In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional ones. We do so in the spirit of quasi reversibility, replacing a classically severely ill-posed PDE problem by a nearby well-posed or only mildly ill-posed one. In order to be able to make use of the known stabilising effect of one-dimensional fractional derivatives of Abel type we work in a particular rectangular (in higher space dimensions cylindrical) geometry. We start with the plain Cauchy problem of reconstructing the values of a harmonic function inside this domain from its Dirichlet and Neumann trace on part of the boundary (the cylinder base) and explore three options for doing this with fractional operators. The two other related problems are the recovery of a free boundary and then this together with simultaneous recovery of the impedance function in the boundary condition. Our main technique here will be Newton's method. The paper contains numerical reconstructions and convergence results for the devised methods.

翻译：本文重新审视了拉普拉斯方程的经典柯西问题以及另外两个相关问题，通过用分数阶导数替代整数阶导数来对该高度病态问题进行正则化。我们采用准可逆性的思想，将一个经典严重病态的偏微分方程问题替换为一个邻近的适定或仅轻度病态的问题。为利用已知的阿贝尔型一维分数阶导数的稳定效应，我们选取特定的矩形（在高维空间中为圆柱形）几何结构。首先研究从边界部分（圆柱底面）上的狄利克雷和诺伊曼迹重构域内调和函数值的标准柯西问题，并探索利用分数阶算子实现该目标的三种方案。另外两个相关问题分别涉及自由边界的恢复，以及该问题与边界条件中阻抗函数同步恢复的联合反演。本文主要采用牛顿法作为核心技术手段，包含数值重建结果及所提出方法的收敛性分析。