二阶椭圆型方程的广义解析解

Analytic solution of non-homogeneous two-order elliptic equation

给出一种刚提出的基于Ｈａｍｉｌｔｏｎ体系的解析法的各个步骤，并用这种方法首次求出了矩形域上二阶非齐次椭圆型方程的广义解析解．这种方法具有一定的普遍意义，还可求解某些尚未获解的偏微分方程．通过算例验证了解答的正确性．

All the steps of a new, Hamiltonian-system - based analytic met hed put forward by the second author,are presented and used to obtain for the first time the analytic solution of general non- homogeneous two-order elliptic equation on rectangular domain. The method is a generalization of classical method of separation of variables based on the adjoint symplectic orthogonality for solving some non-self-adjoint partial differential operators. As a verification,the analytic solution is found very close to the numerical results by Ritz method and finite element methed. The idea can also be used to solve some PDEs of higher dimensions.