A widespread approach to solving the Cauchy problem for Laplace’s equation is to reduce it to an inverse problem. As a rule, an iterative procedure is used to solve this problem. In this study, an efficient direct method for numerically solving the inverse problem in rectangular domains is described. The main idea is to expand the desired solution with respect to a basis consisting of eigenfunctions of a difference analogue of Laplace’s operator.

中文翻译：

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一种数值求解拉普拉斯方程柯西问题的有效直接方法

解决拉普拉斯方程的柯西问题的一种广泛方法是将其简化为反问题。通常，使用迭代过程来解决此问题。在这项研究中，描述了一种用于数值求解矩形域中反问题的有效直接方法。主要思想是相对于由拉普拉斯算子的差分类似物的本征函数组成的基础扩展所需的解。