Green's function

Motivation

Loosely speaking, if such a function G can be found for the operator L, then, if we multiply equation 1 for the Green's function by f(s), and then integrate with respect to s, we obtain, $${\displaystyle \int LG(x,s)f(s)ds=\int \delta (x-s)f(s)ds=f(x).}$$ Because the operator ${\displaystyle L=L(x)}$ is linear and acts only on the variable x (and not on the variable of integration s), one may take the operator ${\displaystyle L}$ outside of the integration, yielding $${\displaystyle L\left(\int G(x,s)f(s)ds\right)=f(x).}$$ This means that

is a solution to the equation ${\displaystyle Lu(x)=f(x).}$

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 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 ${\displaystyle L}$ 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 ${\displaystyle \delta (x-s)}$; 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 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

Let ${\displaystyle L}$ be a Sturm–Liouville operator, a linear differential operator of the form $${\displaystyle L={\displaystyle \frac{d}{dx}}\left[p(x){\displaystyle \frac{d}{dx}}\right]+q(x)}$$ and let ${\displaystyle \mathbf{D}}$ be the vector-valued boundary conditions operator $${\displaystyle \mathbf{D}u=\left[\begin{array}{c}{\alpha }_1{u}^{\prime }(0)+{\beta }_{1}u(0)\\ {\alpha }_2{u}^{\prime }(\ell )+{\beta }_{2}u(\ell )\end{array}\right].}$$

Let ${\displaystyle f(x)}$ be a continuous function in ${\displaystyle [0,\ell ]}$. Further suppose that the problem $${\displaystyle \begin{array}{cc}Lu& =f\\ \mathbf{D}u& =\mathbf{0}\end{array}}$$ is regular, i.e., the only solution for ${\displaystyle f(x)=0}$ for all x is ${\displaystyle u(x)=0}$.

Theorem

There is one and only one solution ${\displaystyle u(x)}$ that satisfies $${\displaystyle \begin{array}{cc}Lu& =f\\ \mathbf{D}u& =\mathbf{0}\end{array}}$$ and it is given by $${\displaystyle u(x)={\int }_0^{\ell }f(s)G(x,s)ds,}$$ where ${\displaystyle G(x,s)}$ is a Green's function satisfying the following conditions:

1. ${\displaystyle G(x,s)}$ is continuous in ${\displaystyle x}$ and ${\displaystyle s}$.
2. For ${\displaystyle x\ne s}$, ${\displaystyle LG(x,s)=0}$.
3. For ${\displaystyle s\ne 0}$, ${\displaystyle \mathbf{D}G(x,s)=\mathbf{0}}$.
4. Derivative "jump": ${\displaystyle G^{\prime }(s_{0+},s)-G^{\prime }(s_{0-},s)=1/p(s)}$.
5. Symmetry: ${\displaystyle G(x,s)=G(s,x)}$.

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 problems involve finding a Green's function that is nonvanishing only for ${\displaystyle s\le x}$; in this case, the solution is sometimes called a retarded Green's function. Similarly, a Green's function that is nonvanishing only for ${\displaystyle s\ge x}$ 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. Both the advanced and retarded Green's functions are called one-sided, while a Green's function that is nonvanishing for all ${\displaystyle x}$ 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 L admits a set of eigenvectors Ψ_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, $${\displaystyle \delta (x-x^{\prime })=\sum _{n=0}^{\mathrm{\infty }}{\mathrm{\Psi }}_n^{†}(x^{\prime }){\mathrm{\Psi }}_n(x).}$$

Then the following holds,

${\displaystyle G(x,x^{\prime })=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{\Psi }}_n^{†}(x^{\prime }){\mathrm{\Psi }}_n(x)}{{\lambda }_n}},}$

where ${\displaystyle †}$ 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.

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

Let ${\displaystyle L}$ be the general linear second-order differential operator defined on ${\displaystyle [a,b]\in \mathbb{R}}$. We write

$${\displaystyle Lu(x)=\alpha (x)\frac{d^2}{d{x}^2}u(x)+\beta (x)\frac{d}{dx}u(x)+\gamma (x)u(x)=f(x).}$$

Suppose that ${\displaystyle u_1}$ and ${\displaystyle u_2}$ together form a basis of linearly independent solutions to the homogeneous problem ${\displaystyle Lu=0.}$ Given homogeneous boundary conditions for the Green's function ${\displaystyle G(a,s)=G(b,s)=0}$, we may construct ${\displaystyle G(x,s)}$ by requiring ${\displaystyle u_1(a)=0}$ and ${\displaystyle u_2(b)=0.}$ The Green's function satisfying these conditions, alongside the continuity of ${\displaystyle G}$ and its derivative "jump", can be written as

$${\displaystyle G(x,s)=\{\begin{array}{ll}{\displaystyle \frac{u_1(x)u_2(s)}{\alpha (s)\mathcal{W}\mathcal{(}\mathcal{s}\mathcal{)}}},& a\le x<s\\ {\displaystyle \frac{u_2(x)u_1(s)}{\alpha (s)\mathcal{W}(s)}},& s<x\le b\end{array}}$$

where ${\displaystyle \mathcal{W}(x)=u_1(x)u_2^{\prime }(x)-u_1^{\prime }(x)u_2(x)}$ is known as the Wronskian determinant of ${\displaystyle u_1}$ and ${\displaystyle u_2}$. 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

