NationMaster - Encyclopedia: Green's function

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. Technically, a Green's function of a linear operator L acting on distributions over a manifold M, at a point x₀, is any solution of Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

(Lf)(x) = δ(x − x₀),

where δ is the Dirac delta function. 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 would give us a unique Green's function. Also, Green's functions in general are distributions, not necessarily proper functions. The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... Partial plot of a function f. ...

Not every operator L admits a Green's function. A Green's function can also be thought of as a one-sided inverse of L.

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. The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... Fig. ... The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ...

The Green's function was named after British mathematician George Green, who first developed the concept in the 1830s. To meet Wikipedias quality standards, this article or section may require cleanup. ... The title page to George Greens original essay on what is now known as Greens theorem. ...

1 Motivation
2 Green's function for solving inhomogeneous boundary value problems
- 2.1 Working frame
- 2.2 Theorem
3 Finding Green's functions
- 3.1 Eigenvalue expansions
4 Green's function for the Laplacian
5 Example
6 Green's functions in condensed matter physics
7 Further examples
8 See also
9 Bibliography
10 External links

Motivation

Convolving with a Green's function gives solutions to inhomogeneous differential-integral equations, most commonly a Sturm-Liouville problem. If g is the Green's function of an operator L, then the solution for f of the equation Lf = h is given by In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles FranÃ§ois Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...

$f(x) = int{ h(s) g(x,s) , ds}.$

This can be thought of as an expansion of h according to a Dirac delta function basis (projecting h over δ(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. The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ... The word projection can mean more than one thing. ... In mathematics, the Fredholm integral equation introduced by Ivar Fredholm gives rises to a Fredholm operator. ... In mathematics, Fredholm theory is a theory of integral equations. ...

Green's function for solving inhomogeneous boundary value problems

The primary use of Green's functions in mathematics is to solve inhomogeneous boundary value problems. In particle 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). In mathematics, a boundary value problem consists of a differential equation to be satisfied at all points in the interior of an interval or a region and a set of boundary conditions specifying the values of the solution or some of its derivatives everywhere on the boundary of the interval... Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ... In quantum field theory, correlation functions generalize the concept of correlation functions in statistics. ...

Working frame

Let $L$ be the Sturm-Liouville operator, a linear differential operator of the form In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles FranÃ§ois Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...

$L = {d over dx}left[ p(x) {d over dx} right] + q(x)$

and let D be the boundary conditions operator In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

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

Let $f (x)$ be a continuous function in $[0, l]-$ . We shall also suppose that the problem In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

$begin{matrix}Lu = f Du = 0 end{matrix}$

is regular, i.e. only the trivial solution exists for the homogeneous problem. In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ... In mathematics, homogeneous may refer to: a homogeneous polynomial, in algebra a homogeneous function a homogeneous differential equation a homogeneous system of linear equations, in linear algebra homogeneous coordinates a homogeneous number a homogeneous space for a Lie group G, or more general transformation group a homogeneous ideal in a...

Theorem

Then there is one and only one solution u(x) which satisfies

$begin{matrix}Lu = f Du = 0 end{matrix}$

and it is given by

$u(x) = int_{0}^{l}{ f(s) g(x,s) , ds}$

where g(x,s) is Green's function and satisfies the following demands:

g(x,s) is continuous in x and s.
For $x ne s$ , $L g (x, s) = 0$ .
For $s ne 0, l$ , $D g (x, s) = 0$ .
Derivative "jump": $g'(s + 0, s) - g'(s - 0, s) = 1 / p (s)$ .
Symmetry: g(x, s) = g(s, x).

In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

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 $Ψ n (x)$ and scalars $λ n$ such that $L Ψ n = λ n Ψ n)$ ) that are complete, then we can construct a Green's function from these eigenvectors and eigenvalues. In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...

By complete, we mean that the set of functions : $Ψ n (x)$ satisfies the following completeness relation:

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

We can prove the following:

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

Now consider acting on this on each side with the operator L. We'll end up with 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. In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ... In mathematics, Fredholm theory is a theory of integral equations. ...

Green's function 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. In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... Greens identities are a set of three identities in vector calculus. ...

To derive Green's theorem, begin with the divergence theorem (otherwise known as Gauss's law): In physics and mathematical analysis, Gausss law gives the relation between the electric flux flowing out a closed surface and the electric charge enclosed in the surface. ...

$int_V nabla cdot hat A dV = int_S hat A cdot dhatsigma$

Let $A = phinablapsi - psinablaphi$ and substitute into Gauss' law. Compute $nablacdothat A$ and apply the chain rule for the $nabla$ operator:

$nablacdothat A = nablacdot(phinablapsi - psinablaphi) = (nablaphi)cdot(nablapsi) + phinabla^2psi - (nablaphi)cdot(nablapsi) - psinabla^2phi = phinabla^2psi - psinabla^2phi$

Plugging this into the divergence theorem, we arrive at Green's theorem: In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special case of the more...

$int_V phinabla^2psi - psinabla^2phi dV = int_S phinablapsi - psinablaphi cdot dhatsigma$

Suppose that our linear differential operator L is the laplacian, $nabla^2$ , and that we have a Green's function G for the laplacian. The defining property of the Green's function still holds: In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

$L G(x,x') = nabla^2 G(x,x') = delta(x-x')$

Let $ψ = G$ in Green's theorem. We get: In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special case of the more...

