双调和方程柯西问题的修正的Tikhonov正则化方法

本文研究了双调和方程柯西问题,这类是不适定的,即问题的解(如果存在)不连续依赖于测量数据.首先在精确解的先验假设下给出问题的条件稳定性结果.接着利用修正的Tikhonov正则化方法求解此不适定问题.在先验和后验正则化参数选取规则下,给出正则解和精确解之间的误差估计式.最后给出几个数值例子验证此正则化方法求解此类反问题的有效性.

A Modified Tikhonov Regularization Method for a Cauchy Problem of the Biharmonic Equation

In this paper, the Cauchy problem of biharmonic equation is considered. This problem is ill-posed, i.e., the solution (if exists) does not depend on the measurable data. Firstly, we give the conditional stability result under the a priori bound assumption for the exact solution. Secondly, a modified Tikhonov regularization method is used to solve this ill-posed problem. Under the a priori and the a posteriori regularization parameter choice rule, the error estimates between the regularization solutions and the exact solution are obtained. Finally, some numerical examples are presented to verify that our method is effective.