If the differential operator ${\displaystyle L}$ can be factored as ${\displaystyle L=L_1{L}_2}$ then the Green's function of ${\displaystyle L}$ can be constructed from the Green's functions for ${\displaystyle L_1}$ and ${\displaystyle L_2}$: $${\displaystyle G(x,s)=\int G_2(x,s_1)G_1(s_1,s)d{s}_1.}$$ The above identity follows immediately from taking ${\displaystyle G(x,s)}$ to be the representation of the right operator inverse of ${\displaystyle L}$, analogous to how for the invertible linear operator ${\displaystyle C}$, defined by ${\displaystyle C=(AB{)}^{-1}=B^{-1}{A}^{-1}}$, is represented by its matrix elements ${\displaystyle C_{i,j}}$.

A further identity follows for differential operators that are scalar polynomials of the derivative, ${\displaystyle L=P_N({\mathrm{\partial }}_x)}$. The fundamental theorem of algebra, combined with the fact that ${\displaystyle {\mathrm{\partial }}_x}$ commutes with itself, guarantees that the polynomial can be factored, putting ${\displaystyle L}$ in the form: $${\displaystyle L=\prod _{i=1}^N\left({\mathrm{\partial }}_x-z_i\right),}$$ where ${\displaystyle z_i}$ are the zeros of ${\displaystyle P_N(z)}$. Taking the Fourier transform of ${\displaystyle LG(x,s)=\delta (x-s)}$ with respect to both ${\displaystyle x}$ and ${\displaystyle s}$ gives: $${\displaystyle \hat{G}(k_x,k_s)=\frac{\delta (k_x-k_s)}{\prod _{i=1}^N(i{k}_x-z_i)}.}$$ The fraction can then be split into a sum using a partial fraction decomposition before Fourier transforming back to ${\displaystyle x}$ and ${\displaystyle s}$ space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if ${\displaystyle L=\left({\mathrm{\partial }}_x+\gamma \right){\left({\mathrm{\partial }}_x+\alpha \right)}^2}$ then one form for its Green's function is: $${\displaystyle \begin{array}{cc}G(x,s)& =\frac{1}{{\left(\gamma -\alpha \right)}^2}\mathrm{\Theta }(x-s)e^{-\gamma (x-s)}-\frac{1}{{\left(\gamma -\alpha \right)}^2}\mathrm{\Theta }(x-s)e^{-\alpha (x-s)}+\frac{1}{\gamma -\alpha }\mathrm{\Theta }(x-s)\left(x-s\right)e^{-\alpha (x-s)}\\ & =\int \mathrm{\Theta }(x-s_1)\left(x-s_1\right)e^{-\alpha (x-s_1)}\mathrm{\Theta }(s_1-s)e^{-\gamma (s_1-s)}d{s}_1.\end{array}}$$ While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial (for example, when ${\displaystyle {\mathrm{\nabla }}^2}$ 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 $r=\sqrt{x^2+y^2+z^2}$, $\rho =\sqrt{x^2+y^2}$, $\mathrm{\Theta }(t)$ is the Heaviside step function, $J_{\nu }(z)$ is a Bessel function, $I_{\nu }(z)$ is a modified Bessel function of the first kind, and $K_{\nu }(z)$ is a modified Bessel function of the second kind. Where time (t) 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), $${\displaystyle {\int }_V\mathrm{\nabla }\cdot \mathbf{A}dV={\int }_S\mathbf{A}\cdot d\hat{\mathit{\sigma }}.}$$

Let ${\displaystyle \mathbf{A}=\phi \mathrm{\nabla }\psi -\psi \mathrm{\nabla }\phi }$ and substitute into Gauss' law.

