格林函數

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在數學中, 格林函數 是一種用來解有邊界條件的 非勻相 微分方程式的函數。

格林函數的名稱是來自於英國 數學家 George Green,他最先在1830年代提出這個概念。

[编辑] 定義以及用法

技術上來說，格林函數，G(x,s)，是一個線性算符 L 作用在distributions over a 流形 M, at a point x₀, is any solution of

where δ is the Dirac delta function. This technique can be used to solve differential equations of the form;

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.

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.

[编辑] Motivation

[编辑] Green's function for solving inhomogeneous boundary value problems

[编辑] Working frame

[编辑] Theorem

[编辑] Finding Green's functions

[编辑] Eigenvalue expansions

[编辑] Green's function for the Laplacian

[编辑] Example

[编辑] Further examples

[编辑] See also

[编辑] References

• 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

[编辑] External links

• 埃立克·魏爾斯史甸在MathWorld中所描述之Green's Function。

• Green's function for differential operator at PlanetMath.
• PlanetMath上Green's function的資料。
• Tutorial on Green's function