热方程侧边值问题的正则化方法及频域中的修改“核”思想

Regularization methods of sideways heat equation and idea of modifying "kernel" in frequency domain

讨论和分析几种著名的求解热方程侧边值问题的正则化方法.在频域中发现了这些方法之间的有趣联系,同时通过对已有文献中相关方法及问题(不同方法应用于同一个问题和一种方法应用于不同问题)的比较分析,提出频域中的修改"核"思想.反之,基于该思想,可以方便地衡量已有文献中的部分正则化方法的好坏和构建某类线性不适定问题的新的正则化方法.以一个二维逆热传导问题和一个反向热传导问题为例说明了修改核思想的部分应用.

Several well-known regularization methods of solving sideways heat equation are discussed and analyzed. In the frequency domain, an interesting relationship among these methods is found, and an idea of modifying kernel is proposed after also analyzing the regularization methods of related ill-posed prob- lems. Based on the idea, one can conveniently review some regularization methods available and construct a new method for a class of linear ill-posed problems. A two-dimensional inverse heat conduction problem and a backward heat conduction problem are taken as the examples to show the partial applications of the proposed idea.