Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

中文翻译：

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一种求解拉普拉斯方程柯西问题的基于Hermite展开的修正Tikhonov正则化方法

摘要 本文研究了拉普拉斯方程的柯西问题。我们开发了一种基于 Hermite 扩展的改进的 Tikhonov 正则化方法来处理问题的不适定性。正则化参数由差异原则确定。对于各种平滑条件，该方法求解过程一致，可以自适应地获得收敛速度。还进行了数值试验以验证该方法的有效性。