Compute ${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}}$ and apply the product rule for the ∇ operator, $${\displaystyle \begin{array}{cc}\mathrm{\nabla }\cdot \mathbf{A}& =\mathrm{\nabla }\cdot \left(\phi \mathrm{\nabla }\psi -\psi \mathrm{\nabla }\phi \right)\\ & =(\mathrm{\nabla }\phi )\cdot (\mathrm{\nabla }\psi )+\phi {\mathrm{\nabla }}^2\psi -(\mathrm{\nabla }\phi )\cdot (\mathrm{\nabla }\psi )-\psi {\mathrm{\nabla }}^2\phi \\ & =\phi {\mathrm{\nabla }}^2\psi -\psi {\mathrm{\nabla }}^2\phi .\end{array}}$$

Plugging this into the divergence theorem produces Green's theorem, $${\displaystyle {\int }_V\left(\phi {\mathrm{\nabla }}^2\psi -\psi {\mathrm{\nabla }}^2\phi \right)dV={\int }_S\left(\phi \mathrm{\nabla }\psi -\psi \mathrm{\nabla }\phi \right)\cdot d\hat{\mathit{\sigma }}.}$$

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, $${\displaystyle LG(\mathbf{x},{\mathbf{x}}^{\prime })={\mathrm{\nabla }}^{2}G(\mathbf{x},{\mathbf{x}}^{\prime })=\delta (\mathbf{x}-{\mathbf{x}}^{\prime }).}$$

Let ${\displaystyle \psi =G}$ in Green's second identity, see Green's identities. Then, $${\displaystyle {\int }_V\left[\phi ({\mathbf{x}}^{\prime })\delta (\mathbf{x}-{\mathbf{x}}^{\prime })-G(\mathbf{x},{\mathbf{x}}^{\prime }){{\mathrm{\nabla }}^{\prime }}^2\phi ({\mathbf{x}}^{\prime })\right]d^3{\mathbf{x}}^{\prime }={\int }_S\left[\phi ({\mathbf{x}}^{\prime }){\mathrm{\nabla }}^{\prime }G(\mathbf{x},{\mathbf{x}}^{\prime })-G(\mathbf{x},{\mathbf{x}}^{\prime }){\mathrm{\nabla }}^{\prime }\phi ({\mathbf{x}}^{\prime })\right]\cdot d{\hat{\mathit{\sigma }}}^{\prime }.}$$

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 $${\displaystyle {\int }_V\phi ({\mathbf{x}}^{\prime })\delta (\mathbf{x}-{\mathbf{x}}^{\prime })d^3{\mathbf{x}}^{\prime }}$$ reduces to simply φ(x) due to the defining property of the Dirac delta function and we have $${\displaystyle \phi (\mathbf{x})=-{\int }_{V}G(\mathbf{x},{\mathbf{x}}^{\prime })\rho ({\mathbf{x}}^{\prime })d^3{\mathbf{x}}^{\prime }+{\int }_S\left[\phi ({\mathbf{x}}^{\prime }){\mathrm{\nabla }}^{\prime }G(\mathbf{x},{\mathbf{x}}^{\prime })-G(\mathbf{x},{\mathbf{x}}^{\prime }){\mathrm{\nabla }}^{\prime }\phi ({\mathbf{x}}^{\prime })\right]\cdot d{\hat{\mathit{\sigma }}}^{\prime }.}$$

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 charge density, and the normal derivative ${\displaystyle \mathrm{\nabla }\phi ({\mathbf{x}}^{\prime })\cdot d{\hat{\mathit{\sigma }}}^{\prime }}$ 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, 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 $${\displaystyle {\int }_S{\mathrm{\nabla }}^{\prime }G(\mathbf{x},{\mathbf{x}}^{\prime })\cdot d{\hat{\mathit{\sigma }}}^{\prime }={\int }_V{\mathrm{\nabla }}^{\prime 2}G(\mathbf{x},{\mathbf{x}}^{\prime })d^3{\mathbf{x}}^{\prime }={\int }_V\delta (\mathbf{x}-{\mathbf{x}}^{\prime })d^3{\mathbf{x}}^{\prime }=1,}$$ 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 1/S, where S is the surface area of the surface. The surface term in the solution becomes $${\displaystyle {\int }_S\phi ({\mathbf{x}}^{\prime }){\mathrm{\nabla }}^{\prime }G(\mathbf{x},{\mathbf{x}}^{\prime })\cdot d{\hat{\mathit{\sigma }}}^{\prime }=⟨\phi {⟩}_S}$$ where ${\displaystyle ⟨\phi {⟩}_S}$ 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.

With no boundary conditions, the Green's function for the Laplacian (Green's function for the three-variable Laplace equation) is $${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=-\frac{1}{4\pi \left|\mathbf{x}-{\mathbf{x}}^{\prime }\right|}.}$$

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

${\displaystyle \phi (\mathbf{x})={\int }_V{\displaystyle \frac{\rho ({\mathbf{x}}^{\prime })}{4\pi \epsilon \left|\mathbf{x}-{\mathbf{x}}^{\prime }\right|}}{d}^3{\mathbf{x}}^{\prime }.}$

Example

Find the Green function for the following problem, whose Green's function number is X11: $${\displaystyle \begin{array}{cc}Lu& =u^{″}+k^{2}u=f(x)\\ u(0)& =0,u\left(\frac{\pi }{2k}\right)=0.\end{array}}$$

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

If ${\displaystyle x\ne s}$, then the delta function gives zero, and the general solution is $${\displaystyle G(x,s)=c_1\cos kx+c_2\sin kx.}$$

For ${\displaystyle x<s}$, the boundary condition at ${\displaystyle x=0}$ implies $${\displaystyle G(0,s)=c_1\cdot 1+c_2\cdot 0=0,c_1=0}$$ if ${\displaystyle x<s}$ and ${\displaystyle s\ne \frac{\pi }{2k}}$.

For ${\displaystyle x>s}$, the boundary condition at ${\displaystyle x=\frac{\pi }{2k}}$ implies $${\displaystyle G\left(\frac{\pi }{2k},s\right)=c_3\cdot 0+c_4\cdot 1=0,c_4=0}$$

The equation of ${\displaystyle G(0,s)=0}$ is skipped for similar reasons.

To summarize the results thus far: $${\displaystyle G(x,s)=\{\begin{array}{ll}{c}_2\sin kx,& \text{for }x<s,\\ c_3\cos kx,& \text{for }s<x.\end{array}}$$

Second step: The next task is to determine ${\displaystyle c_2}$ and ${\displaystyle c_3}$.

Ensuring continuity in the Green's function at ${\displaystyle x=s}$ implies $${\displaystyle c_2\sin ks=c_3\cos ks}$$

One can ensure proper discontinuity in the first derivative by integrating the defining differential equation (i.e., Eq. *) from ${\displaystyle x=s-\epsilon }$ to ${\displaystyle x=s+\epsilon }$ and taking the limit as ${\displaystyle \epsilon }$ goes to zero. Note that we only integrate the second derivative as the remaining term will be continuous by construction. $${\displaystyle c_3\cdot (-k\sin ks)-c_2\cdot (k\cos ks)=1}$$

The two (dis)continuity equations can be solved for ${\displaystyle c_2}$ and ${\displaystyle c_3}$ to obtain $${\displaystyle c_2=-\frac{\cos ks}{k};c_3=-\frac{\sin ks}{k}}$$

So Green's function for this problem is: $${\displaystyle G(x,s)=\{\begin{array}{ll}-\frac{\cos ks}{k}\sin kx,& x<s,\\ -\frac{\sin ks}{k}\cos kx,& s<x.\end{array}}$$

Further examples

• Let n = 1 and let the subset be all of R. Let L be $\frac{d}{dx}$. Then, the Heaviside step function Θ(x − x₀) is a Green's function of L at x₀.
• Let n = 2 and let the subset be the quarter-plane {(x, y) : x, y ≥ 0} and L be the Laplacian. Also, assume a Dirichlet boundary condition is imposed at x = 0 and a Neumann boundary condition is imposed at y = 0. Then the X10Y20 Green's function is $${\displaystyle \begin{array}{cc}G(x,y,x_0,y_0)={\displaystyle \frac{1}{2\pi }}& [\ln \sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2}-\ln \sqrt{{\left(x+x_0\right)}^2+{\left(y-y_0\right)}^2}\\ & +\ln \sqrt{{\left(x-x_0\right)}^2+{\left(y+y_0\right)}^2}-\ln \sqrt{{\left(x+x_0\right)}^2+{\left(y+y_0\right)}^2}].\end{array}}$$
• Let ${\displaystyle a<x<b}$, and all three are elements of the real numbers. Then, for any function ${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ with an ${\displaystyle n}$-th derivative that is integrable over the interval ${\displaystyle [a,b]}$: $${\displaystyle f(x)=\sum _{m=0}^{n-1}\frac{(x-a{)}^m}{m!}{\left[\frac{d^{m}f}{d{x}^m}\right]}_{x=a}+{\int }_a^b\left[\frac{(x-s{)}^{n-1}}{(n-1)!}\mathrm{\Theta }(x-s)\right]{\left[\frac{d^{n}f}{d{x}^n}\right]}_{x=s}ds.}$$ The Green's function in the above equation, ${\displaystyle G(x,s)=\frac{(x-s{)}^{n-1}}{(n-1)!}\mathrm{\Theta }(x-s)}$, is not unique. How is the equation modified if ${\displaystyle g(x-s)}$ is added to ${\displaystyle G(x,s)}$, where ${\displaystyle g(x)}$ satisfies $\frac{d^{n}g}{d{x}^n}=0$ for all ${\displaystyle x\in [a,b]}$ (for example, ${\displaystyle g(x)=-x/2}$ with ${\displaystyle n=2}$)? Also, compare the above equation to the form of a Taylor series centered at ${\displaystyle x=a}$.