψ_{n} and scalars λ_{n} such that LΨ_{n} = λ_{n} Ψ_{n} ) 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 {Ψ_{n}} satisfies the following completeness relation,
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.
If the differential operator $L$ can be factored as $L=L_{1}L_{2}$ then the Green's function of $L$ can be constructed from the Green's functions for $L_{1}$ and $L_{2}$:
The above identity follows immediately from taking $G(x,s)$ to be the representation of the right operator inverse of $L$, analogous to how for the invertible linear operator$C$, defined by $C=(AB)^{-1}=B^{-1}A^{-1}$, is represented by its matrix elements $C_{i,j}$.
A further identity follows for differential operators that are scalar polynomials of the derivative, $L=P_{N}(\partial _{x})$. The fundamental theorem of algebra, combined with the fact that $\partial _{x}$commutes with itself, guarantees that the polynomial can be factored, putting $L$ in the form:
$L=\prod _{i=1}^{N}(\partial _{x}-z_{i}),$
where $z_{i}$ are the zeros of $P_{N}(z)$. Taking the Fourier transform of $LG(x,s)=\delta (x-s)$ with respect to both $x$ and $s$ gives:
The fraction can then be split into a sum using a partial fraction decomposition before Fourier transforming back to $x$ and $s$ space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if $L=(\partial _{x}+\gamma )(\partial _{x}+\alpha )^{2}$ 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 $\nabla ^{2}$ 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 ${\textstyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}$, ${\textstyle \rho ={\sqrt {x^{2}+y^{2}}}}$, ${\textstyle \Theta (t)}$ is the Heaviside step function, ${\textstyle J_{\nu }(z)}$ is a Bessel function, ${\textstyle I_{\nu }(z)}$ is a modified Bessel function of the first kind, and ${\textstyle K_{\nu }(z)}$ is a modified Bessel function of the second kind.^{[2]} Where time (t) appears in the first column, the advanced (causal) Green's function is listed.
Differential operator L
Green's function G
Example of application
$\partial _{t}^{n+1}$
${\frac {t^{n}}{n!}}\Theta (t)$
$\partial _{t}+\gamma$
$\Theta (t)e^{-\gamma t}$
$\left(\partial _{t}+\gamma \right)^{2}$
$\Theta (t)te^{-\gamma t}$
$\partial _{t}^{2}+2\gamma \partial _{t}+\omega _{0}^{2}$ where $\gamma <\omega _{0}$
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 ∇^{2}φ(x) = 0 or Poisson's equation ∇^{2}φ(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
$\int _{V}\varphi (x')\delta (x-x')\,d^{3}x'$
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.
In electrostatics, φ(x) is interpreted as the electric potential, ρ(x) as electric chargedensity, and the normal derivative $\nabla \varphi (x')\cdot d{\widehat {\sigma }}'$ 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 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
where $\langle \varphi \rangle _{S}$ 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
Second step: The next task is to determine $c_{2}$ and $c_{3}$.
Ensuring continuity in the Green's function at $x=s$ implies
$c_{2}\sin ks=c_{3}\cos ks$
One can ensure proper discontinuity in the first derivative by integrating the defining differential equation (i.e., Eq. *) from $x=s-\varepsilon$ to $x=s+\varepsilon$ and taking the limit as $\varepsilon$ goes to zero. Note that we only integrate the second derivative as the remaining term will be continuous by construction.
$c_{3}\cdot (-k\sin ks)-c_{2}\cdot (k\cos ks)=1$
The two (dis)continuity equations can be solved for $c_{2}$ and $c_{3}$ to obtain
Let n = 1 and let the subset be all of R. Let L be ${\textstyle {\frac {d}{dx}}}$. Then, the Heaviside step functionH(x − x_{0}) is a Green's function of L at x_{0}.
Let $a<x<b$, and all three are elements of the real numbers. Then, for any function from reals to reals, $f(x)$, with an $n$-th derivative that is integrable over the interval $[a,b]$:
The Green's function in the above equation, $G(x,s)={\frac {(x-s)^{n-1}}{(n-1)!}}\Theta (x-s)$, is not unique. How is the equation modified if $g(x-s)$ is added to $G(x,s)$, where $g(x)$ satisfies ${\textstyle {\frac {d^{n}g}{dx^{n}}}=0}$ for all $x\in [a,b]$ (for example, $g(x)=-x/2$ with $n=2$)? Also, compare the above equation to the form of a Taylor series centered at $x=a$.
^some examples taken from Schulz, Hermann: Physik mit Bleistift. Frankfurt am Main: Deutsch, 2001. ISBN 3-8171-1661-6 (German)
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. ISBN 0-486-63947-9. Chapter 5 contains a very readable account of using Green's functions to solve boundary value problems in electrostatics.
Polyanin, A.D.; Zaitsev, V.F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC Press. ISBN 1-58488-297-2.
Polyanin, A.D. (2002). Handbook of Linear Partial Differential Equations for Engineers and Scientists. Boca Raton, FL: Chapman & Hall/CRC Press. ISBN 1-58488-299-9.
Mathews, Jon; Walker, Robert L. (1970). Mathematical methods of physics (2nd ed.). New York: W. A. Benjamin. ISBN 0-8053-7002-1.
Folland, G.B.Fourier Analysis and its Applications. Mathematics Series. Wadsworth and Brooks/Cole.
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. ISBN 978-1-4398-1354-6.
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.
Faryad and, M.; Lakhtakia, A. (2018). Infinite-Space Dyadic Green Functions in Electromagnetism. London, UK / San Rafael, CA: IoP Science (UK) / Morgan and Claypool (US). Bibcode:2018idgf.book.....F.
Green functions and conformal mapping at PlanetMath.
Introduction to the Keldysh Nonequilibrium Green Function Technique by A. P. Jauho
Green's Function Library
Tutorial on Green's functions
Boundary Element Method (for some idea on how Green's functions may be used with the boundary element method for solving potential problems numerically)