摘要从物理上看,一个数学物理方程是表示一种特定的"场"和产生这种场的"源"之间的关系.例如,热传导方程表示温度场和热源之间的关系,泊松方程表示静电场和电荷分布的关系,等等.这样,当源被分解成很多点源的叠加时,如果能设法知道点源产生的场,利用叠加原理,我们可以求出同样边界条件下任意源的场,这种求解数学物理方程的方法就叫格林函数法.而点源产生的场就叫做格林函数. 格林函数,又称点源影响函数。格林函数代表一个点源在一定的边界条件和初始条件下所产生的场。知道了点源的场,就可以用迭加的方法计算出任意源所产生的场。7995
关键词 格林函数 狄拉克δ 函数 泊松方程 电像法
Title Green's function with time
Abstract
From the physical point of view,a equation of mathematical physics is that
the relationship between a particular "field" and a "source" which produces
the "field".For example, a heat conduction equation expresses the
relationship between the temperature field and heat source. The Poisson
equation expresses the relationship between electrostatic field and charge
distribution, etc. Thus, when the source is decomposed into a superposition
of many point sources, if we can try to know the field of the point
source ,using the principle of superposition, we can find the field of any
source in the same boundary conditions. This method of solving the
equations of mathematical physics is called Green's function method.
Green's function, is also called the influence function of the point
source. Green's function represents the field generated by a point source
under a certain boundary and the initial conditions. If we have known the
field of a point source, it can use the superposition method to calculate
the field generated by any source.
Keywords Green's function δ function Poisson equation
Method of images
目 次
1 引言1
2 概论 1
2.1 三类典型方程 1
2.2 定解条件和定解问题 2
2.3
函数 2
2.4 冲量定理(齐次化原理)(Duhamel原理) 3
3 格林函数法 4
3.1 泊松方程的格林函数4
3.2 用电像法求格林函数8
3.2.1 无界空间的格林函数 基本解8
3.2.2 电像法求格林函数9
3.3 含时间的格林函数13
3.4 用冲量定理法求格林函数17