Green's function
Named after the physicist and mathematician George Green Green's function is an important aid in the solution of inhomogeneous linear differential equations. Physically, it expresses explicitly the superposition principle, mathematically, the linearity is essential. It is also said, a Green's function " propagated inhomogeneity " and therefore also referred to as a Propagatorfunktion, not only when the key variable is the time.
The concept of Green's functions is very general terms and applies to linear differential equations of all kinds in the potential theory and gravity measurement and is not used, inter alia, to the solution of the first boundary value problem. See also fundamental solution and transfer function.
In theoretical physics, especially in the high-energy and many-body physics, a wealth of different functions is further defined, all of which are referred to as " Green's functions" and are related to the given here functions in one form or another, without this the would be immediately recognizable. These features, especially the propagators of relativistic quantum theories are not meant in the following.
- Table 3.1 Green's functions
- 3.2 Determination of the static electric field
- 3.3 Inhomogeneous wave equation 3.3.1 Green's function by Fourier analysis
- 3.3.2 Alternative derivation
Motivation
An inhomogeneous linear differential equation in a number space with unit element (eg the real or complex numbers ) has the form
Where L is a linear differential operator. The goal is to find the particular solution to the inhomogeneity. One would now like something like a " reversal operator" find, because then you could solve the above equation as write. But if has non-trivial solutions, is not injective, so there can be no left inverse. But well is surjective when the equation has solutions for each. In this case, so you have to look for a right inverse operator, for which:
By then you have found a particular solution:
The general solution is obtained by adding the general solution of the homogeneous problem. In other words: ( ) The Green's function is a special particular solution, especially for the delta distribution as inhomogeneity, f → δ:
Now the question is, can be obtained as for the given inhomogeneity of the Green's function:
Explanation of each step: the first equals sign is the output equation. For each function, the convolution with the delta function is possible and once again provides the function ( is the neutral element of the convolution ). Use, so that solves the differential equation with inhomogeneity. By forming the derivative of a convolution, so the derivative is simply drawn. Finally, from the particular solution are identified, namely as a convolution of the Green's function with the inhomogeneity.
The general solution is obtained by adding the general solution of the homogeneous problem for the particular solution.
Definition
Ordinary Differential Equations
Be
A differential operator with. Then met a Green's function G for this operator the basic equation:
Where the delta function is. It can be useful later adds additional conditions added, eg Retardierungsbedingungen (see below) or the equivalent thereto " Sommerfeld radiation condition " or an initial or boundary condition is uniquely determined by the G. A special solution is obtained in any case by convolution:
As one realizes as follows:
The question remains, how to find a Green's function. In the special case of a differential equation with constant coefficients is obtained, as it functions with their Fourier transforms
Can identify, from the convolution theorem:
Or with the transfer function:
And thus, as also also:
For N = 2 corresponds to the stationary ( " steady " ) response of the system, a damped harmonic oscillator, a ballistic shock unit, ie the special reduced driving force
Partial Differential Equations
Also applies to partial differential equations, the equation defining
And a special solution is again obtained by convolution:
More problematic, however, in the case the discovery of a Green's function and the calculation of multidimensional integrals.
Green's function with boundary conditions
If you know a Green's function to an operator, then one can solve the inhomogeneous part of the differential equation without any problems. For the general solution, but in general nor to satisfy boundary conditions. This can be done in many different species, but an elegant method is the addition of a solution of the homogeneous problem, so that the boundary conditions are met. Intuitively, this corresponds to when solving the Poisson equation adding image charges and removing the edges, so that where the edge was, the previously specified values are accepted. Think of as a simple example of a charged particle in front of a grounded plane. If on the other hand, the level of an oppositely charged charge and removes the intellectual level, is where the plane was, the potential is zero, which satisfies the required boundary condition.
Often one uses this method of solving the Poisson equation ( Gaussian units). Using Green's theorem one finds ():
Depending on whether you have now given the potential or its derivative on the edge, you now select the function you want to be added so that the following holds in the first case and is called G ' usually Dirichlet Green's function. In the second case one chooses not - as would be close - so that disappears, as this would violate the Gaussian set. Instead, you choose so that
Applies (which produces only the average of the potential over the surface in the above integral, a constant around which the solution is indeterminate anyway ) and usually called the Neumann Green's function. The Green's functions are found to be determined for symmetric problems often from geometrical considerations. Alternatively, you can develop after a orthonormal system of the operator. If you have found a solution, so this is uniquely determined, as follows immediately from the maximum principle for elliptic differential equations.
Examples
Table of Green's functions
The following table gives an overview of Green's functions of commonly occurring differential operators, with and.
Determination of the static electric field
According to Maxwell's equations valid for the source strength of the time- constant electric field
Wherein the electric field intensity and the electric charge density. Since it is a conservative system in the electrostatic event shall
Wherein the electric potential. Using the Poisson equation yields
So an inhomogeneous linear partial differential equation of 2nd order. If you know a Green's function of the Laplace operator, so is a particular solution
A (not uniquely determined) Green's function of the Laplace operator in three dimensions is
Which after insertion
Results. Last equation is to clarify the physical interpretation of the Green's function. The Green's function together with the differential represent a " potential impact " that the total potential is then obtained by superposition of all " potential impacts ", ie by performing the integral.
Inhomogeneous wave equation
This case is slightly more difficult and different nature, because you do not have to do it with an elliptical, but with a hyperbolic differential equation. Shown here the above-indicated complications.
Green's function by Fourier analysis
The inhomogeneous wave equation has the form
By Fourier decomposition can be found by executing the operator for the Fourier transform
According to the convolution theorem therefore applies:
The inverse transformation can calculate using the residue calculus and finds
Which in a natural way to two shares (" retarded " or " of advanced " portion) of the Green's function gives rise. The argument in the first delta function, signify that a generated at the time in the "Cause" causes by the finite speed of propagation of the wave only at the time of their "effect" on the spot. For the second delta function results in that the field is ahead in relation to the non-homogeneity to the corresponding time interval. That would be unphysical for causality reasons, if one were to view the inhomogeneity as a cause and as an effect of the field; but it is quite physically when the inhomogeneity acts as an absorber (receiver ) of the shaft.
The retarded Green function in which the inhomogeneity causally a " transmission process " corresponds expiring spherical waves is thus
The retarded solution of the wave equation is then obtained by convolution:
It applies a principle of superposition with retardation: The solution is a superposition of outgoing spherical waves ( Huygens' principle, Sommerfeld radiation condition ), whose formation is similar in electrostatics.
The advanced Green's function in which the inhomogeneity causally a " receiving process " corresponds to incoming spherical waves is,
Alternative derivation
If one presupposes the Green's function of the Laplace operator as known ( see main article Laplacian and Poisson's equation), the retarded Green's function of the wave equation can be obtained without Fourier transformation. First applies to any "smooth" function
The three-dimensional delta function. To see that the left side in the area is always zero, it writes the Laplacian in spherical coordinates with the radial part in the mold. In the immediate vicinity of, the smooth function are considered equal as spatially constant. Application of the Laplacian of the factor to generate the three-dimensional delta function.
The argument can be by developing specify in powers of, the leading power in the application of the Laplace operator must be treated separately.
For particular a Gaussian function can be selected. Since the delta function as the limit can be represented by Gaussian functions, one obtains in the limit of the defining equation for the Green's function of the wave equation.
Comments
The above presentation is for ordinary functions mathematically not strict, for example, one can not interchange the integral and differential operator in general. A more rigorous presentation provides the distribution theory.