Greens function | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-06-04

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Green%27s_function


00:00:29 1 Definition and uses
00:01:45 2 Motivation
00:04:16 3 Green's functions for solving inhomogeneous boundary value problems
00:07:52 3.1 Framework
00:08:25 3.2 Theorem
00:10:13 3.3 Advanced and retarded Green's functions
00:12:16 4 Finding Green's functions
00:12:56 4.1 Units
00:13:06 4.2 Eigenvalue expansions
00:16:09 4.3 Combining Green's functions
00:18:01 4.4 Table of Green's functions
00:19:46 5 Green's functions for the Laplacian
00:28:49 6 Example
00:30:39 7 Further examples
00:35:39 8 See also
00:42:08 9 References
00:48:31 10 External links
00:49:03 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



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SUMMARY
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In mathematics, a Green's function of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions is its impulse response.
This means that if L is the linear differential operator, then

the Green's function is the solution for y of the equation Ly = δ, where δ is Dirac's delta function;
the solution of the initial-value problem Ly = f is the convolution (G * f), where G is the Green's function.Through the superposition principle, given a linear ordinary differential equation (ODE), L(solution) = source, one can first solve L(green) = δs, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.
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 studied largely from the point of view of fundamental solutions instead.
Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.