$int_V phi(x') delta(x - x') - G(x,x') nabla^2phi(x') d^3x' = int_S phi(x')nabla' G(x,x') - G(x,x')nabla'phi(x') cdot dhatsigma'$

Using this expression, we can solve Laplace's equation $nabla^2phi(x)=0$ or Poisson's equation $nabla^2phi(x)=-4pirho(x)$ , 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. In mathematics, Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ... In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ... In mathematics, a Neumann boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values the derivative of a solution is to take on the boundary of the domain. ... In mathematics, a Dirichlet boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values a solution is to take on the boundary of the domain. ...

Suppose we're interested in solving for $φ(x)$ inside the region. Then the integral

$intlimits_V {phi(x')delta(x-x') d^3x'}$

reduces to simply $φ(x)$ due to the defining property of the Dirac delta function and we have: The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

$phi(x) = int_V G(x,x') rho(x') d^3x' + int_S phi(x')nabla' G(x,x') - G(x,x')nabla'phi(x') cdot dhatsigma'$

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 mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ...

In electrostatics, we interpret $φ(x)$ as the electric potential, $ρ(x)$ as electric charge density, and the normal derivative $nablaphi(x')cdot dhatsigma'$ as the normal component of the electric field. Electrostatics is the branch of physics that deals with the forces exerted by a static (i. ... Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... Density (symbol: Ï - Greek: rho) is a measure of mass per volume. ...

If we're interested in solving a Dirichlet boundary value problem, we choose our Green's function such that $G (x, x')$ vanishes when either x or x' is on the bounding surface; conversely, if we're interested in solving a Neumann boundary value problem, we choose our Green's function such that its normal derivative vanishes on the bounding surface. Thus we are left with only one of the two terms in the surface integral.

With no boundary conditions, the Green's function for the Laplacian (Green's function for the three variable Laplace equation) is: + The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

$G(hat x, hat x') = frac{1}{|hat x - hat x'|}$

Supposing that our bounding surface goes out to infinity, and plugging in this expression for the Green's function, we arrive at the familiar expression for electric potential in terms of electric charge density:

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

Example

Given the problem

$begin{matrix}Luend{matrix} = u ' ' + u = f( x )$

$Du = u(0) = 0 quad, quad uleft(frac{pi}{2}right) = 0$

Find Green's function.

First step: From demand-2 we see that

$g(x,s) = c_1 (s) cdot cos x + c_2 (s) cdot sin x$

For x < s we see from demand-3 that the $c 1 (s) = 0$ , while for x > s we see from demand-3 that the $c 2 (s) = 0$ (we leave it to the reader to fill in the in-between steps).

Summarize the results:

$g(x,s)=left{begin{matrix} a(s) sin x, ;; x < s b(s) cos x, ;; s < x end{matrix}right.$

Second step: Now we shall determine a(s) and b(s).

Using demand-1 we get

$a(s) sin s = b(s) cos squad$ .

Using demand-4 we get

$b(s) cdot [ - sin s ] - a(s) cdot cos s = frac{1}{1} = 1, .$

Using Cramer's rule or by intelligent guess solve for a(s) and b(s) and obtain that $a(s) = - cos s quad ; quad b(s) = - sin s$ . Cramers rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. ...

Check that this automatically satisfies demand-5.

So our Green's function for this problem is:

$g(x,s)=left{begin{matrix} -1 cdot cos s cdot sin x, ;; x < s -1 cdot sin s cdot cos x, ;; s < x end{matrix}right.$

Green's functions in condensed matter physics

Green's functions are widely used in condensed matter physics. For a system of non-interacting particles described by a Hamiltonian $H$ , we can define the Green's function $g (E)$ at energy $hbar omega$ as

$g^{pm}(omega) = (hbar omega pm i0^+- H)^{-1},$

where the +(-) signifies the retarded (advanced) Green's function.

Then the density of one particle states at energy $hbar omega$ is written as

$rho(hbar omega)=frac{-1}{pi}int dx textrm{Im}(langle x|g(omega)|x rangle).$

Further examples

Let the manifold be R and L be d/dx. Then, the Heaviside step function H(x − x₀) is a Green's function of L at x₀.
Let the manifold 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 Green's function is

$G(x, y, x_0, y_0)=frac{1}{2pi}left[lnsqrt{(x-x_0)^2+(y-y_0)^2}-lnsqrt{(x+x_0)^2+(y-y_0)^2}right]$

$+frac{1}{2pi}left[lnsqrt{(x-x_0)^2+(y+y_0)^2}-lnsqrt{(x+x_0)^2+(y+y_0)^2}right].$

On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... In mathematics, a Dirichlet boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values a solution is to take on the boundary of the domain. ... In mathematics, a Neumann boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values the derivative of a solution is to take on the boundary of the domain. ...

Bibliography

Eyges, Leonard, The Classical Electromagnetic Field, Dover Publications, New York, 1972. ISBN 0-486-63947-9. (Chapter 5 contains a very readable account of using Green's functions to solve boundary value problems in electrostatics.)
A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9

External links

Weisstein, Eric W., Green's Function at MathWorld.
Green's function for differential operator at PlanetMath.
Green's function on PlanetMath

Categories: Differential equations | Quantum chemistry | Generalized functions MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

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