抛物型偏微分方程初—边值问题的解法

A SOLUTION OF INITIAL BOUNDARY VALUE PROB-LEWS FOR THE PARTIAL DIFFERENTIAL EQUATION

数学物理中的许多问题都可以化为偏微分方程去求解,抛物型方程是其中之一,为研究抛物型方程的Cauchy问题,即初值问题,最自然的是先用Fourier积分方法研究最简单也是十分重要的一维热传导方程,它的解也能被确定。而研究一维热传导方程的初一边值问题则通常用分离变量法.作者探索把初一边值问题中的初始函数在相应的固有函数族下展开成为Fourier级数,把初始函数延拓到整个数轴,把初一边值问题的求解转化成为初值问题的求解,从而使这两类不同的定解问题的求解方法统一起来。对二维或三维热传导方程也可类似得到统一。

To expand initial function among initial-boundary value problems to a Fourier series under relevant eigenfunction system and reduce the solving of initial-boundary value problems to the solving of the initial value problems by the continuation of the initial function outside the given interval, thus the integrating of the two different methods of solving problems was studied. The similar integrating of the two or three dimentional heat conduction equations may be obtained.