一类偏微分方程的解析解及其在稳态晶体生长中的应用

The Analytical Solution for a Class of Partial Differential Equations and Its Application to the Steady State Crystal Growth

本文通过将未知函数展开成复数形式的 Fourier 级数,求出了一类偏微分方程的三角级数形式的 解析解,并研究了其收敛性。最后,把结果用到稳态晶体生长的控制方程上,得到了二维稳态晶 体控制方程的解析解。理论结果表明沿晶体生长方向浓度分布具有周期性震荡衰减性质,而沿横 轴方向,浓度变化呈现为周期性一致变化。理论结果有助于揭示稳态胞晶周期性增长的本质特 性。

This paper studied a class of partial di?erential equations (PDE). By making use of Fourier series in complex number ?eld, the analytical solution with trigonometrical series form of the PDE is presented and its convergence is proved. As its application, an analytical solution of a two-dimension steady state crystal growth concentration governing equation is obtained. The theoretical result shows that the concentration distribution in the direction of crystal growth is exponentially damped oscilla- tion. In the direction of transverse coordinate, the concentration distribution is period. The result is useful for us to reveal the physical mechanism of cellular and dendrite patterns.