In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calder\'{o}n problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradient method is utilized to solve the variational problem. Moreover, we also provide a suggestion to choose the regularization parameter. Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results.

翻译：本文提出了一种基于单次测量确定一维和二维分数阶Calderón问题中势函数的数值方法。采用有限差分格式离散分数阶拉普拉斯算子，并将参数重构问题转化为基于Tikhonov正则化的变分问题，以获得稳定且精确的解。利用共轭梯度法求解该变分问题。此外，本文还提供了正则化参数的选取建议。数值实验验证了所提方法的效率与有效性，并支撑了理论结果。