Laplace方程Green函数法的研究

Study on Green Function Solution of Laplace Equation

利用Green公式及调和函数的性质,系统地研究三元调和函数在空间区域Ω内、外及边界上的积分表达式;简要介绍Green函数的引出及镜像法求解Green函数;讨论求解三维Laplace方程或Poisson方程边值问题的Green函数法,进一步研究球域内利用Green函数求解Dirichlet问题的解的形式,给出更直观易解、便于应用的调和函数积分表示式.

By using the green formula and the character of harmonic function, the integral expression of ternary harmonic function in the spatial region Ω, outside and on the boundary were studied. The definition of Green function and image method is introduced in brief. The solution of Green function is discussed about the boundary value question of three dimensional Laplace equation or the Poisson equation, further study on the ball territory to solve the Dirichlet question using the Green function method was carried out. At last the practical integral expression of harmonic function was given.