三维与二维晶体生长控制方程的精确解

Analytical Solutions of Three-dimension and Two-dimension Crystal Growth Equations

本文通过将未知函数展开成复数形式的Fourier级数,求出了一类二阶偏微分方程的三角级数形式的解析解,并严格证明了其收敛性．三维稳态与二维稳态和二维非稳态晶体生长控制方程都是这类二阶偏微分方程特例．利用这一特点,本文求出了三维稳态与二维稳态和二维非稳态晶体生长控制方程的解析解．理论结果有助于揭示稳态晶体生长的本质特性．本文还给出了三维非稳态晶体生长控制方程的解析解．

A class of second-order partial differential equations is studied （PDE）. By making use of Fourier series in complex number field, the analytical solution with trigonometrical series form of the PDE is established and its convergence is proved. As its applications, analytical solutions of two-dimension and three-two-dimension, steady and non-steady state crystal growth concentration governing equation are all obtained. The theoretical results help us to reveal the physical mechanism of crystal growthing.