二次曲面区域泊松方程第一边值问题的格林函数解法

Green's Function Method for the First Boundary Value Problem of Poisson Equation in the Quadric Surface Region

关于泊松方程第一边值问题,目前大部分研究仅给出了球域、圆域等情况的格林函数解法,而对其他类型的区域讨论甚少.该文从二次曲面成像公式出发,用电像法统一研究椭球面、双曲面、抛物面、球面等二次曲面区域内的泊松方程第一边值问题,旨在给出其各自的格林函数解及相应的第一积分表示式.研究发现,在近轴情况下,二次曲面区域内泊松方程第一边值问题的格林函数解及第一积分表示式有统一形式,该文最终给出了这种统一形式并分别对这几种二次曲面域进行了讨论.

Green's function method is an important way to solve the modern physical problems.The wave equation,the diffusione quation,the Helmholtz equation,the Poisson equation,which is one of the important equations to describe the steady field,and many problems in mode rnengineering can be solved by using Green’s function method.For the first boundary value problem of Poisson equation,most of the research only gives the Green’s function solution to the areas with ellipsoidal surface or spherical surface and soon,but the reislittled is cussion on other types of areas.Based on the quadratic surface imaging for mula,the first bound aryvalue problem of the Poisson equationin the areas with quadratic surfaces such as ellipsoid,hyperboloid,paraboloid and sphereis studied uniformly in this text by using electric image method.The purposeis to give the Green’s function.