Green’s function

In mathematics, a Green’s function is the impulse response of an inhomogeneous linear differential equation defined on a domain, with specified initial conditions or boundary conditions.

Through the superposition principle for linear operator problems, the convolution of a Green’s function with an arbitrary function $f (x)$ on that domain is the solution to the inhomogeneous differential equation for $f (x)$ . In other words, 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 solutionsinstead.

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

A Green’s function, $G (x,s)$ , of a linear differential operator $L = L (x)$ acting on distributions over a subset of the Euclidean space $\displaystyle \mathbb {R} ^{n}$ , at a point $s$ , is any solution of

$\displaystyle LG(x,s)=\delta (s-x),$

(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

$\displaystyle Lu(x)=f(x).$

(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 proper functions.

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.

As a side note, the Green’s function as used in physics is usually defined with the opposite sign, instead, that is,

\displaystyle LG(x,s)=\delta (x-s).

This definition does not significantly change any of the properties of the Green’s 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,

\displaystyle G(x,s)=G(x-s).

In this case, the Green’s function is the same as the impulse response of linear time-invariant system theory.

是何物？

理論工具、形式解！

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,

\displaystyle \int LG(x,s)f(s)\,ds=\int \delta (x-s)f(s)\,ds=f(x).

The right-hand side is now given by the equation (2) to be equal to

L u (x)

, thus

\displaystyle Lu(x)=\int LG(x,s)f(s)\,ds.

Because the operator $\displaystyle L=L(x)$ is linear and acts on the variable $x$ alone (not on the variable of integration $s$ ), one may take the operator $L$ outside of the integration on the right-hand side, yielding

\displaystyle Lu(x)=L\left(\int G(x,s)f(s)\,ds\right),

which suggests

$\displaystyle u(x)=\int G(x,s)f(s)\,ds.$

(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 constitutesFredholm theory.

遇着線性、非時變、因果系統光芒現︰

Time-varying impulse response

The time-varying impulse response h(t₂,t₁) of a linear system is defined as the response of the system at time t = t₂ to a single impulse applied at time t = t₁. In other words, if the input x(t) to a linear system is

\displaystyle x(t)=\delta (t-t_{1})

where δ(t) represents the Dirac delta function, and the corresponding response y(t) of the system is

\displaystyle y(t)|_{t=t_{2}}=h(t_{2},t_{1})

then the function h(t₂,t₁) is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied:

\displaystyle h(t_{2},t_{1})=0,t_{2}<t_{1}

The convolution integral

The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition:

\displaystyle y(t)=\int _{-\infty }^{t}h(t,t')x(t')dt'=\int _{-\infty }^{\infty }h(t,t')x(t')dt'

If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and h() is a function only of the time difference τ = t-t’ which is zero for τ<0 (namely t<t’). By redefinition of h() it is then possible to write the input-output relation equivalently in any of the ways,

\displaystyle y(t)=\int _{-\infty }^{t}h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(\tau )x(t-\tau )d\tau =\int _{0}^{\infty }h(\tau )x(t-\tau )d\tau

Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is:

\displaystyle H(s)=\int _{0}^{\infty }h(t)e^{-st}\,dt.

In applications this is usually a rational algebraic function of s. Because h(t) is zero for negative t, the integral may equally be written over the doubly infinite range and putting s = iω follows the formula for the frequency response function: