Ngô Quốc Anh

March 13, 2009

Green’s function and differential equations

Filed under: Linh Tinh, Nghiên Cứu Khoa Học, PDEs — Tags: — Ngô Quốc Anh @ 12:31

In mathematics, a Green’s function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is also used in physics, specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition; for this sense, see Correlation function (quantum field theory) and Green’s function (many-body theory).

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 largely studied from the point of view of fundamental solutions instead.

Definition and uses

Technically, 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 \mathbb R^n, at a point s, is any solution of

LG(x,s)=\delta(x-s)                         (1)

where \delta is the Dirac delta function.

This technique can be used to solve differential equations of the form

Lu(x)=f(x)                                      (2)

If the kernel of L is nontrivial, 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. Also, Green’s functions in general are distributions, not necessarily proper functions.

Green’s functions are also a useful tool in condensed matter theory, where they allow the resolution of the diffusion equation, and in quantum mechanics, where the Green’s function of the Hamiltonian is a key concept, with important links to the concept of density of states. The Green’s functions used in those two domains are highly similar, due to the analogy in the mathematical structure of the diffusion equation and Schrödinger equation. As a side note, the Green’s function as used in Physics is usually defined with the opposite sign; that is, LG(x,s)=-\delta(x-s). This definition does not significantly change any of the properties of the Green’s function.

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 perform an integration in the s variable, we obtain

\displaystyle\int L G(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 Lu(x), thus

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

Because the operator L=L(x) is linear and acts on the variable x alone (not on the variable of integration s), we can take the operator L outside of the integration on the right hand side obtaining

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

And this implies

\displaystyle u(x) = \int G(x,s) f(s) ds .                 (3)

Thus, we can 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 has resulted from 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 functions for a particular operator, the integral in equation (3), may be quite difficult to perform. 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 is known as a Fredholm integral equation, the study of which constitutes Fredholm theory.

Green’s functions for solving inhomogeneous boundary value problems

The primary use of Green’s functions in mathematics is to solve inhomogeneous boundary value problems. In modern theoretical physics, Green’s functions are also usually used as propagators in Feynman diagrams (and the phrase “Green’s function” is often used for any correlation function).

Framework

Let L be the Sturm-Liouville operator, a linear differential operator of the form

\displaystyle L = {d \over dx}\left( p(x) {d \over dx} \right) + q(x)

and let D be the boundary conditions operator

\displaystyle Du = \left\{\begin{matrix} \alpha _1 u'(0) + \beta _1 u(0) \\ \alpha _2 u'(\ell) + \beta _2 u(\ell). \end{matrix}\right.

Let f(x) be a continuous function in [0,\ell]. We shall also suppose that the problem

\displaystyle\begin{matrix}Lu = f \\ Du = 0 \end{matrix}

is regular, i.e. only the trivial solution exists for the homogeneous problem.

Theorem. There is one and only one solution u(x) which satisfies

\displaystyle\begin{matrix}Lu = f \\ Du = 0 \end{matrix}

and it is given by

\displaystyle u(x) = \int_0^\ell f(s) G(x,s) ds

where G(x,s) is a Green’s function satisfying the following conditions:

  1. G(x,s) is continuous in x and s.
  2. For x\ne s, LG(x,s)=0.
  3. For s \ne 0,\ell, DG(x,s)=0.
  4. Derivative “jump”: G'(s_{+0},s)-G'(s_{-0},s)=\frac{1}{p(s)}.
  5. Symmetry: G(x,s)=G(s,x).

Finding Green’s functions (eigenvalue expansions)

If a differential operator L admits a set of eigenvectors \Psi_n(x) (i.e. a set of functions \Psi_n(x) and scalars \lambda_n such that L\Psi_n =\lambda_n\Psi_n) that are complete, then it is possible to construct a Green’s function from these eigenvectors and eigenvalues.

Complete means that the set of functions \Psi_n(x) satisfies the following completeness relation

\displaystyle\delta(x - x') = \sum_{n=0}^\infty \Psi_n(x) \Psi_n(x').

Then the following holds

\displaystyle G(x, x') = \sum_{n=0}^\infty \frac{\Psi_n(x) \Psi_n(x')}{\lambda_n}.

Applying the operator L to each side of this equation results in the completeness relation, which was assumed true.

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.

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 law)

\displaystyle\int_V \nabla \cdot \hat A\ dV = \int_S \hat A \cdot d\hat\sigma.

Let \hat A = \phi\nabla\psi - \psi\nabla\phi a and substitute into Gauss’ law. Compute $latex \nabla\cdot\hat A$ and apply the chain rule for the \nabla operator

\displaystyle\begin{gathered} \nabla \cdot\hat A = \nabla \cdot(\phi \nabla \psi - \psi \nabla \phi ) \hfill \\ \qquad= (\nabla \phi )\cdot(\nabla \psi ) + \phi {\nabla ^2}\psi - (\nabla \phi )\cdot(\nabla \psi ) - \psi {\nabla ^2}\phi \hfill \\ \qquad= \phi {\nabla ^2}\psi - \psi {\nabla ^2}\phi . \hfill \\ \end{gathered}

Plugging this into the divergence theorem produces Green’s theorem

\displaystyle\int_V (\phi\nabla^2\psi - \psi\nabla^2\phi) dV = \int_S (\phi\nabla\psi - \psi\nabla\phi)\cdot d\hat\sigma.

Suppose that the linear differential operator L is the Laplacian, \nabla^2, and that there is a Green’s function G for the Laplacian. The defining property of the Green’s function still holds

\displaystyle L G(x,x') = \nabla^2 G(x,x') = \delta(x-x').

Let \Psi=G in Green’s theorem. Then

\displaystyle\begin{gathered} \int_V \phi (x')\delta (x - x') - G(x,x'){\nabla ^2}\phi (x')\:{d^3}x' \hfill \\ \qquad\qquad= \int_S {\left[ {\phi (x')\nabla 'G(x,x') - G(x,x')\nabla '\phi (x')} \right]} d\hat \sigma '. \hfill \\ \end{gathered}

Using this expression, it is possible to solve Laplace’s equation \nabla ^2\phi(x)=0 or Poisson’s equation \nabla^2 \phi(x)=-\rho(x), subject to either Neumann or Dirichlet boundary conditions. In other words, we can solve for \phi(x) everywhere inside a volume where either (1) the value of \phi(x) is specified on the bounding surface of the volume (Dirichlet boundary conditions), or (2) the normal derivative of \phi(x) is specified on the bounding surface (Neumann boundary conditions).

Suppose the problem is to solve for \phi(x) inside the region. Then the integral

\displaystyle\int\limits_V {\phi(x')\delta(x-x')\ d^3x'}

reduces to simply \phi(x) due to the defining property of the Dirac delta function and we have

\displaystyle\phi(x) = \int_V G(x,x') \rho(x')\ d^3x' + \int_S \left[\phi(x')\nabla' G(x,x') - G(x,x')\nabla'\phi(x')\right] \cdot d\hat\sigma'.

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, \phi(x) is interpreted as the electric potential, \rho(x) as electric charge density, and the normal derivative \nabla\phi(x')\cdot d\hat\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; conversely, 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. Thus only one of the two terms in the surface integral remains.

With no boundary conditions, the Green’s function for the Laplacian (Green’s function for the three-variable Laplace equation) is

\displaystyle G(\hat x, \hat x') = \frac{1}{|\hat x - \hat x'|}.

Supposing that the bounding surface goes out to infinity, and plugging in this expression for the Green’s function, this gives the familiar expression for electric potential in terms of electric charge density (in the CGS unit system) as

\displaystyle\phi(\hat x) = \int_V \frac{\rho(x')}{|\hat x - \hat x'|} \ d^3x'.

Example. Given the problem

\displaystyle\begin{array}{*{20}{c}} {Lu = u'' + u = f(x)} \\ {u(0) = 0,} \\ {u\left( {\frac{\pi }{2}} \right) = 0.} \\ \end{array}

Find the Green’s function.

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

g''+g=\delta(x-s).

If x\ne s, then the delta function gives zero, and the general solution is

g(x,s)=A\cos x+B \sin x.

For x<s, the boundary condition at x=0 implies g(0,s)=c_1 \cdot 1+c_2 \cdot 0=0 which implies c_1=0. The equation of g(\frac{\pi}{2},s)=0 is skipped because x \ne \frac{\pi}{2} if x<s and s \ne \frac{\pi}{2}. For x>s, the boundary condition at x = \frac{\pi}{2} implies g(\frac{\pi}{2},s)=c_3 \cdot 0 + c_4 \cdot 1=0 which implies c_4=0. The equation of g(0,s)=0 is skipped for similar reasons. To summarize the results thus far

\displaystyle g(x,s)=\left\{\begin{matrix} c_2 \sin x, \;\; x < s \\ c_3 \cos x, \;\; s < x \end{matrix}\right.

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 s = c_3 \cos s.

One can also ensure proper discontinuity in the first derivative by integrating the defining differential equation from x=s-\varepsilon to x=s+\varepsilon and taking the limit as \varepsilon goes to zero

c_3 (-\sin s) - c_2 \cos s =1.

The two (dis)continuity equations can be solved for c_2 and c_3 to obtain

c_2 = -\cos s, \quad c_3 = - \sin s.

So the Green’s function for this problem is

\displaystyle g(x,s)=\left\{\begin{matrix} -\cos s \cdot \sin x, \;\; x < s, \\ - \sin s \cdot \cos x, \;\; s < x. \end{matrix}\right.

Further examples.

  • Let n=1 and let the subset be all of \mathbb R. Let L be \frac{d}{dx}. Then, the Heaviside step function H(x-x_0) is a Green’s function of L at x_0.
  • Let n=2 and let the subset be the quarter-plane \{ (x,y): x,y \geq 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 Green’s function is
  • \displaystyle\begin{gathered} G(x,y;{x_0},{y_0}) = \frac{1}{{2\pi }}\left[ {\ln \sqrt {{{(x - {x_0})}^2} + {{(y - {y_0})}^2}} - \ln \sqrt {{{(x + {x_0})}^2} + {{(y - {y_0})}^2}} } \right] \hfill \\ \qquad\qquad+ \frac{1}{{2\pi }}\left[ {\ln \sqrt {{{(x - {x_0})}^2} + {{(y + {y_0})}^2}} - \ln \sqrt {{{(x + {x_0})}^2} + {{(y + {y_0})}^2}} } \right] \hfill \\ \end{gathered}

    which completes this example.

Source

4 Comments »

  1. Chào NQA,

    Khi mô nị rảnh thì vô đây trà nước với ngộ về cái Green’s function cho vui nhé !

    http://www.violet.vn/brahman/entry/show/entry_id/579890

    Comment by viettran — March 17, 2009 @ 14:56

    • OK nhưng mà hổng bít làm thế nào để comment vào trang web kia, dạo này lại hứng thú với trò hàm Green🙂

      Comment by Ngô Quốc Anh — March 18, 2010 @ 23:07

  2. […] most recent post describes Green’s functions, givining examples. I find the post interesting and […]

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  3. My fellow on Facebook shared this link and I’m not dissapointed that I came to your blog.

    Comment by Samuel L. — April 22, 2009 @ 13:52


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