This paper investigates two analytical methods for solving the inverse Cauchy problem for Laplace’s equation, focusing on the reconstruction of the solution \( u(r, \theta ) \) from boundary data. The first method employs a regularized solution approach based on solving a Fredholm integral equation of the first kind, while the second utilizes the homotopy perturbation method (HPM), a series-based analytical technique. Both methods are tested on a benchmark problem with a known exact solution, allowing a comprehensive comparison of their accuracy, stability, and computational efficiency.