Green's Function

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Green's function

A solution of a partial differential equation for the case of a point source of unit strength within the region under examination. The Green's function is an important mathematical tool that has application in many areas of theoretical physics including mechanics, electromagnetism, acoustics, solid-state physics, thermal physics, and the theory of elementary particles. The underlying physics in each of these areas is generally described by some linear partial differential equation which relates the physical variable of interest (electrostatic potential or pressure amplitude in a sound wave, for example) to a source function. For present purposes the source may be regarded as an independent entity, although in some applications (for example, particle physics) this view masks an inherent nonlinearity. The source may be physically located within the region of interest, it may be simulated by certain boundary conditions on the surface of that region, or it may consist of both possibilities. A Green's function is a solution to the relevant partial differential equation for the particular case of a point source of unit strength in the interior of the region and some designated boundary condition on the surface of the region. Solutions to the partial differential equation for a general source function and appropriate boundary condition can then be written in terms of certain volume and surface integrals of the Green's function.

Green’s Function

 

a function related to the analytic representation of solutions of the boundary value problems of mathematical physics. In many cases, Green’s function makes possible a visual interpretation as a result of the action of a source of force or of a charge concentrated at a point. For this reason. Green’s function is sometimes called the function of a source. Thus, in an electrostatic interpretation, Green’s function represents the potential of the field of a point charge placed within a grounded conducting surface. It can be easily constructed for a number of regions (sphere, half-space, circle, rectangle, and so forth). The function is also used in solving boundary value problems for ordinary differential equations.

Green’s function plays an important role in theoretical physics, particularly in the quantum theory of fields and statistical physics. It describes the propagation of fields from the sources generating them (for this reason it is also called a function of propagation). The function was named after George Green, who first investigated a special case of it in his research on the theory of potential (1828).

REFERENCES

Sobolev, S. L. Uravneniia matematicheskoi fiziki, 4th ed. Moscow, 1966.
Tikhonov, A. N., and A. A. Samarskii. Uravneniia matematicheskoi fiziki, 3rd ed. Moscow, 1966.
Bogoliubov, N. N.. and D. V. Shirkov. Vvedenie v teoriiu kvan tovannykh polei. Moscow, 1957.
Mattuck. R. Feinmanovskie diagrammy v probleme mnogikh tel. Moscow. 1969. (Translated from English.)

Green's function

[′grēnz ‚fəŋk·shən]
(mathematics)
A function, associated with a given boundary value problem, which appears as an integrand for an integral representation of the solution to the problem.
References in periodicals archive ?
Green's functions are obtained for an infinite prestressed thin plate on an elastic foundation under axisymmetric loading [14].
Applications of Green's Functions in Science and Engineering (reprint, 1971)
Duffy, Green's Functions with Applications, Chapman & Hall/CRC, Boca Raton, FL, 2001, p.
Systematic development of the theory begins with notation and conventions, summaries of capacities and Green's functions, and continues with reduction of the variants back to the original theorem with local rationality conditions.
Boundary integral methods [16], which are based on Green's functions for each piecewise constant subdomain, are most powerful for the handling of the scattering in homogeneous unbounded domains.
2], respectively; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are radius-vectors of sources allocated at the surfaces [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are unit vectors of external normals to the surfaces; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the electric and magnetic tensor Green's functions for Hertz vector potentials in the coupled volumes satisfying the vector Helmholtz equation and the boundary conditions on surfaces [S.
By the multidimensional z-transform of FDTD equations in time and spatial domains, they derived the analytical closed form of the discrete Green's functions (DGFs).
Green's functions are obtained by applying the elastodynamic reciprocity principle following a method presented by Achenbach in [6].
Along with tables of Green's functions and related integrals, they cover functions for transient heat conduction in one-dimensional bodies; problems in the rectangular, cylindrical, and spherical coordinate systems; the Galerkin-based method; and the unsteady surface element method.
In [3], there are presented the expressions of the Green's functions for the Neumann's problem for a ball in [R.