变系数热传导反问题的稳定数值边界

Stable Numerical Boundaries for a Heat Conduction Inverse Problem with Variable Coefficient

将研究一类未曾被研究过的含未知边界的变系数反问题,为了计算该类反问题的解及其数值边界,将引入有限差分的方法进行研究,这种方法的基本思想是:把所要研究的问题分成三个子问题,对三个子问题分别构造不同的有限差分格式,以此来计算反问题的解及其两个未知边界,然后对三个差分格式的稳定性进行深刻探讨和证明,最后给出具体的数值试验结果,表明这种方法解出的未知边界数值解不仅结果符合物理解,而且精度远高于其他方法.

An inverse problem of heat conduction with variable coefficient and unknown boundaries was researched in this article. A finite difference technique was introduced to compute unknown boundaries and solution of this problem. In this method, the concerned problem was separated into three parts, and accordingly three new finite difference schemes were constructed, Two unknown boundaries and solution of inverse problem can be solved. Then stability conditions for numerical solution were discussed and proved carefully. In the last section, numerical example shows the method given in this paper is more accurate and effective than others.